On partially observed jump diffusions II: the filtering density

Abstract

A partially observed jump diffusion $Z=(X_t,Y_t)_{t\in [0,T]}$ given by a stochastic differential equation driven by Wiener processes and Poisson martingale measures is considered when the coefficients of the equation satisfy appropriate Lipschitz and growth conditions. Under general conditions it is shown that the conditional density of the unobserved component $X_t$ given the observations $(Y_s)_{s\in [0,t]}$ exists and belongs to $L_p$ if the conditional density of $X_0$ given $Y_0$ exists and belongs to $L_p$.

1 Introduction

As in [4] we consider a partially observed jump diffusion $Z=(X_t,Y_t)_{t\in [0,T]}$ satisfying the system of stochastic differential equations

$$\begin{aligned} \begin{aligned} dX_t&= b(t,Z_t)dt + \sigma (t,Z_t)dW_t + \rho (t,Z_t)dV_t\\&\quad +\int _{{\mathfrak {Z}}_0}\eta (t, Z_{t-},{\mathfrak {z}})\,{{\tilde{N}}}_0(d{\mathfrak {z}},dt) + \int _{{\mathfrak {Z}}_1}\xi (t,Z_{t-},{\mathfrak {z}})\,{{\tilde{N}}}_{1}(d{\mathfrak {z}},dt),\\ dY_t&=B(t,Z_t)dt + dV_t + \int _{{\mathfrak {Z}}_1} {\mathfrak {z}}\,{\tilde{N}}_{1}(d{\mathfrak {z}}, dt), \end{aligned} \end{aligned}$$

(1.1)

driven by a $d_1+d'$-dimensional ${\mathcal {F}}_t$-Wiener process $(W_t,V_t)_{t\geqslant 0}$ and independent $\mathcal {{F}}_t$-Poisson martingale measures ${{\tilde{N}}}_i(d{\mathfrak {z}},dt) = N_i(d{\mathfrak {z}},dt)-\nu _i(d{\mathfrak {z}})dt$ on ${\mathbb {R}}_{+}\times {\mathfrak {Z}}_i$, for $i=0$ and 1, carried by a complete filtered probability space $(\Omega ,{\mathcal {F}},({\mathcal {F}}_t)_{t\in [0,T]},P)$, where $\nu _0$ and $\nu _1$, the characteristic measures of the Poisson random measures $N_0$ and $N_1$, are $\sigma$-finite measures on a separable measurable space $({\mathfrak {Z}}_0,{\mathcal {Z}}_0)$ and on ${\mathfrak {Z}}_1={\mathbb {R}}^{d'}\setminus \{0\}$ equipped with the $\sigma$-algebra ${\mathcal {Z}}_1={\mathcal {B}}({\mathbb {R}}^{d'}\setminus \{0\})$ of its Borel sets, respectively. The mappings $b=(b^i)$, $B=(B^i)$, $\sigma =(\sigma ^{ij})$ and $\rho =(\rho ^{il})$ are Borel functions of $(t,z)=(t,x,y)\in {\mathbb {R}}_+\times {\mathbb {R}}^{d+d'}$, with values in ${\mathbb {R}}^d$, ${\mathbb {R}}^{d'}$, ${\mathbb {R}}^{d\times d_1}$ and ${\mathbb {R}}^{d\times d'}$, respectively, and $\eta =(\eta ^i)$ and $\xi =(\xi ^i)$ are ${\mathbb {R}}^d$-valued ${\mathcal {B}}({\mathbb {R}}_+\times {\mathbb {R}}^{d+d'})\otimes {\mathcal {Z}}_0$-measurable and ${\mathbb {R}}^d$-valued ${\mathcal {B}}({\mathbb {R}}_+\times {\mathbb {R}}^{d+d'})\otimes {\mathcal {Z}}_1$-measurable functions of $(t,z,{\mathfrak {z}}_0)\in {\mathbb {R}}_+\times {\mathbb {R}}^{d+d'}\times {\mathfrak {Z}}_0$ and $(t,z,{\mathfrak {z}}_1)\in {\mathbb {R}}_+\times {\mathbb {R}}^{d+d'}\times {\mathfrak {Z}}_1$, respectively, where ${\mathcal {B}}({\mathbb {U}})$ denotes the Borel $\sigma$-algebra on ${\mathbb {U}}$ for topological spaces ${\mathbb {U}}$.

In [4] we were interested in the equations for the evolution of the conditional distribution $P_t(dx)=P(X_t\in dx|Y_s, s\leqslant t)$ of the unobserved component $X_t$ given the observations $(Y_s)_{s\in [0,T]}$. Our aim in the present paper is to show, under fairly general conditions, that if the conditional distribution of $X_0$ given $Y_0$ has a density $\pi _0$, such that it is almost surely in $L_p$ for some $p\geqslant 2$, then $X_t$ for every t has a conditional density $\pi _t$ given $(Y_s)_{t\in [0,t]}$, which belongs also to $L_p$, almost surely for all t. In a subsequent paper we investigate the regularity properties of the conditional density.

The filtering problem has been the subject of intense research for the past decades and the literature on it is vast. For a brief account on the filtering problem and on the history of the presentation of the filtering equations for partially observed diffusion processes we refer to [9]. Concerning the filtering equations associated to (1.1) we refer the reader to [4] and the references therein.

In the present paper we investigate the existence of the conditional density $\pi _t=dP_t/dx$ under general conditions on the coefficients of the system (1.1). We do not assume any non-degeneracy conditions on $\sigma$ and $\eta$, i.e., they are allowed to vanish. Thus, given the observations, there may not remain any randomness to smooth the conditional distribution $P_t(dx)$ of $X_t$, i.e., if the initial conditional density $\pi _0$ does not exists, then the conditional density $\pi _t$ for $t>0$ may not exist either. Therefore assuming that the initial conditional density $\pi _0$ exists, we are interested in the smoothness and growth conditions which we should require from the coefficients in order to get that $\pi _t$ exists for every $t\in [0,T]$ as well.

For partially observed diffusion processes, i.e., when $\xi =\eta =0$ and the observation process Y does not have jumps, the existence and the regularity properties of the conditional density $\pi _t$ have been extensively studied in the literature. For important results under non-degeneracy conditions see, for example, [9,10,11, 15], and the references therein. Without any non-degeneracy assumptions, in [18] the existence of $\pi _t$ is proved if $\pi _0\in W^2_p\cap W^2_2$ for some $p\geqslant 2$, the coefficients are bounded, $\sigma \sigma ^*+\rho \rho ^*$ has uniformly bounded derivatives in x up to order 3, and b, B have uniformly bounded derivatives in x up to second order. Under these conditions it is also proved that $(\pi _t)_{t\in [0,T]}$ is a weakly continuous process with values in $W^2_p\cap W^2_2$, and that $\pi _t$ has higher regularity if $\pi _0$ and the coefficients are appropriately smoother. By a result in a later work, [13], one knows that if the coefficients $\sigma$, $\rho$, b and B are bounded Lipschitz continuous in x, and $\pi _0\in L_2({\mathbb {R}}^d, dx)$, then $\pi _t$ exists and remains in $L_2$. The approach to obtain this result is based on an $L_2$-estimate for the unnormalised conditional density smoothed by Gaussian kernels. The same method is also used in [1] to prove uniqueness of measure-valued solutions for the Zakai equation in the case where the signal is a diffusion process, the observation contains a jump term and the coefficients are time-independent, globally Lipschitz, except for the observation drift term, which contains a time dependence, but is bounded and globally Lipschitz. The approach from [13] is extended in [14] to partially observed jump diffusions when the Wiener process in the observation process Y is independent of the Wiener process in the unobserved process, to prove, in particular, the existence of the conditional density in $L_2$, if the initial conditional density exists, belongs to $L_2$, the coefficients are bounded Lipschitz functions, the coefficients of the random measures in the unobservable process are differentiable in x and satisfy a condition in terms of their Jacobian. In [16] and [17] the filtering equations for fairly general filtering models with partially observed jump diffusions are obtained and studied, but the existence of the conditional density (in $L_2$) is proved only in [16], in the special case when the equation for the unobserved process is driven by a Wiener process and an $\alpha$-stable additive Lévy process, $\rho =0$, the coefficients b and $\sigma$ are bounded functions of $x\in {\mathbb {R}}^d$, b has bounded first order derivatives, $\sigma$ has bounded derivatives up to second order and $B=B(t,x,y)$ is a bounded Lipschitz function in $z=(x,y)$.

The main theorem, Theorem 2.1, of the present paper reads as follows. Assume that the coefficients b, $\sigma$, $\rho$, B, $\xi$, $\eta$ and $\rho B$ are Lipschitz continuous in $z=(x,y)\in {\mathbb {R}}^{d+d'}$, B is bounded, b, $\sigma$, $\rho$, $\xi$ and $\eta$ satisfy a linear growth condition, $\xi$ and $\eta$ admit uniformly equicontinuous derivatives in $x\in {\mathbb {R}}^d$, $x+\xi (x)$, $x+\eta (x)$ are bijective mappings in $x\in {\mathbb {R}}^d$, and have a Lipschitz continuous inverse with Lipschitz constant independent of the other variables. Assume, moreover, that ${\mathbb {E}}|X_0|^r<\infty$ for some $r>2$. Under these conditions, if the initial conditional density $\pi _0$ exists for some $p\geqslant 2$, then the conditional density $\pi _t$ exists and belongs to $L_p$ for every t. Moreover, $(\pi _t)_{t\in [0,T]}$ is weakly cadlag as $L_p$-valued process.

To prove our main theorem we use the Itô formula from [7] and adapt an approach from [13] to estimate the $L_p$-norm of the smoothed unnormalised conditional distribution for even integers $p\geqslant 2$. Hence we obtain Theorem 2.1 for even integers $p\geqslant 2$. Then we use an interpolation theorem combined with an approximation procedure to get the main theorem for every $p\geqslant 2$. In a follow-up paper we show that if in addition to the above assumptions the coefficients have bounded derivatives up to order $m+1$ and $\pi _0$ belongs to the Sobolev space $W^m_p$ for some $p\geqslant 2$ and integer $m\geqslant 1$, then $\pi _t$ belongs to $W^m_p$ for $t\in [0,T]$.

The paper is organised as follows. In Sect. 2 we formulate our main result. In Sect. 3 we recall important results from [4] together with the filtering equations obtained therein. In Sect. 4 we prove $L_p$ estimates needed for a priori bounds for the smoothed conditional distribution. In Sect. 5 we obtain an Ito formula for the $L_p$-norm of the smoothed conditional distribution and prove our result for the case $p=2$. Section 6 contains existence results for the filtering equation in $L_p$-spaces. In the last section we prove our main theorem.

We conclude with some notions and notations used throughout the paper. For an integer $n\geqslant 0$ the notation $C^n_b({\mathbb {R}}^d)$ means the space of real-valued bounded continuous functions on ${\mathbb {R}}^d$, which have bounded and continuous derivatives up to order n. (If $n=0$, then $C^0_b({\mathbb {R}}^d)=C_b({\mathbb {R}}^d)$ denotes the space of real-valued bounded continuous functions on ${\mathbb {R}}^d$). We use the notation $C^{\infty }_{0}=C^{\infty }_{0}({\mathbb {R}}^d)$ for the space of real-valued compactly supported smooth functions on ${\mathbb {R}}^d$. We denote by ${\mathbb {M}}={\mathbb {M}}({\mathbb {R}}^d)$ the set of finite Borel measures on ${\mathbb {R}}^d$ and by ${\mathfrak {M}}={\mathfrak {M}}({\mathbb {R}}^d)$ the set of finite signed Borel measures on ${\mathbb {R}}^d$. For $\mu \in {\mathfrak {M}}$ we use the notation

$$\begin{aligned} \mu (\varphi )=\int _{{\mathbb {R}}^d}\varphi (x)\,\mu (dx) \end{aligned}$$

for Borel functions $\varphi$ on ${\mathbb {R}}^d$. We say that a function $\nu :\Omega \rightarrow {\mathbb {M}}$ is ${\mathcal {G}}$-measurable for a $\sigma$-algebra ${\mathcal {G}}\subset {\mathcal {F}}$, if $\nu (\varphi )$ is a ${\mathcal {G}}$-measurable random variable for every bounded Borel function $\varphi$ on ${\mathbb {R}}^d$. An ${\mathbb {M}}$-valued stochastic process $\nu =(\nu _t)_{t\in [0,T]}$ is said to be weakly cadlag if almost surely $\nu _t(\varphi )$ is a cadlag function of t for all $\varphi \in C_b({\mathbb {R}}^d)$. An ${\mathfrak {M}}$-valued process $(\nu _t)_{t\in [0,T]}$ is weakly cadlag, if it is the difference of two ${\mathbb {M}}$-valued weakly cadlag processes. For processes $U=(U_t)_{t\in [0,T]}$ we use the notation ${\mathcal {F}}_t^{U}$ for the P-completion of the $\sigma$-algebra generated by $\{U_s: s\leqslant t\}$. By an abuse of notation, we often write ${\mathcal {F}}_t^U$ when referring to the filtration $({\mathcal {F}}^U_t)_{t\in [0,T]}$, whenever this is clear from the context. For a measure space $({\mathfrak {Z}},{\mathcal {Z}},\nu )$ and $p\geqslant 1$ we use the notation $L_p({\mathfrak {Z}})$ for the $L_p$-space of ${\mathbb {R}}^d$-valued ${\mathcal {Z}}$-measurable mappings defined on ${\mathfrak {Z}}$. However, if not otherwise specified, the function spaces are considered to be over ${\mathbb {R}}^d$. We always use without mention the summation convention, by which repeated integer valued indices imply a summation. For a multi-index $\alpha =(\alpha _1,\dots ,\alpha _d)$ of nonnegative integers $\alpha _i, i=1,\dots ,d$, a function $\varphi$ of $x=(x_1,\dots ,x_d)\in {\mathbb {R}}^d$ and a nonnegative integer k we use the notation

$$\begin{aligned} D^\alpha \varphi (x)=D_1^{\alpha _1}D_2^{\alpha _2}\ldots D_d^{\alpha _d}\varphi (x), \quad \text {as well as} \quad |D^k\varphi |^2=\sum _{|\gamma |=k}|D^\gamma \varphi |^2, \end{aligned}$$

where $D_i=\tfrac{\partial }{\partial {x^i}}$ and $|\cdot |$ denotes an appropriate norm. We also use the notation $D_{ij}=D_iD_j$. If we want to stress that the derivative is taken in a variable x, we write $D^\alpha _x$. If the norm $|\cdot |$ is not clear from the context, we sometimes use appropriate subscripts, as in $|\varphi |_{L_p}$ for the $L_p({\mathbb {R}}^d)$-norm of $\varphi$. For $p\geqslant 1$ and integers $m\geqslant 0$ the space of functions from $L_p$, whose generalized derivatives up to order m are also in $L_p$, is denoted by $W^m_p$. The norm $|f|_{W^m_p}$ of f in $W^m_p$ is defined by

$$\begin{aligned} |f|_{W^m_p}^p:=\sum _{k=0}^m \int _{{\mathbb {R}}^d}|D^kf(x)|^p\,dx<\infty . \end{aligned}$$

For real-valued functions f and g defined on ${\mathbb {R}}^d$ the notation (f, g) means the Lebesgue integral of fg over ${\mathbb {R}}^d$ when it is well-defined. Throughout the paper we work on the finite time interval [0, T], where $T>0$ is fixed but arbitrary, as well as on a given complete probability space $(\Omega ,{\mathcal {F}},P)$ equipped with a filtration $({\mathcal {F}}_t)_{t\geqslant 0}$ such that ${\mathcal {F}}_0$ contains all the P-null sets. For $p,q\geqslant 1$ and integers $m\geqslant 1$ we denote by ${\mathbb {W}}^m_p= L_p((\Omega ,{\mathcal {F}}_0,P),W^m_p({\mathbb {R}}^d))$ and ${\mathbb {L}}_{p,q}\subset L_p(\Omega ,L_q([0,T],L_p({\mathbb {R}}^d)))$ the set of ${\mathcal {F}}_0\otimes {\mathcal {B}}({\mathbb {R}}^d)$-measurable real-valued functions $f=f(\omega ,x)$ and ${\mathcal {F}}_t$-optional $L_p$-valued functions $g=g_t(\omega ,x)$ such that

$$\begin{aligned} |f|_{{\mathbb {W}}^m_p}^p:={\mathbb {E}}|f|_{W^m_p}^p<\infty \quad \text {and}\quad |g|_{{\mathbb {L}}_{p,q}}^p:={\mathbb {E}}\Big (\int _0^T |g_t|_{L_p}^q dt\Big )^{p/q}<\infty \end{aligned}$$

respectively. If $m=0$ we set ${\mathbb {L}}_p={\mathbb {W}}^0_p$.

2 Formulation of the main results

We fix nonnegative constants $K_0$, $K_1$, L, K and functions ${\bar{\xi }}\in L_2({\mathfrak {Z}}_1)=L_2({\mathfrak {Z}}_1,{\mathcal {Z}}_1,\nu _1)$, ${\bar{\eta }}\in L_2({\mathfrak {Z}}_0)=L_2({\mathfrak {Z}}_0,{\mathcal {Z}}_0,\nu _0)$, used throughout the paper, and make the following assumptions.

Assumption 2.1

• (i) For $z_j=(x_j,y_j)\in {\mathbb {R}}^{d+d'}$ ($j=1,2$), $t\geqslant 0$ and ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$ ($i=0,1$),

  $$\begin{aligned}{} & {} |b(t, z_1)-b(t,z_2)| + |B(t,z_1)-B(t,z_2)| +|\sigma (t,z_1)-\sigma (t, z_2)| \\{} & {} \quad + |\rho (t,z_1)-\rho (t,z_2)|\leqslant L|z_1-z_2|, \\{} & {} |\eta (t,z_1,{\mathfrak {z}}_0)-\eta (t,z_2,{\mathfrak {z}}_0)|\leqslant \bar{\eta }({\mathfrak {z}}_0)|z_1-z_2|, \\{} & {} |\xi (t,z_1,{\mathfrak {z}}_1)-\xi (t,z_2,{\mathfrak {z}}_1)|\leqslant \bar{\xi }({\mathfrak {z}}_1)|z_1-z_2|. \end{aligned}$$

• (ii) For all $z=(x,y)\in {\mathbb {R}}^{d+d'}$, $t\geqslant 0$ and ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$ for $i=0,1$ we have

  $$\begin{aligned}{} & {} |b(t,z)| +|\sigma (t,z)|+ |\rho (t,z)|\leqslant K_0+K_1|z|, \quad |B(t,z)|\leqslant K,\\{} & {} |\eta (t,z,{\mathfrak {z}}_0)|\leqslant \bar{\eta }({\mathfrak {z}}_0)(K_0+K_1|z|), \quad |\xi (t,z,{\mathfrak {z}}_1)| \leqslant \bar{\xi }({\mathfrak {z}}_1)( K_0+K_1|z|),\\{} & {} \int _{{\mathfrak {Z}}_1}|{\mathfrak {z}}|^2\,\nu _1(d{\mathfrak {z}})\leqslant K_0^2. \end{aligned}$$

• (iii) The initial condition $Z_0=(X_0,Y_0)$ is an ${\mathcal {F}}_0$-measurable random variable with values in ${\mathbb {R}}^{d+d'}$.

Assumption 2.2

The functions ${\bar{\eta }}\in L_2({\mathfrak {Z}}_0)$ and ${\bar{\xi }}\in L_2({\mathfrak {Z}}_1)$ are bounded in magnitude by constants $K_{\eta }$ and $K_{\xi }$, respectively.

Assumption 2.3

For some $r>2$ we have ${\mathbb {E}}|X_0|^r<\infty$, and the measure $\nu _1$ satisfies

$$\begin{aligned} K_r:=\int _{{\mathfrak {Z}}_1} |{\mathfrak {z}}|^{r}\,\nu _1(d{\mathfrak {z}})<\infty . \end{aligned}$$

By a well-known theorem of Itô one knows that Assumption 2.1 ensures the existence and uniqueness of a solution $(X_t,Y_t)_{t\geqslant 0}$ to (1.1) for any given ${\mathcal {F}}_0$-measurable initial value $Z_0=(X_0,Y_0)$, and for every $T>0$,

$$\begin{aligned} {\mathbb {E}}\sup _{t\leqslant T}(|X_t|^q+|Y_t|^q)\leqslant N(1+{\mathbb {E}}|X_0|^q+{\mathbb {E}}|Y_0|^q) \end{aligned}$$

(2.1)

holds for $q=2$ with a constant N depending only on T, $K_0$, K, $K_1$, $|\bar{\xi }|_{L_2}$, $|\bar{\eta }|_{L_2}$ and $d+d'$. If in addition to Assumption 2.1 we assume Assumptions 2.2 and 2.3, then it is known, see, e.g., [2], that the moment estimate (2.1) holds with $q:=r$ for every $T>0$, where now the constant N depends also on r, $K_r$ $K_{\xi }$ and $K_{\eta }$.

We also need the following additional assumption.

Assumption 2.4

(i) The functions $f_0(t,x,y,{\mathfrak {z}}_0):{=}\eta (t,x,y,{\mathfrak {z}}_0)$ and $f_1(t,x,y, {\mathfrak {z}}_1):=\xi (t, x,y,{\mathfrak {z}}_1)$ are continuously differentiable in $x\in {\mathbb {R}}^d$ for each $(t,y,{\mathfrak {z}}_i)\in {\mathbb {R}}_+\times {{\mathbb {R}}^{d'}}\times {\mathfrak {Z}}_i$, for $i=0$ and $i=1$, respectively, such that

$$\begin{aligned} \lim _{\varepsilon \downarrow 0} \sup _{t\in [0,T]}\sup _{{\mathfrak {z}}\in {\mathfrak {Z}}_i}\sup _{|y|\leqslant R} \sup _{|x|\leqslant R, |x'|\leqslant R, |x-x'|\leqslant \varepsilon } |D_xf_i(t,x,y,{\mathfrak {z}}_i)-D_xf_i(t,x',y,{\mathfrak {z}}_i)|=0 \end{aligned}$$

for every $R>0$.

(ii) There is a constant $\lambda >0$ such that for $\theta \in [0,1]$, $(t,y,{\mathfrak {z}}_i)\in {\mathbb {R}}_+\times {{\mathbb {R}}^{d'}}\times {\mathfrak {Z}}_i$ for $i=0,1$ we have

$$\begin{aligned} \lambda |x_1-x_2|\leqslant |x_1-x_2+\theta (f_i(t,x_1,y,{\mathfrak {z}}_i)-f_i(t,x_2,y,{\mathfrak {z}}_i))| \quad \text {for }x_1,x_2\in {\mathbb {R}}^d. \end{aligned}$$

(iii) The function $\rho B=(\rho ^{ik}B^k)$ is Lipschitz in $x\in {\mathbb {R}}^d$, uniformly in (t, y), i.e.,

$$\begin{aligned} |(\rho B)(t,x_1,y)-(\rho B)(t,x_2,y)|\leqslant L|x_1-x_2| \quad \text {for all }x_1,x_2\in {\mathbb {R}}^d\text { and }(t,y)\in [0,T]\times {\mathbb {R}}^{d'}. \end{aligned}$$

Recall that ${\mathcal {F}}_t^Y$ denotes the completion of the $\sigma$-algebra generated by $(Y_s)_{s\leqslant t}$. Then the main result of the paper reads as follows.

Theorem 2.1

Let Assumptions 2.1, 2.2 and 2.4 hold. If $K_1\ne 0$ in Assumption 2.1, then let additionally Assumption 2.3 hold. Assume the conditional density $\pi _0=P(X_0\in dx|{\mathcal {F}}_0^Y)/dx$ exists and ${\mathbb {E}}|\pi _0|_{L_p}^p<\infty$ for some $p\geqslant 2$. Then almost surely the conditional density $\pi _t=P(X_t\in dx|{\mathcal {F}}^Y_t)/dx$ exists for all $t\in [0,T]$. Moreover, $(\pi _t)_{t\in [0,T]}$ is an $L_p$-valued weakly cadlag process.

3 The filtering equations

To describe the evolution of the conditional distribution $P_t(dx)=P(X_t\in dx|Y_s,s\leqslant t)$ for $t\in [0,T]$, we introduce the random differential operators

$$\begin{aligned} {\mathcal {L}}_t=a^{ij}_t(x)D_{ij}+b^i_t(x)D_i, \quad {\mathcal {M}}^k_t =\rho _t^{ik}(x)D_i+B^k_t(x), \quad k=1,2,\ldots ,d', \end{aligned}$$

where

$$\begin{aligned}{} & {} a^{ij}_t(x):=\tfrac{1}{2}\sum _{k=1}^{d_1}(\sigma ^{ik}_t\sigma ^{jk}_t)(x) +\tfrac{1}{2}\sum _{l=1}^{d'}(\rho _t^{il}\rho _t^{jl})(x),\\{} & {} \sigma _t^{ik}(x):=\sigma ^{ik}(t,x,Y_t),\quad \rho _t^{il}(x):=\rho ^{il}(t,x,Y_t), \\{} & {} b^i_t(x):=b^i(t,x,Y_t), \quad B^k_t(x):= B^k(t,x,Y_t) \end{aligned}$$

for $\omega \in \Omega$, $t\geqslant 0$, $x=(x^1,\ldots ,x^d)\in {\mathbb {R}}^d$, and $D_i=\partial /\partial x^i$, $D_{ij}=\partial ^2/(\partial x^i\partial x^j)$ for $i,j=1,2\ldots ,d$. Moreover for every $t\geqslant 0$ and ${\mathfrak {z}}\in {\mathfrak {Z}}_1$ we introduce the random operators $I_t^{\xi }$ and $J_t^{\xi }$ defined by

$$\begin{aligned} \varphi (x,{\mathfrak {z}}){} & {} =\varphi (x+\xi _t(x,{\mathfrak {z}}), {\mathfrak {z}})-\varphi (x,{\mathfrak {z}}), \quad J_t^{\xi }\phi (x, {\mathfrak {z}})=I_t^{\xi }\phi (x, {\mathfrak {z}})\nonumber \\{} & {} \quad -\sum _{i=1}^d\xi _t^i(x,{\mathfrak {z}})D_i\phi (x,{\mathfrak {z}}) \end{aligned}$$

(3.1)

for functions $\varphi =\varphi (x,{\mathfrak {z}})$ and $\phi =\phi (x,{\mathfrak {z}})$ of $x\in {\mathbb {R}}^d$ and ${\mathfrak {z}}\in {\mathfrak {Z}}_1$, and furthermore the random operators $I_t^{\eta }$ and $J_t^{\eta }$, defined as $I_t^{\xi }$ and $J_t^{\xi }$, respectively, with $\eta _t(x,{\mathfrak {z}})$ in place of $\xi _t(x,{\mathfrak {z}})$, where

$$\begin{aligned} \xi _t(x,{\mathfrak {z}}_{1}):=\xi (t,x,Y_{t-},{\mathfrak {z}}_{1}), \quad \eta _t(x,{\mathfrak {z}}_{0}):=\eta (t,x,Y_{t-},{\mathfrak {z}}_{0}) \end{aligned}$$

for $\omega \in \Omega$, $t\geqslant 0$, $x\in {\mathbb {R}}^d$ and ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$ for $i=0,1$.

Define the processes

$$\begin{aligned}{} & {} \gamma _t =\exp \left( -\int _0^tB_s(X_s)\,dV_s-\tfrac{1}{2}\int _0^t|B_s(X_s)|^2\,ds\right) , \quad t\in [0,T],\nonumber \\{} & {} {{\tilde{V}}}_t=\int _0^tB_s(X_s)\,ds+V_t, \quad t\in [0,T]. \end{aligned}$$

(3.2)

Since by Assumption 2.1 (ii) B is bounded in magnitude by a constant, it is well-known that $(\gamma _t)_{t\in [0,T]}$ is an ${\mathcal {F}}_t$-martingale and hence by Girsanov’s theorem the measure Q defined by $dQ=\gamma _TdP$ is a probability measure equivalent to P, and under Q the process $(W_t,{{\tilde{V}}}_t)_{t\in [0,T]}$ is a $d_1+d'$-dimensional ${\mathcal {F}}_t$-Wiener process.

The equations governing the conditional distribution and the unnormalised conditional distributions of $X_t$, given the observations $\{Y_s:s\in [0,t]\}$ are given by the following result. We denote by $({\mathcal {F}}^Y_t)_{t\in [0,T]}$ the completed filtration generated by $(Y_t)_{t\in [0,T]}$.

Theorem 3.1

Assume that $Z=(X_t,Y_t)_{t\in [0,T]}$ satisfies equation (1.1) and let Assumption 2.1(ii) hold. Assume also ${\mathbb {E}}|X_0|^2<\infty$ if $K_1\ne 0$ in Assumption 2.1(ii). Then there exist measure-valued ${\mathcal {F}}^Y_t$-adapted weakly cadlag processes $(P_t)_{t\in [0,T]}$ and $(\mu _t)_{t\in [0,T]}$ such that

$$\begin{aligned} P_t(\varphi )= & {} \mu _t(\varphi )/\mu _t(\textbf{1}), \quad \text {for }\omega \in \Omega ,\,\, t\in [0,T], \\ P_t(\varphi )= & {} {\mathbb {E}}(\varphi (X_t)|{\mathcal {F}}^Y_t),\quad \mu _t(\varphi ) ={\mathbb {E}}_{Q}(\gamma _t^{-1}\varphi (X_t)|{\mathcal {F}}^Y_t) \quad \text {(a.s.) for each }t\in [0,T], \end{aligned}$$

for bounded Borel functions $\varphi$ on ${\mathbb {R}}^d$, and for every $\varphi \in C^{2}_b({\mathbb {R}}^d)$ almost surely

$$\begin{aligned} \begin{aligned} \mu _t(\varphi )&= \mu _0(\varphi ) + \int _0^t\mu _{s}({\mathcal {L}}_s\varphi )\,ds + \int _0^t \mu _{s}({\mathcal {M}}_s^k\varphi )\,d{{\tilde{V}}}^k_s + \int _0^t\int _{{\mathfrak {Z}}_0}\mu _{s}(J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})ds\\&\quad + \int _0^t\int _{{\mathfrak {Z}}_1}\mu _{s}(J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})ds +\int _0^t\int _{{\mathfrak {Z}}_1}\mu _{s-}(I_s^{\xi }\varphi )\,{\tilde{N}}_1(d{\mathfrak {z}},ds), \end{aligned} \end{aligned}$$

(3.3)

and

$$\begin{aligned} \begin{aligned} P_t(\varphi )&= P_0(\varphi ) + \int _0^tP_{s}({\mathcal {L}}_s\varphi )\,ds + \int _0^t \big (P_{s}({\mathcal {M}}_s^k\varphi )-P_{s}(\varphi )P_s(B^k_s)\big )\,d{{\bar{V}}}^k_s\\&\quad + \int _0^t\int _{{\mathfrak {Z}}_0}P_{s}(J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})ds + \int _0^t\int _{{\mathfrak {Z}}_1}P_{s}(J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})ds\\&\quad +\int _0^t\int _{{\mathfrak {Z}}_1}P_{s-}(I_s^{\xi }\varphi )\,{{\tilde{N}}}_1(d{\mathfrak {z}},ds)\\ \end{aligned} \end{aligned}$$

(3.4)

for all $t\in [0,T]$, where $({{\tilde{V}}}_t)_{t\in [0,T]}$ is given in (3.2), and the process $({{\bar{V}}}_t)_{t\in [0,T]}$ is defined by

$$\begin{aligned} d{{\bar{V}}}_t=d{{\tilde{V}}}_t-P_t(B_t)\,dt=dV_t+(B_t(X_t)-P_t(B_t))\,dt, \quad {{\bar{V}}}_0=0. \end{aligned}$$

Proof

This theorem, under more general assumptions, is proved in [4]. $\square$

Remark 3.1

Clearly, equation (3.3) can be rewritten as

$$\begin{aligned} \begin{aligned} \mu _t(\varphi )&= \mu _0(\varphi ) + \int _0^t\mu _{s}({\tilde{{\mathcal {L}}}}_s\varphi )\,ds + \int _0^t \mu _{s}({\mathcal {M}}_s^k\varphi )\,dV^k_s + \int _0^t\int _{{\mathfrak {Z}}_0}\mu _{s}(J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})ds\\&\quad + \int _0^t\int _{{\mathfrak {Z}}_1}\mu _{s}(J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})ds +\int _0^t\int _{{\mathfrak {Z}}_1}\mu _{s-}(I_s^{\xi }\varphi )\,{\tilde{N}}_1(d{\mathfrak {z}},ds), \end{aligned} \end{aligned}$$

(3.5)

where ${\tilde{{\mathcal {L}}}}_s={\mathcal {L}}_s+B_s(X_s){\mathcal {M}}_s$. Moreover, if $d\mu _t/dx$ exists for $P\otimes dt$-a.e. $(\omega ,t)\in \Omega \times [0,T]$, and $u=u_t(x)$ is an ${\mathcal {F}}_t$-adapted $L_p$-valued weakly cadlag process, for $p>1$, such that almost surely $u_t=d\mu _t/dx$ for all $t\in [0,T]$, then for each $\varphi \in C_b^2({\mathbb {R}}^d)$ we have that almost surely

$$\begin{aligned} \begin{aligned} (u_t,\varphi )&=(u_0,\varphi ) + \int _0^t(u_{s},{\tilde{{\mathcal {L}}}}_s\varphi )\,ds + \int _0^t(u_{s},{\mathcal {M}}_s^k\varphi )\,dV^k_s \\&\quad + \int _0^t\int _{{\mathfrak {Z}}_0}(u_{s},J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})ds\\&\quad + \int _0^t\int _{{\mathfrak {Z}}_1}(u_{s},J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})ds +\int _0^t\int _{{\mathfrak {Z}}_1}(u_{s-},I_s^{\xi }\varphi )\,{\tilde{N}}_1(d{\mathfrak {z}},ds). \end{aligned} \end{aligned}$$

(3.6)

holds for all $t\in [0,T]$.

Remark 3.2

We also recall from [4] that there exists a cadlag ${\mathcal {F}}^Y_t$-adapted positive process $( ^o\!\gamma _t)_{t\in [0,T]}$, the optional projection of $(\gamma _t)_{t\in [0,T]}$ under P with respect to $({\mathcal {F}}^Y_t)_{t\in [0,T]}$, such that for every $\mathcal {{F}}^Y_t$-stopping time $\tau \leqslant T$ we have

$$\begin{aligned} {\mathbb {E}}(\gamma _\tau |{\mathcal {F}}^Y_\tau ) = {^o\!\gamma }_\tau ,\quad \text {almost surely.} \end{aligned}$$

(3.7)

Since for each $t\in [0,T]$, by known properties of conditional expectations (see i.e. [19, Thm. 6.1]), almost surely

$$\begin{aligned} \mu _t(\textbf{1}) = {\mathbb {E}}_Q(\gamma _t^{-1}|{\mathcal {F}}^Y_t) = 1/{\mathbb {E}}(\gamma _t|{\mathcal {F}}^Y_t) = 1/{^o\!\gamma }_t, \end{aligned}$$

we also have that for each $\varphi \in C_b^2$ almost surely $P_t(\varphi ) = \mu _t(\varphi ){^o\!\gamma }_t$ for all $t\in [0,T]$.

Definition 3.1

An ${\mathfrak {M}}$-valued weakly cadlag ${\mathcal {F}}_t$-adapted process $(\mu _t)_{t\in [0,T]}$ is said to be an ${\mathfrak {M}}$-solution to the equation

$$\begin{aligned} d\mu _t&=\,{\tilde{{\mathcal {L}}}}_t^{*}\mu _tdt+{\mathcal {M}}_t^{k*}\mu _t\,dV^k_t +\int _{{\mathfrak {Z}}_0}J_t^{\eta *}\mu _t\,\nu _0(d{\mathfrak {z}})dt \nonumber \\&\quad +\int _{{\mathfrak {Z}}_1}J_t^{\xi *}\mu _t\,\nu _1(d{\mathfrak {z}})dt +\int _{{\mathfrak {Z}}_1}I_t^{\xi *}\mu _{t-}\,{{\tilde{N}}}_1(d{\mathfrak {z}},dt) \end{aligned}$$

(3.8)

with initial value $\mu _0$, if for each $\varphi \in C^2_b$ almost surely equation (3.5) holds for all $t\in [0,T]$. If $(\mu _t)_{t\in [0,T]}$ is an ${\mathfrak {M}}$-solution to Eq. (3.8), such that it takes values in ${\mathbb {M}}$, then we call it a measure-valued solution.

Definition 3.2

Let $p\geqslant 1$ and let $\psi$ be an $L_p$-valued ${\mathcal {F}}_0$-measurable random variable. Then we say that an $L_p$-valued ${\mathcal {F}}_t$-adapted weakly cadlag process $(u_t)_{t\in [0,T]}$ is an $L_p$-solution of the equation

$$\begin{aligned} du_t&={\tilde{{\mathcal {L}}}}_t^{*}u_tdt+{\mathcal {M}}_t^{k*}u_t\,dV^k_t +\int _{{\mathfrak {Z}}_0}J_t^{\eta *}u_t\,\nu _0(d{\mathfrak {z}})dt \nonumber \\&\quad +\int _{{\mathfrak {Z}}_1}J_t^{\xi *}u_t\,\nu _1(d{\mathfrak {z}})dt +\int _{{\mathfrak {Z}}_1}I_t^{\xi *}u_{t-}\,{{\tilde{N}}}_1(d{\mathfrak {z}},dt) \end{aligned}$$

(3.9)

with initial condition $\psi$, if for every $\varphi \in C_0^{\infty }$ almost surely (3.6) holds for all $t\in [0,T]$ and $u_0=\psi$ (a.s.).

Lemma 3.2

Let Assumption 2.1 hold, and assume also ${\mathbb {E}}|X_0|^2<\infty$ if $K_1\ne 0$ in Assumptions 2.1(ii). Let $(\mu _t)_{t\in [0,T]}$ be the measure-valued process from Theorem 3.1. Then we have

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}\mu _t(\textbf{1})\leqslant N, \end{aligned}$$

(3.10)

with a constant N depending only on d, K and T.

Proof

Taking $\textbf{1}$ instead of $\varphi$ in the Zakai equation (3.3) yields for $(\beta _t)_{t\in [0,T]}:=(B_{t}(X_t))_{t\in [0,T]}$,

$$\begin{aligned} \mu _t(\textbf{1}) = 1 + \int _0^t\mu _s(\beta _{s}^kB_{s}^k)\,ds + \int _0^t\mu _s (B_{s}^k)\,dV_s^k. \end{aligned}$$

(3.11)

Taking here $t\wedge \tau _n$ in place of t, with the stopping times

$$\begin{aligned} \tau _n:=\inf \{t\geqslant 0:\mu _t(\textbf{1})\geqslant n\},\quad n\geqslant 1, \end{aligned}$$

and using $|B|\leqslant K$ after taking expectations on both sides, we get

$$\begin{aligned} {\mathbb {E}}\mu _{t\wedge \tau _n}(\textbf{1}) \leqslant 1 + dK^2\int _0^{t}{\mathbb {E}}\mu _{s\wedge \tau _n}(\textbf{1})\,ds. \end{aligned}$$

Using Gronwall’s inequality and Fatou’s lemma, we obtain for each n,

$$\begin{aligned} \sup _{t\in [0,T]}{\mathbb {E}}\mu _t(\textbf{1})\leqslant N, \end{aligned}$$

(3.12)

with a constant $N=N(d,K,T)$. Moreover, by Davis’ and Young’s inequalities and due to (3.12) we have

$$\begin{aligned}{} & {} {\mathbb {E}}\sup _{t\in [0,T]} \int _0^{t\wedge \tau _n}\mu _s(B_{s}^k)\,dV_s^k \leqslant 3{\mathbb {E}}\Big ( \sum _k\int _0^{T\wedge \tau _n} \mu _s(B_{s}^k)^2\,ds\Big )^{1/2} \\{} & {} \quad \leqslant N{\mathbb {E}}\Big (\sup _{t\in [0,T]}\mu _{t\wedge \tau _n}(\textbf{1})\int _0^T\mu _s(\textbf{1})\,ds\Big )^{1/2} \leqslant \tfrac{1}{2}{\mathbb {E}}\sup _{t\in [0,T]}\mu _{t\wedge \tau _n}(\textbf{1})\\{} & {} \qquad + N'E\int _0^T\mu _s(\textbf{1})\,ds<\infty , \end{aligned}$$

with constants N and $N'$ depending only on d, K, T. Using this and (3.12), from (3.11) we get

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}\mu _{t\wedge \tau _n}(\textbf{1}) \leqslant N,\quad \text {for all }n\geqslant 1 \end{aligned}$$

with a constant N(d, K, T). By Fatou’s lemma we then obtain (3.10) $\square$

4 $L_p$-estimates

Recall that ${\mathbb {M}}={\mathbb {M}}({\mathbb {R}}^d)$ denotes the set of finite measures on ${\mathcal {B}}({\mathbb {R}}^d)$, and ${\mathfrak {M}}:=\{\mu -\nu :\mu ,\nu \in {\mathbb {M}}\}$. For $\nu \in {\mathfrak {M}}$ we use the notation $|\nu |:=\nu ^{+}+\nu ^{-}$ for the total variation and set $\Vert \nu \Vert =|\nu |({\mathbb {R}}^d)$, where $\nu ^{+}\in {\mathbb {M}}$ and $\nu ^{-}\in {\mathbb {M}}$ are the positive and negative parts of $\nu$. For $\varepsilon >0$ we use the notation $k_{\varepsilon }$ for the Gaussian density function on ${\mathbb {R}}^d$ with mean 0 and covariance matrix $\varepsilon I$. For linear functionals $\Phi$, acting on a real vector space V containing ${\mathcal {S}}={\mathcal {S}}({\mathbb {R}}^d)$, the rapidly decreasing smooth functions on ${\mathbb {R}}^d$, the mollification $\Phi ^{(\varepsilon )}$ is defined by

$$\begin{aligned} \Phi ^{(\varepsilon )}(x)=\Phi (k_{\varepsilon }(x-\cdot )), \quad x\in {\mathbb {R}}^d. \end{aligned}$$

In particular, when $\Phi =\mu$ is a (signed) measure from ${\mathcal {S}}^{*}$, the dual of ${\mathcal {S}}$, or $\Phi =f$ is a function from ${\mathcal {S}}^{*}$, then

$$\begin{aligned} \mu ^{(\varepsilon )}(x)=\int _{{\mathbb {R}}^d}k_{\varepsilon }(x-y)\,\mu (dy), \quad f^{(\varepsilon )}(x)=\int _{{\mathbb {R}}^d}k_{\varepsilon }(x-y)f(y)\,dy,\quad x\in {\mathbb {R}}^d, \end{aligned}$$

and

$$\begin{aligned}{} & {} (L^{*}\mu )^{(\varepsilon )}(x):=\int _{{\mathbb {R}}^d}L_yk_{\varepsilon }(x-y)\mu (dy),\quad x\in \mathbb {{R}}^d, \\{} & {} (L^{*}f)^{(\varepsilon )}(x):=\int _{{\mathbb {R}}^d}(L_yk_{\varepsilon }(x-y))f(y)\,dy,\quad x\in \mathbb {{R}}^d, \end{aligned}$$

when L is a linear operator on V such that the integrals are well-defined for every $x\in {\mathbb {R}}^d$. The subscript y in $L_y$ indicates that the operator L acts in the y-variable of the function ${{\bar{k}}}_{\varepsilon }(x,y):= k_{\varepsilon }(x-y)$. For example, if L is a differential operator of the form $a^{ij}D_{ij}+b^iD_i+c$, where $a^{ij}$, $b^i$ and c are functions defined on ${\mathbb {R}}^d$, then

$$\begin{aligned} (L^*\mu )^{(\varepsilon )}(x)=\int _{{\mathbb {R}}^d}(a^{ij}(y)\tfrac{\partial ^2}{\partial y^i\partial y^j} +b^i(y)\tfrac{\partial }{\partial y^i}+c(y))k_{\varepsilon }(x-y)\mu (dy). \end{aligned}$$

We will often use the following well-known properties of mollifications with $k_{\varepsilon }$:

• (i) $|\varphi ^{(\varepsilon )}|_{L_p} \leqslant |\varphi |_{L_p}$ for $\varphi \in L_p({\mathbb {R}}^d)$, $p\in [1,\infty )$;

• (ii) $\mu ^{(\varepsilon )}(\varphi ):=\int _{{\mathbb {R}}^d}\mu ^{(\varepsilon )}(x)\varphi (x)\,dx =\int _{{\mathbb {R}}^d}\varphi ^{(\varepsilon )}(x)\mu (dx) =:\mu (\varphi ^{(\varepsilon )})$ for $\mu \in {\mathfrak {M}}$ and $\varphi \in L_p({\mathbb {R}}^d)$, $p\geqslant 1$;

• (iii) $|\mu ^{(\delta )}|_{L_p}\leqslant |\mu ^{(\varepsilon )}|_{L_p}$ for $0\leqslant \varepsilon \leqslant \delta$, $\mu \in {\mathfrak {M}}$ and $p\geqslant 1$. This property follows immediately from (i) and the “semigroup property" of the Gaussian kernel,

  $$\begin{aligned} k_{r+s}(y-z)=\int _{{\mathbb {R}}^d}k_{r}(y-x)k_{s}(x-z)\,dx, \quad y,z\in {\mathbb {R}}^d\text { and }r,s\in (0,\infty ).\nonumber \\ \end{aligned}$$

  (4.1)

The following generalization of (iii) is also useful: for integers $p\geqslant 2$ we have

$$\begin{aligned} \rho _{\varepsilon }(y):= & {} \int _{{\mathbb {R}}^d}\Pi _{r=1}^pk_{\varepsilon }(x-y_r)\,dx =c_{p,\varepsilon }e^{-\sum _{1\leqslant r<s\leqslant p}|y_r-y_s|^2/(2\varepsilon p)}, \nonumber \\ y= & {} (y_1,\ldots ,y_p)\in {\mathbb {R}}^{pd}, \end{aligned}$$

(4.2)

for $\varepsilon >0$, with a constant $c_{p,\varepsilon }=c_{p,\varepsilon }(d)=p^{-d/2}(2\pi \varepsilon )^{(1-p)d/2}$. This can be seen immediately by noticing that for $x,y_k\in {\mathbb {R}}^d$ and $y=(y_k)_{k=1}^p\in {\mathbb {R}}^{pd}$ we have

$$\begin{aligned} \sum _{k=1}^p(x-y_k)^2=p\Big (x-\sum _ky_k/p\Big )^2 +\tfrac{1}{p}\sum _{1\leqslant k<l\leqslant p}(y_k-y_l)^2. \end{aligned}$$

Clearly, for every $r=1,2,\ldots ,p$ and $i=1,2,\ldots ,d$,

$$\begin{aligned} \partial _{y^i_r}\rho _{\varepsilon }(y)= & {} \tfrac{1}{\varepsilon p} \sum _{s=1}^p(y^i_s-y^i_r)\rho _{\varepsilon }(y), \quad y=(y_1,\ldots ,y_p)\in {\mathbb {R}}^d, \nonumber \\ y_r= & {} (y^1_r\ldots ,y^d_r)\in {\mathbb {R}}^d. \end{aligned}$$

(4.3)

It is easy to see that

$$\begin{aligned} \sum _{r=1}^p\partial _{y_r^j}\rho _{\varepsilon }(y)=0 \quad \text {for }y\in {\mathbb {R}}^{pd}, j=1,2,\ldots ,d. \end{aligned}$$

We will often use this in the form

$$\begin{aligned} \partial _{y^j_r}\rho _{\varepsilon }(y) =-\sum _{s\ne r}^p\partial _{y_s^j}\rho _{\varepsilon }(y) \quad \text {for }r=1,\ldots ,p\text { and }j=1,2,\ldots ,d. \end{aligned}$$

(4.4)

In order for the left-hand side of the following $L_p$-estimates for $\mu \in {\mathfrak {M}}$ in this section to be well-defined, we require that

$$\begin{aligned} K_1\int _{{\mathbb {R}}^d}|x|^2\,|\mu |(dx)<\infty , \end{aligned}$$

(4.5)

where we use the formal convention that $0\cdot \infty = 0$, i.e., if $K_1=0$, then condition (4.5) is satisfied. The following lemma generalises a lemma from [13].

Lemma 4.1

Let $p\geqslant 2$ be an integer. Let $\sigma =(\sigma ^{ik})$ and $b=(b^{i})$ be Borel functions on ${\mathbb {R}}^d$ with values in ${\mathbb {R}}^{d\times m}$ and ${\mathbb {R}}^d$, respectively, such that for some nonnegative constants $K_0$, $K_1$ and L we have

$$\begin{aligned} |\sigma (x)|+|b(x)|\leqslant K_0+K_1|x| \quad |\sigma (x)-\sigma (y)|\leqslant L|x-y|, \quad |b(x)-b(y)|\leqslant L|x-y|\nonumber \\ \end{aligned}$$

(4.6)

for all $x,y\in {\mathbb {R}}^d$. Set $a^{ij}=\sigma ^{ik}\sigma ^{jk}/2$ for $i,j=1,2,\ldots ,d$. Let $\mu \in {\mathfrak {M}}$ such that it satisfies (4.5). Then we have

$$\begin{aligned}{} & {} p((\mu ^{(\varepsilon )})^{p-1}, ((a^{ij}D_{ij})^*\mu )^{(\varepsilon )}) +\tfrac{p(p-1)}{2} ((\mu ^{(\varepsilon )})^{p-2} ((\sigma ^{ik}D_i)^*\mu )^{(\varepsilon )}, ((\sigma ^{jk}D_j)^*\mu )^{(\varepsilon )})\nonumber \\{} & {} \quad \leqslant NL^2||\mu |^{(\varepsilon )}|^p_{L_p}, \end{aligned}$$

(4.7)

$$\begin{aligned}{} & {} \quad ((\mu ^{(\varepsilon )})^{p-1}, ((b^{i}D_{i})^*\mu )^{(\varepsilon )}) \leqslant NL^2||\mu |^{(\varepsilon )}|^p_{L_p}\nonumber \\ \end{aligned}$$

(4.8)

with a constant $N=N(d,p)$.

Proof

Let A and B denote the left-hand side of the inequalities (4.7) and (4.8), respectively. Note first that using

$$\begin{aligned} \sup _{x\in {\mathbb {R}}^d}\sum _{k=0}^2|D^kk_{\varepsilon }(x)|<\infty ,\quad \text {and} \quad \int _{{\mathbb {R}}^d}(1+|x|+K_1|x|^2)\,|\mu |(dx)<\infty , \end{aligned}$$

(4.9)

as well as the conditions on on $\sigma$ and b, it is easy to verify that A and B are well-defined. Then by Fubini’s theorem and the symmetry of the Gaussian kernel

$$\begin{aligned} A=\int _{{\mathbb {R}}^{(p+1)d}}f(x,y)\,\mu _p(dy)\,dx \end{aligned}$$

with

$$\begin{aligned} f(x,y):=\left( pa^{ij}(y_p)\partial _{y^i_p}\partial _{y^j_p} +\tfrac{p(p-1)}{2} \sigma ^{ik}(y_{p-1})\sigma ^{jk}(y_{p})\partial _{y^i_{p-1}}\partial _{y^j_p}\right) \Pi _{k=1}^pk_{\varepsilon }(x-y_k), \end{aligned}$$

where $x\in {\mathbb {R}}^d$, $y=(y_k)_{k=1}^p\in {\mathbb {R}}^{dp}$, and $y^i_k$ denotes the i-th coordinate of $y_k\in {\mathbb {R}}^d$ for $k=1,\ldots ,p$, and $\mu _p(dy):=\mu ^{\otimes p}(dy)=\mu (dy_1)\ldots \mu (dy_p)$. Hence by Fubini’s theorem and symmetry again

$$\begin{aligned} A=\int _{{\mathbb {R}}^{pd}} \left( pa^{ij}(y_p)\partial _{y^i_p}\partial _{y^j_p} +\tfrac{p(p-1)}{2} \sigma ^{ik}(y_{p-1})\sigma ^{jk}(y_{p})\partial _{y^i_{p-1}}\partial _{y^j_p}\right) \rho _{\varepsilon }(y)\,\mu _{p}(dy) \end{aligned}$$

(4.10)

$$\begin{aligned} =\tfrac{1}{2}\sum _{r=1}^p\int _{{\mathbb {R}}^{pd}} \Big (2a^{ij}(y_r)\partial _{y^i_r}\partial _{y^j_r} +\sum _{s\ne r} \sigma ^{ik}(y_{r})\sigma ^{jk}(y_{s})\partial _{y^i_{r}}\partial _{y^j_s}\Big ) \rho _{\varepsilon }(y)\,\mu _{p}(dy),\nonumber \\ \end{aligned}$$

(4.11)

where $\rho _{\varepsilon }$ is given in (4.2). Using here (4.4) and symmetry of expressions in $y_k$ and $y_l$, we obtain

$$\begin{aligned} A{} & {} =-\tfrac{1}{2}\sum _{r=1}^p\sum _{s\ne r}\int _{{\mathbb {R}}^{pd}} \left( 2a^{ij}(y_r)\partial _{y^i_r}\partial _{y^j_s} - \sigma ^{ik}(y_{r})\sigma ^{jk}(y_{s})\partial _{y^i_{r}}\partial _{y^j_s}\right) \rho _{\varepsilon }(y)\,\mu _p(dy) \\{} & {} =-\tfrac{1}{2}\sum _{r=1}^p\sum _{s\ne r}\int _{{\mathbb {R}}^{pd}} \left( (a^{ij}(y_r)+a^{ij}(y_s))\partial _{y^i_r}\partial _{y^j_s} - \sigma ^{ik}(y_{r})\sigma ^{jk}(y_{s})\partial _{y^i_{r}}\partial _{y^j_s}\right) \rho _{\varepsilon }(y)\,\mu _{p}(dy)\\{} & {} =-\tfrac{1}{2}\sum _{r=1}^p\sum ^p_{s=1}\int _{{\mathbb {R}}^{pd}} a^{ij}(y_r,y_s)\partial _{y^i_{r}}\partial _{y^j_s} \rho _{\varepsilon }(y)\,\mu _{p}(dy)\\{} & {} =-\tfrac{1}{2}\sum _{r=1}^p\sum _{s=1}^p\int _{{\mathbb {R}}^{pd}} a^{ij}(y_r,y_s)l^{ij,rs}_{\varepsilon }(y) \rho _{\varepsilon }(y)\,\mu _{p}(dy), \end{aligned}$$

where

$$\begin{aligned} a^{ij}(x,z)=\frac{1}{2}(\sigma ^{ik}(x)-\sigma ^{ik}(z))(\sigma ^{jk}(x)-\sigma ^{jk}(z)) \quad \text {for }x,z\in {\mathbb {R}}^d \end{aligned}$$

(4.12)

and

$$\begin{aligned} l^{ij,rs}_{\varepsilon }(y)=\rho ^{-1}_{\varepsilon }(y)\partial _{y^i_{r}}\partial _{y^j_s} \rho _{\varepsilon }(y) =\tfrac{1}{(p\varepsilon )^2}\sum ^p_{k=1}\sum _{l=1}^p(y^i_k-y^i_r)(y^j_l-y^j_s) +\tfrac{\delta _{ij}}{p\varepsilon }. \end{aligned}$$

Making use of the Lipschitz condition on $\sigma$ and using for $q=1,2$ that

$$\begin{aligned} \varepsilon ^{-q}\sum _{s\ne r}|y_s-y_r|^{2q}\rho _{\varepsilon } (y)\leqslant N\rho _{2\varepsilon }(y), \quad y\in {\mathbb {R}}^{pd}, \quad q\geqslant 0 \end{aligned}$$

(4.13)

with a constant $N=N(d,p,q)$, we have

$$\begin{aligned}{} & {} \big |\sum _{r=1}^p\sum _{s=1}^p a^{ij}(y_r,y_s)l^{ij,rs}_{\varepsilon }(y)\big | \leqslant \tfrac{NL^2}{(p\varepsilon )^2}\sum _{1\leqslant r<s\leqslant p}|y_r-y_s|^4\rho _{\varepsilon }(y)\\{} & {} \quad + \tfrac{NL^2}{p\varepsilon }\sum _{1\leqslant r<s\leqslant p}|y_r-y_s|^2\rho _{\varepsilon }(y) \leqslant N'L^2\rho _{2\varepsilon }(y)\quad \text {for }y\in {\mathbb {R}}^{pd} \end{aligned}$$

with constants $N=N(d,p)$ and $N'=N'(d,p)$. Hence

$$\begin{aligned}{} & {} A\leqslant N'L^2\int _{{\mathbb {R}}^{pd}}\rho _{2\varepsilon }(y)|\mu _{p}|(dy)=N'L^2 \int _{{\mathbb {R}}^{pd}}\int _{{\mathbb {R}}^d}\Pi _{r=1}^pk_{2\varepsilon }(x-y_r)\,dx|\mu _{p}|(dy) \\{} & {} \quad =N'L^2\int _{{\mathbb {R}}^{d}}\Pi _{r=1}^p\int _{{\mathbb {R}}^d}k_{2\varepsilon }(x-y_r)|\mu |(dy_r)\,dx =N'L^2||\mu |^{(2\varepsilon )}|^p_{L_p}. \end{aligned}$$

To prove (4.8) we proceed similarly. By Fubini’s theorem and symmetry

$$\begin{aligned} pB{} & {} =\int _{{\mathbb {R}}^{pd}}pb^i(y_p)\partial _{y^i_p}\rho _{\varepsilon }(y)\,\mu _{p}(dy) =\sum _{r=1}^p\int _{{\mathbb {R}}^{pd}}b^i(y_r)\partial _{y^i_r}\rho _{\varepsilon }(y)\,\mu _{p}(dy) \\{} & {} =-\sum _{r=1}^p\sum _{s\ne r}\int _{{\mathbb {R}}^{pd}}b^i(y_r)\partial _{y^i_s}\rho _{\varepsilon }(y)\,\mu _{p}(dy) =-\sum _{r=1}^p\sum _{s\ne r}\int _{{\mathbb {R}}^{pd}}b^i(y_s)\partial _{y^i_r}\rho _{\varepsilon }(y)\,\mu _{p}(dy). \end{aligned}$$

Thus

$$\begin{aligned} B{} & {} =\tfrac{p-1}{p}\sum _{r=1}^p\int _{{\mathbb {R}}^{pd}} b^i(y_r)\partial _{y^i_r}\rho _{\varepsilon }(y)\,\mu _{p}(dy) -\tfrac{1}{p}\sum _{r=1}^p\sum _{s\ne r} \int _{{\mathbb {R}}^{pd}}b^i(y_s)\partial _{y^i_r}\rho _{\varepsilon }(y)\,\mu _{p}(dy) \nonumber \\{} & {} =\tfrac{1}{p}\sum _{r=1}^p \int _{{\mathbb {R}}^{pd}}\sum _{s\ne r}(b^i(y_r)-b^i(y_s))\partial _{y^i_r}\rho _{\varepsilon }(y)\,\mu _{p}(dy)\nonumber \\{} & {} =\tfrac{1}{\varepsilon p^2}\sum _{r=1}^p \int _{{\mathbb {R}}^{pd}}\sum _{s\ne r}(b^i(y_r)-b^i(y_s))\sum _{l\ne r}(y^i_l-y^i_r) \rho _{\varepsilon }(y)\,\mu _{p}(dy). \end{aligned}$$

(4.14)

Using the Lipschitz condition on b and the inequality (4.13), we obtain

$$\begin{aligned}{} & {} B\leqslant \tfrac{NL}{\varepsilon }\int _{{\mathbb {R}}^{pd}}\sum _{s\ne r}|y_r-y_s|^2\rho _{\varepsilon }(y) |\mu _p|(dy) \leqslant N'L\int _{{\mathbb {R}}^{pd}}\rho _{2\varepsilon }(y)|\mu _p|(dy)\\{} & {} \quad =N'L||\mu |^{(2\varepsilon )}|^p_{L_p} \end{aligned}$$

with constants $N=N(p,d)$ and $N'=N(p,d)$, which completes the proof of the lemma. $\square$

Lemma 4.2

Let $p\geqslant 2$ be an integer and let $\sigma = (\sigma ^{i})$ and b be Borel functions on ${\mathbb {R}}^d$ with values in ${\mathbb {R}}^d$ and ${\mathbb {R}}$ respectively. Assume furthermore that there exist constants $K\geqslant 1$, $K_0$ and L such that

$$\begin{aligned}{} & {} |b(x)|\leqslant K, \quad |\sigma (x)|\leqslant K_0+K_1|x|, \quad |\sigma (x)-\sigma (y)|\leqslant L|x-y|,\quad |b\sigma (x)\\{} & {} \quad -b\sigma (y)|\leqslant L|x-y|\end{aligned}$$

for all $x,y\in {\mathbb {R}}^d$. Let $\mu \in {\mathfrak {M}}$ such that it satisfies (4.5). Then we have

$$\begin{aligned} \big ((\mu ^{{(\varepsilon )}})^{p-2}(b\mu )^{(\varepsilon )},(b\mu )^{(\varepsilon )}\big ) \leqslant K^2||\mu |^{(\varepsilon )}|_{L_p}^p, \end{aligned}$$

(4.15)

$$\begin{aligned} \big ((\mu ^{(\varepsilon )})^{p-2},((\sigma ^iD_i)^*\mu )^{(\varepsilon )}(b\mu )^{(\varepsilon )}\big ) \leqslant NKL||\mu |^{(\varepsilon )}|_{L_p}^p \end{aligned}$$

(4.16)

for every $\varepsilon >0$ with a constant $N=N(d,p)$.

Proof

We note again that by (4.9) together with the conditions on $\sigma$ and b, the left-hand sides of (4.15) and (4.16) are well-defined. Rewriting products of integrals as multiple integrals and using Fubini’s theorem for the left-hand side of the inequality (4.15) we have

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^{dp}}b(y_r)b(y_{s})\int _{{\mathbb {R}}^d}\Pi _{j=1}^pk_{\varepsilon }(x-y_j)\,dx \mu _p(dy)\\{} & {} \leqslant K^2 \int _{{\mathbb {R}}^{d(p+1)}}\Pi _{k=1}^pk_{\varepsilon }(x-y_k) \,dx|\mu _p|(dy) = K^2||\mu |^{{(\varepsilon )}}|_{L_p}^p, \end{aligned}$$

for any $r,s\in \{1,2,\ldots ,p\}$, where $y=(y_1,\ldots ,y_p)\in {\mathbb {R}}^{pd}$, $y_j\in {\mathbb {R}}^d$ for $j=1,2,\ldots ,p$, and the notation $\mu _p(dy)=\mu (dy_1)\ldots \mu (dy_p)$ is used. This proves (4.15).

Rewriting products of integrals as multiple integrals, using Fubini’s theorem, interchanging the order of taking derivatives and integrals, and using equation (4.2), for the left-hand side R of the inequality (4.16) we have

$$\begin{aligned} R&= \int _{{\mathbb {R}}^{dp}}b(y_{k})\sigma ^i(y_{r})\partial _{y_{r}^i} \int _{{\mathbb {R}}^d}\Pi _{j=1}^pk_{\varepsilon }(x-y_j)\,dx \mu _p(dy)\nonumber \\&= \int _{{\mathbb {R}}^{dp}} b(y_{k})\sigma ^i(y_{r})\partial _{y_{r}^i}\rho _\varepsilon (y)\, \mu _p(dy) \end{aligned}$$

(4.17)

for any $r, k\in \{1,2,\ldots ,p\}$ such that $r\ne k$. Hence

$$\begin{aligned} p(p-1)^2R= & {} \sum _{s=1}^p\sum _{r\ne s}\sum _{k\ne s} \int _{{\mathbb {R}}^{dp}}b(y_k)\sigma ^i(y_{s})\partial _{y_s^i}\rho _\varepsilon (y)\, \mu _p(dy) \nonumber \\= & {} \sum _{s=1}^p\sum _{r\ne s}\sum _{k\ne r} \int _{{\mathbb {R}}^{dp}} b(y_k)\sigma ^i(y_{s})\partial _{y_s^i}\rho _\varepsilon (y)\, \mu _p(dy)\nonumber \\{} & {} +\sum _{s=1}^p\sum _{r\ne s} \int _{{\mathbb {R}}^{dp}} (b(y_r)-b(y_s))\sigma ^i(y_{s})\partial _{y_s^i}\rho _\varepsilon (y)\, \mu _p(dy), \nonumber \\ \end{aligned}$$

(4.18)

and using (4.4) from (4.17) we obtain

$$\begin{aligned} p(p-1)R= -\sum _{r=1}^p\sum _{k\ne r}\sum _{s\ne r} \int _{{\mathbb {R}}^{dp}}b(y_k)\sigma ^i(y_{r})\partial _{y_s^i}\rho _\varepsilon (y)\, \mu _p(dy)\nonumber \\ =-\sum _{s=1}^p\sum _{r\ne s}\sum _{k\ne r} \int _{{\mathbb {R}}^{dp}}b(y_k)\sigma ^i(y_{r})\partial _{y_s^i}\rho _\varepsilon (y)\, \mu _p(dy) \end{aligned}$$

(4.19)

Adding up Eqs. (4.18) and (4.19), and taking into account the equation

$$\begin{aligned} (b(y_r)-b(y_s))\sigma ^i(y_{s})=b(y_r)\sigma ^i(y_r)-b(y_s)\sigma ^i(y_{s}) -b(y_r)(\sigma ^i(y_{r})-\sigma ^i(y_{s})) \end{aligned}$$

we get

$$\begin{aligned} p^2(p-1)R= & {} \sum _{s=1}^p\sum _{r\ne s}\sum ^p_{k=1} \int _{{\mathbb {R}}^{dp}}f^i(y_k,y_s,y_r)\partial _{y_s^i}\rho _\varepsilon (y)\, \mu _p(dy)\nonumber \\{} & {} +\sum _{s=1}^p\sum _{r\ne s} \int _{{\mathbb {R}}^{dp}}g^i(y_r,y_s)\partial _{y_s^i}\rho _\varepsilon (y)\, \mu _p(dy) \end{aligned}$$

(4.20)

with functions

$$\begin{aligned} f^i(x,u,v):=b(x)(\sigma ^i(u)-\sigma ^i(v)), \quad g^i(u,v):=b(u)\sigma ^i(u)-b(v)\sigma ^i(v) \nonumber \\ \end{aligned}$$

(4.21)

defined for $x,u,v\in {\mathbb {R}}^d$ for each $i=1,2,\ldots ,d$. By the boundedness of |b| and the Lipschitz condition on $\sigma$ and $b\sigma$ we have

$$\begin{aligned} |f^{i}(x,u,v)|\leqslant KL|u-v|, \quad |g^i(u,v)|\leqslant L|u-v|\quad x,u,v\in {\mathbb {R}}^d, i=1,2,\ldots ,d. \end{aligned}$$

Thus, taking into account (4.3) and (4.13), from (4.20) we obtain

$$\begin{aligned} p^2(p-1)R\leqslant KLN\int _{{\mathbb {R}}^{dp}}\rho _{2\varepsilon }(y)|\mu _p|(dy) =KL N||\mu |^{(2\varepsilon )}|^p_{L_p} \leqslant NKL||\mu |^{(\varepsilon )}|^p_{L_p} \end{aligned}$$

with a constant $N=N(d,p )$, that finishes the proof of (4.16). $\square$

For ${\mathbb {R}}^d$-valued functions $\xi$ on ${\mathbb {R}}^d$ we define the linear operators $I^{\xi }$, $J^{\xi }$ and $T^\xi$ by

$$\begin{aligned}{} & {} T^\xi \varphi (x) = \varphi (x+\xi (x)),\quad I^{\xi }\varphi (x):=T^\xi \varphi (x)-\varphi (x), \nonumber \\{} & {} J^{\xi }\psi (x):=I^{\xi }\psi (x)-\xi ^i(x)D_i\psi (x),\quad x\in {\mathbb {R}}^d \end{aligned}$$

(4.22)

acting on functions $\varphi$ and differentiable functions $\psi$ on ${\mathbb {R}}^d$. If $\xi$ depends also on some parameters, then $I^{\xi }\phi$ and $J^{\xi }\psi$ are defined for each fixed parameter as above.

Lemma 4.3

Let $\xi$ be an ${\mathbb {R}}^d$-valued function of $x\in {\mathbb {R}}^d$ such that for some constants $\lambda >0$, $K_0$ and L

$$\begin{aligned} |\xi (x)-\xi (y)|\leqslant L|x-y| \quad \text {for all }x,y\in {\mathbb {R}}^d \end{aligned}$$

and

$$\begin{aligned} \lambda |x-y|\leqslant |x-y+\theta (\xi (x)-\xi (y))| \quad \text {for all }x,y\in {\mathbb {R}}^d\text { and }\theta \in [0,1].\nonumber \\ \end{aligned}$$

(4.23)

Let $\mu \in {\mathfrak {M}}$ such that it satisfies (4.5), let $p\geqslant 2$ be an integer, and for $\varepsilon >0$ set

$$\begin{aligned} C{} & {} {:=} \int _{{\mathbb {R}}^d}p(\mu ^{(\varepsilon )})^{p-1}(J^{\xi *}\mu )^{(\varepsilon )} +(\mu ^{(\varepsilon )}+(I^{\xi *}\mu )^{(\varepsilon )})^p-(\mu ^{(\varepsilon )})^p\\{} & {} \quad -p(\mu ^{(\varepsilon )})^{p-1}(I^{\xi *}\mu )^{(\varepsilon )}\,dx, \end{aligned}$$

where, to ease notation, the argument $x\in {\mathbb {R}}^d$ is suppressed in the integrand. Then

$$\begin{aligned} |C|\leqslant N(1+L^2)L^2||\mu |^{(\varepsilon )}|^p_{L_p} \quad \text {for all }\varepsilon >0, \end{aligned}$$

(4.24)

with a constant $N=N(d,p,\lambda )$.

Remark 4.1

Notice that in the special case $p=2$ the estimate (4.24) can be rewritten as

$$\begin{aligned} 2(\mu ^{(\varepsilon )},(J^{\xi *}\mu )^{(\varepsilon )}) +((I^{\xi *}\mu )^{(\varepsilon )},(I^{\xi *}\mu )^{(\varepsilon )}) \leqslant N(1+L^2)L^2||\mu |^{(\varepsilon )}|^2_{L_2} \quad \text {for all }\varepsilon >0. \end{aligned}$$

Proof of Lemma 4.3

Again we note that by (4.9), together with the conditions on $\xi$ and that by Taylor’s formula

$$\begin{aligned} I^\xi k_{\varepsilon }(x)= & {} \int _0^1 (D_ik_{\varepsilon })(x-r-\theta \xi (r))\,d\theta \,\xi ^i(r), \\ J^\xi k_{\varepsilon }(x)= & {} \int _0^1(1-\theta )(D_{ij}k_{\varepsilon })(x-r-\theta \xi (r))\,d\theta \,\xi ^i(r)\xi ^j(r), \end{aligned}$$

as well as that $\sup _{x\in {\mathbb {R}}^d}\sum _{k=0}^2|D^k\rho _\varepsilon (x)|<\infty$, it is easy to verify that C is well-defined. Notice that

$$\begin{aligned} \mu ^{(\varepsilon )}+(I^{\xi *}\mu )^{(\varepsilon )}=(T^{\xi *}\mu )^{(\varepsilon )} \end{aligned}$$

and

$$\begin{aligned} p(\mu ^{(\varepsilon )})^{p-1}(J^{\xi *}\mu )^{(\varepsilon )} -p(\mu ^{(\varepsilon )})^{p-1}(I^{\xi *}\mu )^{(\varepsilon )} =-p(\mu ^{(\varepsilon )})^{p-1}((\xi ^iD_i)^*\mu )^{(\varepsilon )}. \end{aligned}$$

Hence

$$\begin{aligned} C=\int _{{\mathbb {R}}^d}((T^{\xi *}\mu )^{(\varepsilon )})^p-(\mu ^{(\varepsilon )})^p -p(\mu ^{(\varepsilon )})^{p-1}((\xi ^iD_i)^*\mu )^{(\varepsilon )}\,dx. \end{aligned}$$

Rewriting here the product of integrals as multiple integrals and using the product measure $\mu _p(dy):=\mu (dy_1)\ldots \mu (dy_p)$ by Fubini’s theorem we get

$$\begin{aligned} ((T^{\xi *}\mu )^{(\varepsilon )})^p(x)&=\int _{{\mathbb {R}}^{pd}} \Pi _{r=1}^pT^{\xi }_{y_r}\Pi _{r=1}^pk_{\varepsilon }(x-y_r)\,\mu _p(dy), \nonumber \\ (\mu ^{(\varepsilon )})^p&=\int _{{\mathbb {R}}^{pd}}\Pi _{r=1}^pk_{\varepsilon }(x-y_r)\,\mu _p(dy),\nonumber \\ p(\mu ^{(\varepsilon )})^{p-1}((\xi ^iD_i)^*\mu )^{(\varepsilon )}&=p\int _{{\mathbb {R}}^{pd}}\Pi _{r=1}^{p-1}k_{\varepsilon }(x-y_r) \xi ^i(y_p)\partial _{y^i_p}k_{\varepsilon }(x-y_p) \mu _p(dy), \nonumber \\&=p\int _{{\mathbb {R}}^{pd}}\xi ^i(y_p)\partial _{y^i_p} \Pi _{r=1}^{p}k_{\varepsilon }(x-y_r)\mu _p(dy) \nonumber \\&=\int _{{\mathbb {R}}^{pd}}\sum _{r=1}^p\xi ^i(y_r)\partial _{y^i_r} \Pi _{r=1}^{p}k_{\varepsilon }(x-y_r)\,\mu _p(dy) \end{aligned}$$

(4.25)

where the last equation is due to the symmetry of the function $\Pi _{r}^pk_{\varepsilon }(x-y_r)$ and the measure $\mu _p(dy)$ in $y=(y_1,\ldots ,y_p)\in {\mathbb {R}}^{pd}$. Thus

$$\begin{aligned} C=\int _{{\mathbb {R}}^d}\int _{{\mathbb {R}}^{pd}}L_y\Pi _{r=1}^pk_{\varepsilon }(x-y_r)\,\mu _p(dy)\,dx \end{aligned}$$

with the operator

$$\begin{aligned} L^\xi _y=\Pi _{r=1}^pT^{\xi }_{y_r}-{\mathbb {I}}-\sum _{r=1}^p\xi ^i(y_r)\partial _{y^i_r}. \end{aligned}$$

Using here Fubini’s theorem then changing the order of the operator $L_y^{\xi }$ and the integration against dx we have

$$\begin{aligned} C= & {} \int _{{\mathbb {R}}^{pd}}\int _{{\mathbb {R}}^d}L_y^\xi \Pi _{r=1}^pk_{\varepsilon }(x-y_r)\,dx\,\mu _p(dy) \nonumber \\= & {} \int _{{\mathbb {R}}^{pd}}L_y^\xi \int _{{\mathbb {R}}^d}\Pi _{r=1}^pk_{\varepsilon }(x-y_r)\,dx\,\mu _p(dy) =\int _{{\mathbb {R}}^{pd}}L^{\xi }_y\rho _{\varepsilon }(y)\,\mu _p(dy), \nonumber \\ \end{aligned}$$

(4.26)

where, see (4.2),

$$\begin{aligned} \rho _{\varepsilon }(y)= c_{p,\varepsilon }e^{-\sum _{1\leqslant r<s\leqslant p}|y_r-y_s|^2/(2\varepsilon p)} \end{aligned}$$

(4.27)

with $c_{p,\varepsilon }=p^{-d/2}(2\pi \varepsilon )^{(1-p)d/2}$.

Introduce for $\varepsilon >0$ the function

$$\begin{aligned} \psi _{\varepsilon }(z)=c_{p,\varepsilon }e^{-\sum _{1\leqslant r<s\leqslant p}z^2_{rs}/(2p\varepsilon )}, \quad z=(z_{rs})_{1\leqslant r<s\leqslant p}\in {\mathbb {R}}^{p(p-1)d/2}. \end{aligned}$$

Then clearly, $\rho _{\varepsilon }(y)=\psi _{\varepsilon }({{\tilde{y}}})$ with

$$\begin{aligned}{} & {} {{\tilde{y}}}:=(y_{rs})_{1\leqslant r<s\leqslant p}:=(y_r-y_s)_{1\leqslant r<s\leqslant p}\in {\mathbb {R}}^{p(p-1)d/2}, \\{} & {} \Pi _{r=1}^pT^{\xi }_{y_r}\rho _{\varepsilon }(y)=\psi _{\varepsilon }({\tilde{y}}+{\tilde{\xi }}(y)) \quad \text {with }\tilde{\xi }(y)=({\tilde{\xi }}_{rs}(y))_{1\leqslant r<s\leqslant p}, \,\,{\tilde{\xi }}_{rs}(y)=\xi (y_r)-\xi (y_s), \end{aligned}$$

and by the chain rule

$$\begin{aligned}{} & {} \sum _{r=1}^p\xi ^i(y_r)\partial _{y^i_r}\rho _{\varepsilon } =\sum _{r=1}^p\xi ^i(y_r)\sum _{1\leqslant k<l\leqslant p}(\delta _{kr}-\delta _{lr}) (\partial _{z_{kl}^i}\psi _{\varepsilon })({{\tilde{y}}}) \\{} & {} \quad =\sum _{1\leqslant k<l\leqslant p}\sum _{r=1}^p (\delta _{kr}-\delta _{lr})\xi ^i(y_r) (\partial _{z_{kl}^i}\psi _{\varepsilon })({{\tilde{y}}}) = \sum _{1\leqslant k<l\leqslant p} (\xi ^i(y_k)-\xi ^i(y_l)) (\partial _{z_{kl}^i}\psi _{\varepsilon })({{\tilde{y}}}). \end{aligned}$$

Consequently,

$$\begin{aligned} L^{\xi }_y\rho _{\varepsilon }(y)=\psi _{\varepsilon }({{\tilde{y}}}+\tilde{\xi }(y))-\psi _{\varepsilon }({{\tilde{y}}}) -\sum _{1\leqslant k<l\leqslant p} {\tilde{\xi }}^i_{kl} (y)(\partial _{z_{kl}^i}\psi _{\varepsilon })({\tilde{y}}), \end{aligned}$$

which by Taylor’s formula gives

$$\begin{aligned} L^{\xi }_y\rho _{\varepsilon }(y) =\int _0^1(1-\theta )\sum _{1\leqslant k<l\leqslant p}\sum _{1\leqslant r<s\leqslant p} (\partial _{z_{kl}^i}\partial _{z_{rs}^j}\psi _{\varepsilon })({\tilde{y}}+\theta {\tilde{\xi }}(y)) {\tilde{\xi }}^i_{kl}(y){\tilde{\xi }}^j_{rs}(y)\,d\theta , \end{aligned}$$

where the summation convention is used with respect to the repeated indices $i,j=1,2,\ldots ,d$. Note that

$$\begin{aligned} (\partial _{z_{kl}^i}\partial _{z_{rs}^j}\psi _{\varepsilon })({\tilde{y}}+\theta {\tilde{\xi }}(y)) =\psi _{\varepsilon }({{\tilde{y}}}+\theta {\tilde{\xi }}(y))l^{ij,rs,kl}_{\varepsilon }({{\tilde{y}}}+\theta {\tilde{\xi }}(y)) \end{aligned}$$

with

$$\begin{aligned} l^{ij,rs,kl}_{\varepsilon }(z):=\tfrac{1}{(p\varepsilon )^2}z^i_{kl}z^j_{rs}-\tfrac{1}{p\varepsilon }\delta _{rk}\delta _{sl}\delta _{ij} \quad \text {for }z=(z_{kl})_{1\leqslant k<l\leqslant p}. \end{aligned}$$

Due to the condition (4.23) there is a constant $\kappa =\kappa (d,\lambda )>1$ such that for $\theta \in [0,1]$

$$\begin{aligned} \kappa ^{-1}|x_1-x_2|^2\leqslant |x_1-x_2+\theta (\xi (x_1)-\xi (x_2))|^2 \quad \text {for }x_1,x_2\in {\mathbb {R}}^d, \end{aligned}$$

(4.28)

which implies

$$\begin{aligned} \psi _{\varepsilon }({{\tilde{y}}}+\theta {\tilde{\xi }}(y)) \leqslant N\psi _{\varepsilon }({{\tilde{y}}}/\kappa ) =N\rho _{\kappa \varepsilon }(y), \quad y\in {\mathbb {R}}^{dp}, \end{aligned}$$

and together with the Lipschitz condition on $\xi$,

$$\begin{aligned} |l^{ij,rs,kl}_{\varepsilon }({{\tilde{y}}}+\theta {\tilde{\xi }}(y))| \leqslant \tfrac{N}{\varepsilon ^2}(1+L^2) \sum _{1\leqslant r<s\leqslant p}|y_r-y_s|^2+\tfrac{N}{\varepsilon } \end{aligned}$$

for all $1\leqslant r<s\leqslant p$, $1\leqslant k<l\leqslant p$ and $i,j=1,2,\ldots ,d$ with a constant $N=N(d,p,\lambda )$. Moreover,

$$\begin{aligned} |{\tilde{\xi }}^i_{kl}(y){\tilde{\xi }}^j_{rs}(y)|\leqslant NL^2\sum _{1\leqslant r<s\leqslant p}|y_s-y_p|^2 \quad \text {for }y=(y_r)_{r=1}^p\in {\mathbb {R}}^{pd} \end{aligned}$$

with a constant $N=N(d,p)$. Consequently, taking into account (4.13) we have

$$\begin{aligned} |L^{\xi }_y\rho _{\varepsilon }(y)|&\leqslant \tfrac{N}{\varepsilon ^2}L^2(1+L^2) \big (\sum _{1\leqslant r<s\leqslant p}|y_r-y_s|^2\big )^2 \rho _{\kappa \varepsilon }(y)+ \tfrac{N}{\varepsilon }L^2\\&\sum _{1\leqslant r<s\leqslant p}|y_r-y_s|^2\rho _{\kappa \varepsilon }(y) \leqslant N'L^2(1+L^2)\rho _{2\kappa \varepsilon }(y) \quad \text {for }y\in {\mathbb {R}}^{dp} \end{aligned}$$

with constants $N=N(d,p,\lambda )$ and $N'=N'(d,p,\lambda )$. Using this we finish the proof by noting that (4.26) implies

$$\begin{aligned} |C|{} & {} \leqslant N'L^2(1+L^2)\int _{{\mathbb {R}}^{pd}}\rho _{2\kappa \varepsilon }(y)\,|\mu _p|(dy) =N'L^2(1+L^2)||\mu |^{(2\kappa \varepsilon )}|^p_{L_p} \\{} & {} \leqslant N{L^2}(1+L^2)||\mu |^{(\varepsilon )}|^p_{L_p}. \end{aligned}$$

$\square$

Corollary 4.4

Let the conditions of Lemma 4.3 hold. Then for every even integer $p\geqslant 2$ there is a constant $N=N(d,p,\lambda )$ such that for $\varepsilon >0$ and $\mu \in {\mathfrak {M}}$ we have

$$\begin{aligned} \int _{{\mathbb {R}}^d}(\mu ^{(\varepsilon )})^{p-1}(x) (J^{\xi *}\mu )^{(\varepsilon )}(x)\,dx \leqslant NL^2(1+L^2)||\mu |^{(\varepsilon )}|^p_{L_p}. \end{aligned}$$

(4.29)

Proof

Notice that $|a+b|^p-|b|^p-p|a|^{p-2}ab\geqslant 0$ for $p\geqslant 2$ for any $a,b\in {\mathbb {R}}$ by the convexity of the function $f(a)=|a|^p$, $a\in {\mathbb {R}}$. Using this with $a=\mu ^{(\varepsilon )}$ and $b=(I^{\xi }\mu )^{(\varepsilon )}$ we have

$$\begin{aligned} (\mu ^{(\varepsilon )}+(I^{\xi *}\mu )^{(\varepsilon )})^p-(\mu ^{(\varepsilon )})^p -p(\mu ^{(\varepsilon )})^{p-1}(I^{\xi *}\mu )^{(\varepsilon )}\geqslant 0 \quad \text {for }x\in {\mathbb {R}}^d, \end{aligned}$$

which shows that (4.24) implies (4.29), since

$$\begin{aligned} \int _{{\mathbb {R}}^d}|\mu ^{(\varepsilon )}(x)|^{p-1}|(J^{\xi *}\mu )^{(\varepsilon )}(x)|\,dx<\infty . \end{aligned}$$

$\square$

Lemma 4.5

Let the conditions of Lemma 4.3 hold. Let $p\geqslant 2$ be an even integer. Then there is a constant $N=N(d,p,\lambda )$ such that for $\varepsilon >0$ and $\mu \in {\mathfrak {M}}$ we have

$$\begin{aligned} \Big |\int _{{\mathbb {R}}^d} (\mu ^{(\varepsilon )}+(I^{\xi *}\mu )^{(\varepsilon )})^p-(\mu ^{(\varepsilon )})^p \,dx\Big | \leqslant N(1+L)L||\mu |^{(\varepsilon )}|^p_{L_p}, \end{aligned}$$

(4.30)

where the argument $x\in {\mathbb {R}}^d$ is suppressed in the integrand.

Proof

By the same arguments as in the proof of Lemma 4.3 we see that the left-hand side of (4.30) is well-defined. Clearly,

$$\begin{aligned} D:=\int _{{\mathbb {R}}^d}((T^{\xi *}\mu )^{(\varepsilon )})^p-(\mu ^{(\varepsilon )})^p\,dx =\int _{{\mathbb {R}}^d}\int _{{\mathbb {R}}^{pd}}M^{\xi }_y\Pi _{r=1}^pk_{\varepsilon }(x-y_r)\,\mu _p(dy)\,dx \end{aligned}$$

with the operator

$$\begin{aligned} M^\xi _y=\Pi _{r=1}^pT^{\xi }_{y_r}-{\mathbb {I}}, \end{aligned}$$

where $\mu _p(dy)=\Pi _{r=1}^p\mu (dy_r)$, $y=(y_1,\ldots ,y_p)\in {\mathbb {R}}^{pd}$. Hence by Fubini’s theorem, then changing the order of the operator $M_y^{\xi }$ and the integration against dx and by taking into account (4.2) we have

$$\begin{aligned} D{} & {} =\int _{{\mathbb {R}}^{pd}}\int _{{\mathbb {R}}^d}M_y^\xi \Pi _{r=1}^pk_{\varepsilon }(x-y_r)\,dx\,\mu _p(dy)\nonumber \\{} & {} =\int _{{\mathbb {R}}^{pd}}M_y^\xi \int _{{\mathbb {R}}^d}\Pi _{r=1}^pk_{\varepsilon }(x-y_r)\,dx\,\mu _p(dy)\nonumber \\{} & {} =\int _{{\mathbb {R}}^{pd}}M^{\xi }_y\rho _{\varepsilon }(y)\,\mu _p(dy). \end{aligned}$$

(4.31)

As in the proof of Lemma 4.3 we introduce for $\varepsilon >0$ the function

$$\begin{aligned} \psi _{\varepsilon }(z)=c_{p,\varepsilon }e^{-\sum _{1\leqslant r<s\leqslant p}z^2_{rs}/(2p\varepsilon )},\quad z=(z_{rs})_{1\leqslant r<s\leqslant p}\in {\mathbb {R}}^{p(p-1)d/2}, \end{aligned}$$

such that $\rho _{\varepsilon }(y)=\psi _{\varepsilon }({{\tilde{y}}})$ with ${{\tilde{y}}}:=(y_{rs})_{1\leqslant r<s\leqslant p}:=(y_r-y_s)_{1\leqslant r<s\leqslant p}\in {\mathbb {R}}^{p(p-1)d/2},$ and

$$\begin{aligned} \Pi _{r=1}^pT^{\xi }_{y_r}\rho _{\varepsilon }(y) =\psi _{\varepsilon }({{\tilde{y}}}+{\tilde{\xi }}(y)) \quad \text {with }\tilde{\xi }(y)=({\tilde{\xi }}_{rs}(y))_{1\leqslant r<s\leqslant p}, \,\, {\tilde{\xi }}_{rs}(y)=\xi (y_r)-\xi (y_s). \end{aligned}$$

By Taylor’s formula

$$\begin{aligned} M^{\xi }_y\rho _{\varepsilon }(y) =\psi _{\varepsilon }({{\tilde{y}}}+\tilde{\xi }(y))-\psi _{\varepsilon }({{\tilde{y}}}) =\int _0^1\sum _{1\leqslant k<l\leqslant p} (\partial _{z_{kl}^i}\psi _{\varepsilon })({{\tilde{y}}}+\theta {\tilde{\xi }}(y)) {\tilde{\xi }}^i_{kl}(y)\,d\theta , \end{aligned}$$

where the summation convention is used with respect to the repeated indices $i=1,2,\ldots ,d$. Note that

$$\begin{aligned} (\partial _{z_{kl}^i}\psi _{\varepsilon })({{\tilde{y}}}+\theta {\tilde{\xi }}(y)) =\psi _{\varepsilon } ({{\tilde{y}}}+\theta {\tilde{\xi }}(y))l^{kl,i}_{\varepsilon }({{\tilde{y}}}+\theta {\tilde{\xi }}(y)) \end{aligned}$$

with

$$\begin{aligned} l^{kl,i}_{\varepsilon }(z):=\tfrac{1}{p\varepsilon }z^i_{kl} \quad \text {for }z=(z_{kl})_{1\leqslant k<l\leqslant p}\in {\mathbb {R}}^{p(p-1)d/2}. \end{aligned}$$

By (4.28) we have

$$\begin{aligned} \psi _{\varepsilon }({{\tilde{y}}}+\theta {\tilde{\xi }}(y)) \leqslant N\psi _{\varepsilon }({{\tilde{y}}}/\kappa ) =N\rho _{\kappa \varepsilon }(y), \quad y\in {\mathbb {R}}^{dp}, \end{aligned}$$

and due to the Lipschitz condition on $\xi$,

$$\begin{aligned} |l^{kl,i}_{\varepsilon }({{\tilde{y}}}+\theta {\tilde{\xi }}(y))| \leqslant \tfrac{N}{\varepsilon }(1+L)|y_k-y_l| \end{aligned}$$

for all $i=1,2,\ldots ,d$ with constants $\kappa (d,\lambda )>1$ and $N=N(d,p,\lambda )$. Moreover, we get

$$\begin{aligned} |{\tilde{\xi }}^i_{kl}(y)|\leqslant NL|y_k-y_l| \quad \text {for }y=(y_r)_{r=1}^p\in {\mathbb {R}}^{pd} \end{aligned}$$

with a constant $N=N(d,p)$. Consequently, taking into account (4.13) we have

$$\begin{aligned} |M^{\xi }_y\rho _{\varepsilon }(y)| \leqslant \tfrac{N}{\varepsilon }(1+L)L \sum _{1\leqslant k<l\leqslant p}|y_k-y_l|^2 \rho _{\kappa \varepsilon }(y)\leqslant N' (1+L)L\rho _{2\kappa \varepsilon }(y) \quad \text {for }y\in {\mathbb {R}}^{dp} \end{aligned}$$

with constants N and $N'$ depending only on d, p and $\lambda$. Using this, from (4.31) we obtain

$$\begin{aligned} |D|{} & {} \leqslant N'(1+L)L\int _{{\mathbb {R}}^{pd}}\rho _{2\kappa \varepsilon }(y)\,|\mu _p|(dy) =N'(1+L)L||\mu |^{(2\kappa \varepsilon )}|^p_{L_p} \\{} & {} \leqslant N'(1+L)L||\mu |^{(\varepsilon )}|^p_{L_p}, \end{aligned}$$

which completes the proof of the lemma. $\square$

5 The smoothed measures

We use the notations introduced in Sect. 4. Recall that an ${\mathbb {M}}$-valued stochastic process $(\nu _t)_{t\in [0,T]}$ is said to be weakly cadlag if almost surely $\nu _t(\varphi )$ is a cadlag function of t for every $\varphi \in C_b({\mathbb {R}}^d)$. For such a process $(\nu _t)_{t\in [0,T]}$ there is a set $\Omega '\subset \Omega$ of full probability and there is uniquely defined (up to indistinguishability) ${\mathbb {M}}$-valued processes $(\nu _{t-})_{t\geqslant 0}$ such that for every $\omega \in \Omega '$

$$\begin{aligned} \nu _{t-}(\varphi )=\lim _{s\uparrow t}\nu _{s}(\varphi ) \quad \text {for all }\varphi \in C_b({\mathbb {R}}^d)\text { and }t\in [0,T], \end{aligned}$$

and for each $\omega \in \Omega '$ we have $\nu _{t-}=\nu _t$, for all but at most countably many $t\in (0,T]$. We say that an ${\mathfrak {M}}$-valued process $(\nu _t)_{t\in [0,T]}$ is weakly cadlag if it is the difference of weakly cadlag ${\mathbb {M}}$-valued processes. An ${\mathfrak {M}}$-valued process $(\nu _t)_{t\geqslant 0}$ is said to be adapted to a filtration $({\mathcal {G}}_t)_{t\in [0,T]}$ if $\nu _t(\varphi )$ is a ${\mathcal {G}}_t$-measurable random variable for every $t\in [0,T]$ and bounded Borel function $\varphi$ on ${\mathbb {R}}^d$.

We present first a version of an Itô formula, Theorem 2.1 from [6], for $L_p$-valued processes. To formulate it, let $\psi =(\psi (x))$, $f=(f_t(x))$, $g=(g_t^{j}(x))$ and $h=(h_t(x,{\mathfrak {z}}))$ be functions with values in ${\mathbb {R}}$, ${\mathbb {R}}$, ${\mathbb {R}}^{m}$ and ${\mathbb {R}}$, respectively, defined on $\Omega \times {\mathbb {R}}^d$, $\Omega \times H_T$, $\Omega \times H_T$ and $\Omega \times H_T\times {\mathfrak {Z}}$, respectively, where $H_T:=[0,T]\times {\mathbb {R}}^d$ and $({\mathfrak {Z}}, {\mathcal {Z}},\nu )$ is a measure space with a $\sigma$-finite measure $\nu$ and countably generated $\sigma$-algebra ${\mathcal {Z}}$. Assume that $\psi$ is ${\mathcal {F}}_0\otimes {\mathcal {B}}({\mathbb {R}}^d)$-measurable, f and g are ${\mathcal {O}}\otimes {\mathcal {B}}({\mathbb {R}}^d)$-measurable and h is ${\mathcal {P}}\otimes {\mathcal {B}}({\mathbb {R}}^d)\otimes {\mathcal {Z}}$-measurable, such that almost surely

$$\begin{aligned} \int _0^T|f_t(x)|\,dt<\infty ,\quad \int _0^T\sum _j|g^j_t(x)|^2\,dt<\infty , \quad \int _0^T\int _{{\mathfrak {Z}}}|h_t(x,{\mathfrak {z}})|^2\,\nu (d{\mathfrak {z}})\,dt<\infty \nonumber \\ \end{aligned}$$

(5.1)

for each $x\in {\mathbb {R}}^d$, and for each bounded Borel set $\Gamma \subset {\mathbb {R}}^d$ almost surely

$$\begin{aligned} \int _{\Gamma }\int _0^T|f_t(x)|\,dt\,dx<\infty ,\quad \int _{\Gamma }\Big (\int _0^T\sum _j|g^j_t(x)|^2\,dt\Big )^{1/2}\,dx<\infty , \end{aligned}$$

$$\begin{aligned} \int _{\Gamma } \Big (\int _0^T\int _{{\mathfrak {Z}}}\sum _j|h_t(x,{\mathfrak {z}})|^2\,\nu (d{\mathfrak {z}})\,dt\Big )^{1/2}\,dx<\infty . \end{aligned}$$

(5.2)

Assume, moreover, that for a number $p\in [2,\infty )$ almost surely

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^d}|\psi (x)|^p\,dx<\infty , \quad \int _0^T\int _{{\mathbb {R}}^d}|f_t(x)|^p\,dx\,dt\\{} & {} \quad<\infty , \quad \int _0^T\int _{{\mathbb {R}}^d}\Big (\sum _{j}|g^j_t(x)|^2\Big )^{p/2}\,dx\,dt<\infty , \end{aligned}$$

$$\begin{aligned}{} & {} \int _0^T\int _{{\mathbb {R}}^d}\int _{{\mathfrak {Z}}}|h_t(x,{\mathfrak {z}})|^p\,\nu (d{\mathfrak {z}})\,dx\,dt<\infty ,\nonumber \\{} & {} \quad \int _0^T\int _{{\mathbb {R}}^d}\Big (\int _{{\mathfrak {Z}}}|h_t(x,{\mathfrak {z}})|^2\,\nu (d{\mathfrak {z}})\Big )^{p/2}\,dx\,dt <\infty . \end{aligned}$$

(5.3)

Theorem 5.1

Let conditions (5.1), (5.2) and (5.3) with a number $p\geqslant 2$ hold. Assume there is an ${\mathcal {O}}\otimes {\mathcal {B}}({\mathbb {R}}^d)$-measurable real-valued function v on $\Omega \times H_T$ such that almost surely

$$\begin{aligned} \int _{{\mathbb {R}}^d}|v_t(x)|^p\,dx<\infty \quad \text {for all }t\in [0,T] \end{aligned}$$

(5.4)

and for every $x\in {\mathbb {R}}^d$ almost surely

$$\begin{aligned} v_t(x)=\psi (x)+\int _0^tf_s(x)\,ds+\int _0^tg^j_s(x)\,dw^j_s +\int _0^t\int _{{\mathfrak {Z}}}h_s(x,{\mathfrak {z}})\,{\tilde{\pi }}(d{\mathfrak {z}},ds)\nonumber \\ \end{aligned}$$

(5.5)

holds for all $t\in [0,T]$, where $(w_t)_{t\geqslant 0}$ is an m-dimensional ${\mathcal {F}}_t$-Wiener process, $\pi (d{\mathfrak {z}},ds)$ is an ${\mathcal {F}}_t$-Poisson measure with characteristic measure $\nu$, and $\tilde{\pi }(d{\mathfrak {z}},ds)=\pi (d{\mathfrak {z}},ds)-\nu (d{\mathfrak {z}})ds$ is the compensated martingale measure. Then almost surely $(v_t)_{t\in [0,T]}$ is an $L_p$-valued ${\mathcal {F}}_t$-adapted cadlag process and almost surely

$$\begin{aligned} |v_t|^p_{L_p}&= |\psi |_{L_p}^p +p\int _0^t\int _{{\mathbb {R}}^d}|v_s|^{p-2} v_sg^{j}_s\,dx\, dw^j_s \nonumber \\&\quad +\tfrac{p}{2}\int _0^t\int _{{\mathbb {R}}^d}\big ( 2|v_s|^{p-2}v_sf_s +(p-1)|v_s|^{p-2}\sum _{j}|g^j_s|^2\big )\,dx\,ds \nonumber \\&\quad +p\int _0^t\int _{{\mathfrak {Z}}}\int _{{\mathbb {R}}^d}|v_{s-}|^{p-2}v_{s-}h_s \,dx\,\tilde{\pi }(d{\mathfrak {z}},ds) \nonumber \\&\quad +\int _0^t\int _{{\mathfrak {Z}}}\int _{{\mathbb {R}}^d} (|v_{s-}+h_s|^p-|v_{s-}|^p-p|v_{s-}|^{p-2}v_{s-}h_s) \,dx\,\pi (d{\mathfrak {z}},ds) \end{aligned}$$

(5.6)

for all $t\in [0,T]$, where $v_{s-}$ means the left-hand limit in $L_p$ at s of v.

Proof

By a truncation and stopping time argument it is not difficult to see that without loss of generality we may assume that the random variables in (5.3) have finite expectation. From (5.5) we get that for each $\varphi \in C^{\infty }_0$ almost surely

$$\begin{aligned} (v_t,\varphi )=(\psi ,\varphi )+\int _0^t(f_s,\varphi )\,ds+\int _0^t(g^j_s,\varphi )\,dw^j_s +\int _0^t\int _{{\mathfrak {Z}}}(h_s({\mathfrak {z}}),\varphi )\,{\tilde{\pi }}(d{\mathfrak {z}},ds)\nonumber \\ \end{aligned}$$

(5.7)

holds for all $t\in [0,T]$. This we can see if we multiply both sides of equation (5.5) with $\varphi$ and then, making use of our measurability conditions and the conditions (5.1) and (5.2), we integrate over ${\mathbb {R}}^d$ with respect to dx and use deterministic and stochastic Fubini theorems from [12] and [6] to change the order of integrations. Due to (5.7), the measurability conditions on $\psi$, f, g, h and v and to their integrability conditions, (5.3) and (5.4), by virtue of Theorem 2.1 from [6] there is an $L_p$-valued ${\mathcal {F}}_t$-adapted cadlag process $({\bar{v}}_t)_{t\in [0,T]}$ such that for each $\varphi \in C^{\infty }_0$ almost surely (5.7) holds with ${{\bar{v}}}$ in place of v, and almost surely (5.6) holds with ${{\bar{v}}}$ in place of v. Thus for each $\varphi \in C_0^{\infty }$ almost surely $(v_t,\varphi )=({{\bar{v}}}_t,\varphi )$ for all $t\in [0,T]$, which implies that almost surely $v={{\bar{v}}}$ as $L_p$-valued processes, and finishes the proof of the theorem. $\square$

Lemma 5.2

Let Assumption 2.1 hold. Assume $(\mu _t)_{t\in [0,T]}$ is an ${\mathfrak {M}}$-solution to Eq. (3.8) If $K_1\ne 0$ in Assumption 2.1 (ii) then assume also

$$\begin{aligned} \mathrm{ess\,sup}_{t\in [0,T]}\int _{{\mathbb {R}}^d}|y|^2\,|\mu _t|(dy)<\infty \,\,(\mathrm {a.s.)}. \end{aligned}$$

(5.8)

Then for each $x\in {\mathbb {R}}^d$ and $\varepsilon >0$,

$$\begin{aligned} \mu ^{(\varepsilon )}_t(x)&= \mu ^{(\varepsilon )}_0(x) + \int _0^t(\tilde{{\mathcal {L}}}^*_s\mu _s)^{(\varepsilon )}(x)\,ds +\int _0^t({\mathcal {M}}^{j*}_s\mu _s)^{(\varepsilon )}(x)\,dV^j_s \nonumber \\&\quad + \int _0^t\int _{{\mathfrak {Z}}_0}(J^{\eta *}_s\mu _s)^{(\varepsilon )}(x)\,\nu _0(d{\mathfrak {z}})\,ds + \int _0^t\int _{{\mathfrak {Z}}_1}(J^{\xi *}_s\mu _s)^{(\varepsilon )}(x)\,\nu _1(d{\mathfrak {z}})\,ds\nonumber \\&\quad + \int _0^t\int _{{\mathfrak {Z}}_1}(I^{\xi *}_s\mu _{s-})^{(\varepsilon )}(x)\,{\tilde{N}}_{1}(d{\mathfrak {z}},ds) \end{aligned}$$

(5.9)

holds almost surely for all $t\in [0,T]$. Moreover, for each $\varepsilon >0$ and $p\geqslant 2$

$$\begin{aligned} |\mu ^{(\varepsilon )}_t|_{L_p}^p&= |\mu ^{(\varepsilon )}_0|_{L_p}^p + p\int _0^t \big (|\mu ^{(\varepsilon )}_s|^{p-2}\mu ^{(\varepsilon )}_s,({\mathcal {M}}_s^{k*}\mu _s)^{(\varepsilon )}\big )\,dV^k_s \nonumber \\&\quad + p\int _0^t \big (|\mu ^{(\varepsilon )}_s|^{p-2}\mu ^{(\varepsilon )}_s,(\tilde{{\mathcal {L}}}^*_s\mu _s)^{(\varepsilon )}\big )\,ds \nonumber \\&\quad + \tfrac{p(p-1)}{2}\sum _k \int _0^t\big (|\mu _s^{(\varepsilon )}|^{p-2}, |({\mathcal {M}}_s^{k*}\mu _s)^{(\varepsilon )}|^2 \big )\,ds + p\int _0^t\int _{{\mathfrak {Z}}_0} \nonumber \\&\quad \big (|\mu ^{(\varepsilon )}_s|^{p-2}\mu ^{(\varepsilon )}_s,(J^{\eta *}_s\mu _s)^{(\varepsilon )}\big )\,\nu _0(d{\mathfrak {z}})ds \nonumber \\&\quad + p\int _0^t\int _{{\mathfrak {Z}}_1} \big (|\mu ^{(\varepsilon )}_s|^{p-2}\mu ^{(\varepsilon )}_s,(J^{\xi *}_s\mu _s)^{(\varepsilon )}\big )\,\nu _1(d{\mathfrak {z}})ds +p\int _0^t\int _{{\mathfrak {Z}}_1} \nonumber \\&\quad \big (|\mu ^{(\varepsilon )}_{s-}|^{p-2}\mu ^{(\varepsilon )}_{s-},(I^{\xi *}_s\mu _{s-})^{(\varepsilon )}\big )\,{{\tilde{N}}}_1(d{\mathfrak {z}},ds)\nonumber \\&\quad +\int _0^t\int _{{\mathfrak {Z}}_1}\int _{{\mathbb {R}}^d} \Big \{ \big |\mu _{s-}^{(\varepsilon )}+ (I_s^{\xi *}\mu _{s-})^{(\varepsilon )}\nonumber \\&\quad \big |^p - |\mu _{s-}^{(\varepsilon )}|^p - p|\mu _{s-}^{(\varepsilon )}|^{p-2}\mu ^{(\varepsilon )}_{s-}(I_s^{\xi *}\mu _{s-})^{(\varepsilon )}\Big \}\,dxN_1(d{\mathfrak {z}},ds) \end{aligned}$$

(5.10)

holds almost surely for all $t\in [0,T]$.

Proof

Let $\psi \in C^{\infty }_0({\mathbb {R}}^d)$ such that $\psi (0)=1$, and for integers $r\geqslant 1$ define $\psi _r$ by dilation, $\psi _r(x)=\psi (x/r)$, $x\in {\mathbb {R}}^d$. Then substituting $k_{\varepsilon }(x-\cdot )\psi _r(\cdot )\in C^{\infty }_0$ in place of $\varphi$ in (3.5), for each $x\in {\mathbb {R}}^d$ we get

$$\begin{aligned} \mu _t(k_{\varepsilon }(x-\cdot )\psi _r)= & {} \mu _0(k_{\varepsilon }(x-\cdot )\psi _r) + \int _0^t\mu _{s}(\tilde{{\mathcal {L}}_s}(k_{\varepsilon }(x-\cdot )\psi _r))\,ds \nonumber \\{} & {} \quad + \int _0^t \mu _{s}({\mathcal {M}}^k_s(k_{\varepsilon }(x-\cdot )\psi _r))\,dV^k_s \nonumber \\{} & {} \quad + \int _0^t\int _{{\mathfrak {Z}}_0}\mu _{s}(J_s^{\eta }(k_{\varepsilon }(x-\cdot )\psi _r))\,\nu _0(d{\mathfrak {z}})ds \nonumber \\{} & {} \quad + \int _0^t\int _{{\mathfrak {Z}}_1}\mu _{s}(J_s^{\xi }(k_{\varepsilon }(x-\cdot )\psi _r))\,\nu _1(d{\mathfrak {z}})ds\nonumber \\{} & {} \quad +\int _0^t\int _{{\mathfrak {Z}}_1}\mu _{s-}(I_s^{\xi }(k_{\varepsilon }(x-\cdot )\psi _r))\,{\tilde{N}}_1(d{\mathfrak {z}},ds), \end{aligned}$$

(5.11)

almost surely for all $t\in [0,T]$. Clearly, $\lim _{r\rightarrow \infty }k_{\varepsilon }(x-y)\psi _r(y)=k_{\varepsilon }(x-y)$ and there is a constant N, independent of r, such that

$$\begin{aligned} |k_{\varepsilon }(x-y)\psi _r(y)|\leqslant N\quad \text {for all }x,y\in {\mathbb {R}}^d. \end{aligned}$$

Hence almost surely

$$\begin{aligned} \lim _{r\rightarrow \infty }\mu _t(k_{\varepsilon }(x-\cdot )\psi _r)=\mu _t(k_{\varepsilon }(x-\cdot )) \quad \text {for each }x\in {\mathbb {R}}^d\text { and }t\in [0,T].\nonumber \\ \end{aligned}$$

(5.12)

It is easy to see that for every $\omega \in \Omega$, $x,y\in {\mathbb {R}}^d$, $s\in [0,T]$ and ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$ $(i=0,1)$ we have

$$\begin{aligned} \lim _{r\rightarrow \infty }A_s(k_{\varepsilon }(x-y)\psi _r(y))=A_s(k_{\varepsilon }(x-y)) \end{aligned}$$

(5.13)

with $\tilde{{\mathcal {L}}}$, ${\mathcal {M}}^k$, $J^{\eta }$, $J^{\xi }$ and $I^{\xi }$ in place of A. Clearly,

$$\begin{aligned}{} & {} \sup _{x\in {\mathbb {R}}^d}|\psi _r(x)|=\sup _{x\in {\mathbb {R}}^d}|\psi (x)|<\infty , \quad \sup _{x\in {\mathbb {R}}^d}|D\psi _r(x)|=r^{-1}\sup _{x\in {\mathbb {R}}^d}|D\psi (x)|<\infty ,\\{} & {} \sup _{x\in {\mathbb {R}}^d}|D^2\psi _r(x)|=r^{-2}\sup _{x\in {\mathbb {R}}^d}|D^2\psi (x)|<\infty , \end{aligned}$$

and there is a constant N depending only on d and $\varepsilon$ such that for all $x,y\in {\mathbb {R}}^d$

$$\begin{aligned} |k_{\varepsilon }(x-y)| +|Dk_{\varepsilon }(x-y)| +|D^2k_{\varepsilon }(x-y)|\leqslant N. \end{aligned}$$

(5.14)

Hence, due to Assumption 2.1, we have a constant $N=N(\varepsilon , d,K,K_0,K_1)$ such that

$$\begin{aligned}{} & {} |\tilde{{\mathcal {L}}_s}(k_{\varepsilon }(x-y)\psi _r(y))| \leqslant N(K^2_0+K_1^2|y|^2+K^2_1|Y_s|^2), \end{aligned}$$

(5.15)

$$\begin{aligned}{} & {} \sum _k|{\mathcal {M}}^k_s(k_{\varepsilon }(x-y)\psi _r(y))|^2\leqslant N(K^2_0+K^2_1|y|^2+K^2_1|Y_s|^2) \end{aligned}$$

(5.16)

for $x,y\in {\mathbb {R}}^d$, $s\in [0,T]$, $r\geqslant 1$ and $\omega \in \Omega$. Similarly, applying Taylor’s formula to

$$\begin{aligned} A_s(k_{\varepsilon }(x-y)\psi _r(y))\quad \text {with }J^{\eta }, J^{\xi }\text { and }I^{\xi }\text { in place of }A, \end{aligned}$$

we have a constant $N=N(\varepsilon , d, K_0,K_1)$ such that

$$\begin{aligned} |J_s^{\eta }(k_{\varepsilon }(x-y)\psi _r(y))|&\leqslant \sup _{v\in {\mathbb {R}}^d}|D^2_v(k_{\varepsilon }(x-v)\psi _r(v))||\eta _{s}(y,{\mathfrak {z}}_0)|^2 \leqslant N|\eta _{s}(y,{\mathfrak {z}}_0)|^2, \end{aligned}$$

(5.17)

$$\begin{aligned} |J_s^{\xi }(k_{\varepsilon }(x-y)\psi _r(y))|&\leqslant \sup _{v\in {\mathbb {R}}^d}|D^2_v(k_{\varepsilon }(x-v)\psi _r(v))||\xi _{s}(y,{\mathfrak {z}}_1)|^2 \leqslant N|\xi _{s}(y,{\mathfrak {z}}_1)|^2 \end{aligned}$$

(5.18)

and

$$\begin{aligned} |I^{\xi }_{s}(k_{\varepsilon }(x-y)\psi _r(y))|^2 \leqslant \sup _{v\in {\mathbb {R}}^d}|D_v(k_{\varepsilon }(x-v)\psi _r(v))|^2|\xi _{s}(y,{\mathfrak {z}}_1)|^2 \leqslant N|\xi _{s}(y,{\mathfrak {z}}_1)|^2,\nonumber \\ \end{aligned}$$

(5.19)

respectively, for all $x,y\in {\mathbb {R}}^d$, $s\in [0,T]$, ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$, $i=0,1$ and $\omega \in \Omega$. Using (5.13) (with $A:={\mathcal {L}}$), (5.15) and (5.8), by Lebesgue’s theorem on dominated convergence we get for each $x\in {\mathbb {R}}^d$

$$\begin{aligned} \lim _{r\rightarrow \infty }\int _0^t\mu _{s}(\tilde{{\mathcal {L}}}_s(k_{\varepsilon }(x-\cdot )\psi _r))\,ds =\int _0^t\mu _{s}(\tilde{{\mathcal {L}}}_sk_{\varepsilon }(x-\cdot ))\,ds \quad \text {almost surely,} \end{aligned}$$

uniformly in $t\in [0,T]$. Using Jensen’s inequality, (5.13) (with $A:={\mathcal {M}}$), (5.16) and (5.8), by Lebesgue’s theorem on dominated convergence we obtain

$$\begin{aligned}{} & {} \limsup _{r\rightarrow \infty }\int _0^T \sum _k|\mu _{s}({\mathcal {M}}^k_s(k_{\varepsilon }(x-\cdot )\psi _r))-\mu _{s}({\mathcal {M}}^k_s(k_{\varepsilon }(x-\cdot ))|^2\,ds \\{} & {} \quad \leqslant \mathrm{ess\,sup}_{s\in [0,T]}\Vert \mu _s\Vert \limsup _{r\rightarrow \infty }\int _0^T\int _{{\mathbb {R}}^d} \sum _k|{\mathcal {M}}^k_s(k_{\varepsilon }(x-\cdot )\psi _r))\\{} & {} \qquad -{\mathcal {M}}^k_s(k_{\varepsilon }(x-\cdot ))|^2\,|\mu _s|(dy)\,ds=0 \end{aligned}$$

almost surely, which implies that for $r\rightarrow \infty$

$$\begin{aligned} \int _0^t\mu _{s}({\mathcal {M}}^k_s(k_{\varepsilon }(x-\cdot )\psi _r))\,dV^k_s \rightarrow \int _0^t\mu _{s}({\mathcal {M}}^k_sk_{\varepsilon }(x-\cdot ))\,dV^k_s \end{aligned}$$

in probability, for each $x\in {\mathbb {R}}^d$, uniformly in $t\in [0,T]$. Since by Assumption 2.1(ii) and (5.8)

$$\begin{aligned}{} & {} \int _0^T\int _{{\mathbb {R}}^d}\int _{{\mathfrak {Z}}_0}|\eta _s(y,{\mathfrak {z}}_0)|^2\nu _0(d{\mathfrak {z}}_0)\,|\mu _s|(dy)\,ds \leqslant 2K^2_0|{\bar{\eta }}|_{L_2}^2\int _0^T\Vert \mu _s\Vert \,ds\\{} & {} \quad +2K^2_1|{\bar{\eta }}|_{L_2}^2 \int _0^T\int _{{\mathbb {R}}^d}|y|^2\,|\mu _s|(dy)\,ds +2K^2_1|{\bar{\eta }}|_{L_2}^2\\{} & {} \quad \int _0^T\int _{{\mathbb {R}}^d}|Y_s|^2\,|\mu _s|(dy)\,ds<\infty \quad \text {(a.s.)}, \end{aligned}$$

from (5.13) (with $A:=J^{\eta }$) and (5.17) by Lebesgue’s theorem on dominated convergence we get

$$\begin{aligned} \lim _{r\rightarrow \infty }\int _0^t\int _{{\mathfrak {Z}}_0}\mu _{s}(J_s^{\eta }(k_{\varepsilon }(x-\cdot )\psi _r))\,\nu _0(d{\mathfrak {z}})ds =\int _0^t\int _{{\mathfrak {Z}}_0}\mu _{s}(J_s^{\eta }k_{\varepsilon }(x-\cdot ))\,\nu _0(d{\mathfrak {z}})ds \quad \text {(a.s.)}, \end{aligned}$$

uniformly in $t\in [0,T]$. In the same way we obtain this with $J^{\xi }$, $\nu _1$ and ${\mathfrak {Z}}_1$ in place of $J^{\eta }$, $\nu _0$ and ${\mathfrak {Z}}_0$, respectively. Similarly, using first Jensen’s inequality and Fubini’s theorem we have

$$\begin{aligned}{} & {} \limsup _{r\rightarrow \infty }\int _0^T\int _{{\mathfrak {Z}}_1} |\mu _{s}(I_s^{\xi }(k_{\varepsilon }(x-\cdot )\psi _r))-\mu _{s}(I_s^{\xi }(k_{\varepsilon }(x-\cdot ))|^2\,\nu _1(d{\mathfrak {z}})\,ds \\{} & {} \quad \leqslant \mathrm{ess\,sup}_{s\in [0,T]}\Vert \mu _s\Vert \limsup _{r\rightarrow \infty }\int _0^T\int _{{\mathbb {R}}^d}\int _{{\mathfrak {Z}}_1} |I_s^{\xi }(k_{\varepsilon }(x-y)\psi _r(y)))\\{} & {} \qquad -I_s^{\xi }(k_{\varepsilon }(x-y))|^2\,\nu _1(d{\mathfrak {z}})\,|\mu _s|(dy)\,ds=0\,\,\text {(a.s.)}, \end{aligned}$$

which implies that for $r\rightarrow \infty$ for each $x\in {\mathbb {R}}^d$ we have

$$\begin{aligned} \int _0^t\int _{{\mathfrak {Z}}_1} \mu _{s}(I_{s-}^{\xi }(k_{\varepsilon }(x-\cdot )\psi _r))\,{{\tilde{N}}}_1(d{\mathfrak {z}},ds)\rightarrow \int _0^t\int _{{\mathfrak {Z}}_1} \mu _{s}(I_{s-}^{\xi }(k_{\varepsilon }(x-\cdot )))\,{\tilde{N}}_1(d{\mathfrak {z}},ds) \end{aligned}$$

in probability, uniformly in $t\in [0,T]$. Consequently, letting $r\rightarrow \infty$ in equation (5.11), we obtain that (5.9) holds almost surely for each $t\in [0,T]$.

To prove (5.10) we are going to verify that

$$\begin{aligned}{} & {} f_t(x):=(\tilde{{\mathcal {L}}}^*_t\mu _t)^{(\varepsilon )}(x) +\int _{{\mathfrak {Z}}_0}(J^{\eta *}_t\mu _s)^{(\varepsilon )}(x)\,\nu _0(d{\mathfrak {z}}_0) +\int _{{\mathfrak {Z}}_1}(J^{\xi *}_t\mu _s)^{(\varepsilon )}(x)\,\nu _1(d{\mathfrak {z}}_1),\\{} & {} g^j_t(x):=({\mathcal {M}}^{j*}_t\mu _t)^{(\varepsilon )}(x), \quad h_t(x,{\mathfrak {z}}):=(I^{\xi *}_t\mu _{t-})^{(\varepsilon )}(x), \quad v_t(x):=\mu ^{(\varepsilon )}_t(x), \end{aligned}$$

($\omega \in \Omega$, $t\in [0,T]$, $x\in {\mathbb {R}}^d$, ${\mathfrak {z}}\in {\mathfrak {Z}}_1$, $j=1,2,\ldots ,d'$) satisfy the conditions of Theorem 5.1 with the ${\mathcal {F}}_t$-Wiener process $w:=V$ and ${\mathcal {F}}_t$-Poisson martingale measure ${\tilde{\pi }}:={{\tilde{N}}}_1$, carried by the probability space $(\Omega , {\mathcal {F}},Q)$ equipped with the filtration $({\mathcal {F}}_t)_{t\geqslant 0}$. To see that f, g, h satisfy the required measurability properties first we claim that for bounded ${\mathcal {O}}\otimes {\mathcal {B}}({\mathbb {R}}^d)\otimes {\mathcal {B}}({\mathbb {R}}^{d})\otimes {\mathcal {Z}}_0$-measurable functions $A=A_t(x,y,{\mathfrak {z}})$ and bounded ${\mathcal {P}}\otimes {\mathcal {B}}({\mathbb {R}}^d)\otimes {\mathcal {B}}({\mathbb {R}}^{d})\otimes {\mathcal {Z}}_0$-measurable $A=A_t(x,y,{\mathfrak {z}})$, the functions

$$\begin{aligned} \int _{{\mathbb {R}}^d}A_t(x,y,{\mathfrak {z}})\mu _t(dy) \quad \text {and}\ \quad \int _{{\mathbb {R}}^d}B_t(x,y,{\mathfrak {z}})\mu _{t-}(dy) \end{aligned}$$

(5.20)

are ${\mathcal {O}}\otimes {\mathcal {B}}({\mathbb {R}}^d)\otimes {\mathcal {Z}}_0$- and ${\mathcal {P}}\otimes {\mathcal {B}}({\mathbb {R}}^d)\otimes {\mathcal {Z}}_0$-measurable, in $(\omega ,t,x,{\mathfrak {z}})\in \Omega \times [0,T]\times {\mathbb {R}}^d\times {\mathfrak {Z}}_0$, respectively. Indeed, this is obvious if $A_t(x,y,{\mathfrak {z}})=\alpha _t\varphi (x)\phi (y)\kappa ({\mathfrak {z}})$ and $B_t(x,y,{\mathfrak {z}})=\beta _t\varphi (x)\phi (y)\kappa ({\mathfrak {z}})$ with $\varphi ,\phi \in C_b({\mathbb {R}}^d)$, bounded ${\mathcal {Z}}_0$-measurable function $\kappa$ on ${\mathfrak {Z}}_0$, and bounded ${\mathcal {O}}$-measurable function $\alpha$ and bounded ${\mathcal {P}}$-measurable $\beta$ on $\Omega \times [0,T]$. Thus our claim follows by a standard application of the monotone class lemma for functions. Hence one can easily see that our claim remains valid if we replace the boundedness condition with the existence of the integrals in (5.20). Using this and taking into account (5.15) and (5.16) and the estimates obtained by Taylor’s formula,

$$\begin{aligned}{} & {} |J_s^{\eta }k_{\varepsilon }(x-y)|\leqslant N|\eta _{s}(y,{\mathfrak {z}}_0)|^2, \quad |J_s^{\xi }k_{\varepsilon }(x-y)|\leqslant N|\xi _{s}(y,{\mathfrak {z}}_1)|^2, \end{aligned}$$

(5.21)

$$\begin{aligned}{} & {} |I^{\xi }_{s}k_{\varepsilon }(x-y)|^2\leqslant N|\xi _{s}(y,{\mathfrak {z}}_1)|^2 \end{aligned}$$

(5.22)

for $x,y\in {\mathbb {R}}^d$, $s\in [0,T]$, ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$ and $\omega \in \Omega$, where $N=N(\varepsilon , d)$, it is not difficult to show that $(\tilde{{\mathcal {L}}}^*_t\mu _t)^{(\varepsilon )}(x)$, $({\mathcal {M}}^{j*}\mu _t)^{(\varepsilon )}(x)$ are ${\mathcal {O}}\otimes {\mathcal {B}}({\mathbb {R}}^d)$-measurable in $(\omega ,t)$, $(J_t^{\eta *}\mu _t)^{(\varepsilon )}(x)$ and $(J_t^{\xi *}\mu _t)^{(\varepsilon )}(x)$ are ${\mathcal {P}}\otimes {\mathcal {B}}({\mathbb {R}}^d)\otimes {\mathcal {Z}}_0$- and ${\mathcal {P}}\otimes {\mathcal {B}}({\mathbb {R}}^d)\otimes {\mathcal {Z}}_1$-measurable in $(\omega ,t,x,{\mathfrak {z}}_0)$ and $(\omega ,t,x,{\mathfrak {z}}_1)$, respectively, and $(I_t^{\xi *}\mu _{t-})^{(\varepsilon )}(x)$ is ${\mathcal {P}}\otimes {\mathcal {B}}({\mathbb {R}}^d)\otimes {\mathcal {Z}}_1$-measurable in $(\omega ,t,x,{\mathfrak {z}}_1)$. Finally, integrating $(J_t^{\eta *}\mu _t)^{(\varepsilon )}(x)$ and $(J_t^{\xi *}\mu _t)^{(\varepsilon )}(x)$ over ${\mathfrak {Z}}_0$ and ${\mathfrak {Z}}_1$, respectively, by Fubini’s theorem we get that f is ${\mathcal {O}}\otimes {\mathcal {B}}({\mathbb {R}}^d)$-measurable. Using the estimates (5.15), (5.16) together with (5.21) and (5.22) it is easy to see that due to $\mathrm{ess\,sup}_{t\in [0,T]}|\mu _t|({\mathbb {R}}^d)<\infty$ (a.s.) and (5.8) the conditions (5.1), (5.2) hold. By Minkowski’s inequality for every $x\in {\mathbb {R}}^d$, $t\in [0,T]$ and $\omega \in \Omega$ we have

$$\begin{aligned} |\mu _t^{(\varepsilon )}|^p_{L_p} =\int _{{\mathbb {R}}^d}\left| \int _{{\mathbb {R}}^d}k_{\varepsilon }(x-y)\mu _t(dy)\right| ^p\,dx \leqslant |k_{\varepsilon }|^p_{L_p}|\mu _t|^p({\mathbb {R}}^d)<\infty , \end{aligned}$$

which shows that condition (5.4) holds. To complete the proof of the lemma it remains to show that almost surely

$$\begin{aligned}{} & {} A:=\int _0^T\int _{{\mathbb {R}}^d}|(\tilde{{\mathcal {L}}}^*_s\mu _s)^{(\varepsilon )}(x)|^p\,dxds<\infty ,\\{} & {} \quad B:=\int _0^T\int _{{\mathbb {R}}^d} \big (\sum _k|({\mathcal {M}}^{k*}_s\mu _s)^{(\varepsilon )}(x)|^2\big )^{p/2}\,dxds<\infty ,\\{} & {} C_{\eta }:=\int _0^T \int _{{\mathbb {R}}^d}\Big |\int _{{\mathfrak {Z}}_0} (J^{\eta *}_s\mu _s)^{(\varepsilon )}(x,{\mathfrak {z}})\,\nu _0(d{\mathfrak {z}})\Big |^pdxds<\infty , \\{} & {} C_{\xi }:=\int _0^T \int _{{\mathbb {R}}^d}\big |\int _{{\mathfrak {Z}}_1} (J^{\xi *}_s\mu _s)^{(\varepsilon )}(x,{\mathfrak {z}})\,\nu _1(d{\mathfrak {z}})\big |^pdxds<\infty , \\{} & {} G:=\int _0^T\int _{{\mathbb {R}}^d} \int _{{\mathfrak {Z}}_1}|(I^{\xi *}_s\mu _s)^{(\varepsilon )}(x,{\mathfrak {z}})|^p\,\nu _1(d{\mathfrak {z}})dxds<\infty , \\{} & {} H:=\int _0^T \int _{{\mathbb {R}}^d} \Big (\int _{{\mathfrak {Z}}_1} |(I^{\xi *}_s\mu _s)^{(\varepsilon )}(x,{\mathfrak {z}})|^2\,\nu _1(d{\mathfrak {z}})\Big )^{p/2}dxds<\infty . \end{aligned}$$

To this end note first that with a constant $N=N(\varepsilon ,d)$

$$\begin{aligned} |k_{\varepsilon }(x-y)| +|Dk_{\varepsilon }(x-y)| +|D^2k_{\varepsilon }(x-y)|\leqslant Nk_{2\varepsilon }(x-y)\quad \text {for all }x,y\in {\mathbb {R}}^d. \nonumber \\ \end{aligned}$$

(5.23)

Thus, using Minkowski’s inequality and Assumption 2.1(ii), we have a constant N, depending on $\varepsilon$, d, $K_0$, K and $K_1$, such that almost surely

$$\begin{aligned}{} & {} A\leqslant \int _0^T\Big (\int _{{\mathbb {R}}^d} \Big (\int _{{\mathbb {R}}^d}|\tilde{{\mathcal {L}}}_s k_{\varepsilon }(x-y)|^pdx\Big )^{1/p}|\mu _s|(dy)\Big )^pds\\{} & {} \quad \leqslant N |k_{2\varepsilon }|^p_{L_p} \int _0^T\Big (\int _{{\mathbb {R}}^d}(K^2_0+K_1^2|y|^2+ K_1^2|Y_s|^2)|\mu _s|(dy)\Big )^p ds. \end{aligned}$$

Hence taking into account $\mathrm{ess\,sup}_{s\in [0,T]}\Vert \mu _s\Vert <\infty$ (a.s.), (5.8) (if $K_1\ne 0$), as well as the cadlagness of $(Y_t)_{t\in [0,T]}$, we get $A<\infty$ (a.s.). In the same way we have $B<\infty$ (a.s.). By Taylor’s formula and (5.23) for each $x\in {\mathbb {R}}^d$ we have

$$\begin{aligned}{} & {} |J^{\eta }_yk_{\varepsilon }(x-y)| \leqslant \int _0^1|D^2_yk_{\varepsilon }|(x-y-\theta \eta (y,{\mathfrak {z}}))|\eta (y,{\mathfrak {z}})|^2\,d\theta , \\{} & {} \quad \leqslant N\int _0^1k_{2\varepsilon }(x-y-\theta \eta (y,{\mathfrak {z}}))\,d\theta \,|\eta (y,{\mathfrak {z}})|^2, \end{aligned}$$

for all $y\in {\mathbb {R}}^d$, $s\in [0,T]$, ${\mathfrak {z}}\in {\mathfrak {Z}}_0$ and $\omega \in \Omega$. Here, and often later on, the variable s is suppressed, and the subscript y in $J^{\eta }_y$ indicates that the operator $J^{\eta }$ acts in the variable y. Hence Minkowski’s inequality gives

$$\begin{aligned} \Big (\int _{{\mathbb {R}}^d}|J^{\eta }_yk_{\varepsilon }(x-y)|^p\,dx\Big )^{1/p} \leqslant N|k_{2\varepsilon }|_{L_p}|\eta (y,{\mathfrak {z}})|^2 \end{aligned}$$

with a constant $N=N(d,\varepsilon )$. Thus by the Minkowski inequality and Fubini’s theorem,

$$\begin{aligned}{} & {} C_{\eta }\leqslant \int _0^T \left( \int _{{\mathfrak {Z}}_0} \int _{{\mathbb {R}}^d} \left( \int _{{\mathbb {R}}^d}|J^{\eta }_yk_{\varepsilon }(x-y)|^p\,dx \right) ^{1/p}\,|\mu _s|(dy) \,\nu _0(d{\mathfrak {z}}) \right) ^pds \\{} & {} \quad \leqslant N^p|k_{2\varepsilon }|^p_{L_p} \int _0^T \Big ( \int _{{\mathfrak {Z}}_0} \int _{{\mathbb {R}}^d}|\eta _s(y,{\mathfrak {z}})|^{2}\,|\mu _s|(dy) \,\nu _0(d{\mathfrak {z}}) \Big )^pds \\{} & {} \quad \leqslant 2^pN^p|{\bar{\eta }}|^{2p}_{L_2}|k_{2\varepsilon }|^p_{L_p} \int _0^T \Big ( \int _{{\mathbb {R}}^d}(K_0^2+K_1^2|y|^{2}+K_1^2|Y_s|^{2})\,|\mu _s|(dy) \Big )^p\,ds<\infty \,\,\text {(a.s.)}. \end{aligned}$$

In the same way we get $C_{\xi }<\infty$ (a.s.). By Taylor’s formula and (5.23) for each $x\in {\mathbb {R}}^d$ we have

$$\begin{aligned}{} & {} |I^{\xi }_yk_{\varepsilon }(x-y)| \leqslant \int _0^1 |D_yk_{\varepsilon }|(x-y-\theta \xi (y,{\mathfrak {z}}))|\xi (y,{\mathfrak {z}})|\,d\theta , \nonumber \\{} & {} \quad \leqslant N\int _0^1k_{2\varepsilon }(x-y-\theta \xi (y,{\mathfrak {z}}))\,d\theta \,|\xi (y,{\mathfrak {z}})|, \end{aligned}$$

(5.24)

for all $y\in {\mathbb {R}}^d$, $s\in [0,T]$, ${\mathfrak {z}}\in {\mathfrak {Z}}_0$ and $\omega \in \Omega$, with a constant $N=N(d,p,\varepsilon )$. Hence similarly to above we obtain

$$\begin{aligned} G\leqslant NK_{\xi }^{p-2}|{\bar{\xi }}|^2_{L_2}|k_{2\varepsilon }|^p_{L_p} \int _0^T\Big ( \int _{{\mathbb {R}}^d}(K_0+K_1|y|+K_1|Y_s|)\,|\mu _s|(dy) \Big )^p\,ds<\infty \,\,\text {(a.s.)}. \end{aligned}$$

with a constant $N=N(d,p,\varepsilon )$. By Minkowski’s inequality, taking into account (5.24) and Assumption 2.2 we have

$$\begin{aligned}{} & {} H\leqslant \int _0^T \left( \int _{{\mathfrak {Z}}_1}\Big (\int _{{\mathbb {R}}^d}|(I_t^{\xi *}\mu _{t})^{{(\varepsilon )}}|^p\,dx\Big )^{2/p}\,\nu _1(d{\mathfrak {z}}) \right) ^{p/2}\,dt \\{} & {} \quad \leqslant \int _0^T \left( \int _{{\mathfrak {Z}}_1}\Big (\int _{{\mathbb {R}}^d} \Big ( \int _{{\mathbb {R}}^d}|I_t^\xi k_{\varepsilon }(x-y)|^p\,dx\Big )^{1/p}|\mu _t|(dy) \Big )^{2}\,\nu _1(d{\mathfrak {z}})\right) ^{p/2}\,dt \\{} & {} \quad \leqslant N |{\bar{\xi }}|^p_{L_2}|k_{2\varepsilon }|^p_{L_p} \int _0^T \Big ( \int _{{\mathbb {R}}^d}(K_0+K_1|y|+K_1|Y_t|)\,|\mu _t|(dy) \Big )^{p}\,dt<\infty \end{aligned}$$

almost surely, with a constant $N=N(d,p,\varepsilon )$. $\square$

Lemma 5.3

Let Assumption 2.1 hold. Assume $(u_t)_{t\in [0,T]}$ is an $L_p$-solution of equation (3.9) for a given $p\geqslant 2$ such that $\mathrm{ess\,sup}_{t\in [0,T]}|u_t|_{L_1}<\infty$ (a.s.), and if $K_1\ne 0$ in Assumption 2.1 (ii), then

$$\begin{aligned} \mathrm{ess\,sup}_{t\in [0,T]}\int _{{\mathbb {R}}^d}|y|^2\,|u_t|(dy)<\infty \,\,(\mathrm {a.s.)}. \end{aligned}$$

(5.25)

Then for each $x\in {\mathbb {R}}^d$ and $\varepsilon >0$,

$$\begin{aligned} u^{(\varepsilon )}_t(x)&= u^{(\varepsilon )}_0(x) + \int _0^t(\tilde{{\mathcal {L}}}^*_su_s)^{(\varepsilon )}(x)\,ds +\int _0^t({\mathcal {M}}^{j*}_su_s)^{(\varepsilon )}(x)\,dV^j_s \nonumber \\&\quad +\int _0^t\int _{{\mathfrak {Z}}_0}(J^{\eta *}_su_s)^{(\varepsilon )}(x )\,\nu _0(d{\mathfrak {z}})\,ds + \int _0^t\int _{{\mathfrak {Z}}_1}(J^{\xi *}_su_s)^{(\varepsilon )}(x)\,\nu _1(d{\mathfrak {z}})\,ds\nonumber \\&\quad + \int _0^t\int _{{\mathfrak {Z}}_1}(I^{\eta *}_su_{s-})^{(\varepsilon )}(x)\,{\tilde{N}}_{1}(d{\mathfrak {z}},ds) \end{aligned}$$

(5.26)

holds almost surely for all $t\in [0,T]$. Moreover, for each $\varepsilon >0$ and $p\geqslant 2$

$$\begin{aligned} |u^{(\varepsilon )}_t|_{L_p}^p&= |u^{(\varepsilon )}_0|_{L_p}^p + p\int _0^t \big (|u^{(\varepsilon )}_s|^{p-2}u^{(\varepsilon )}_s,({\mathcal {M}}_s^{k*}u_s)^{(\varepsilon )}\big )\,dV^k_s \nonumber \\&\quad + p\int _0^t \big (|u^{(\varepsilon )}_s|^{p-2}u^{(\varepsilon )}_s,(\tilde{{\mathcal {L}}}^*_s u_s)^{(\varepsilon )}\big )\,ds \nonumber \\&\quad + \tfrac{p(p-1)}{2}\sum _k\int _0^t\big (|u_s^{(\varepsilon )}|^{p-2}, |({\mathcal {M}}_s^{k*}u_s)^{(\varepsilon )}|^2 \big )\,ds \nonumber \\&\quad + p\int _0^t\int _{{\mathfrak {Z}}_0} \big (|u^{(\varepsilon )}_s|^{p-2}u^{(\varepsilon )}_s,(J^{\eta *}_su_s)^{(\varepsilon )}\big )\,\nu _0(d{\mathfrak {z}})ds \nonumber \\&\quad + p\int _0^t\int _{{\mathfrak {Z}}_1} \big (|u^{(\varepsilon )}_s|^{p-2}u^{(\varepsilon )}_s,(J^{\xi *}_su_s)^{(\varepsilon )}\big )\,\nu _1(d{\mathfrak {z}})ds +p\int _0^t\int _{{\mathfrak {Z}}_1} \nonumber \\&\quad \big ( |u^{(\varepsilon )}_{s-}|^{p-2}u^{(\varepsilon )}_{s-},(I^{\xi *}u_{s-})^{(\varepsilon )}\big )\,{\tilde{N}}_1(d{\mathfrak {z}},ds)\nonumber \\&\quad +\int _0^t\int _{{\mathfrak {Z}}_1}\int _{{\mathbb {R}}^d} \Big \{\big |u_{s-}^{(\varepsilon )}+ (I_s^{\xi *}u_{s-})^{(\varepsilon )}\nonumber \\&\quad \big |^p - |u_{s-}^{(\varepsilon )}|^p - p|u_{s-}^{(\varepsilon )}|^{p-2}u^{(\varepsilon )}_{s-}(I_s^{\xi *}u_{s-})^{(\varepsilon )}\Big \}\,dxN_1(d{\mathfrak {z}},ds) \end{aligned}$$

(5.27)

holds almost surely for all $t\in [0,T]$, where $u_{s-}$ denotes the left limit in $L_p$.

Proof

Notice that Eqs. (5.26) and (5.27) can be formally obtained from equations (5.9) and (5.10), respectively, by substituting $u_t(x)dx$ and $u_{t-}(x)dx$ in place of $\mu _t(dx)$ and $\mu _{t-}(dx)$, respectively. Note, however, that $u_t(x)dx$, defines a signed measure only for $P\otimes dt$-almost every $(\omega ,t)\in \Omega \times [0,T]$. Thus this lemma does not follow directly from Lemma 5.2. We can copy, however, the proof of Lemma 5.2 by replacing $\mu _t(dx)$ and $\mu _{t-}(dx)$ with $u_t(x)dx$ and $u_{t-}(x)dx$, respectively. We need also take into account that since $(u_{t})_{t\in [0,T]}$ is an $L_p$-valued weakly cadlag process, we have have a set $\Omega '$ of full probability such that $u_{t-}(\omega )=u_t(\omega )$ for all but countably many $t\in [0,T]$, and $\sup _{t\in [0,T]}|u_t(\omega )|_{L_p}<\infty$ for $\omega \in \Omega '$. $\square$

Lemma 5.4

Let Assumptions 2.1, 2.2 and 2.4 hold. Let $(\mu _t)_{t\in [0,T]}$ be a measure-valued solution to (3.8). If $K_1\ne 0$ in Assumption 2.1, then assume additionally (5.8). Then for $\varepsilon >0$ and even integers $p\geqslant 2$ we have

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|\mu ^{(\varepsilon )}_t|^p_{L_p} \leqslant N{\mathbb {E}}|\mu ^{(\varepsilon )}_0|^p_{L_p} \end{aligned}$$

(5.28)

with a constant $N=N(p,d,T,K, K_{\xi }, K_{\eta },L,\lambda , |{\bar{\xi }}|_{L_2}, |{\bar{\eta }}|_{L_2})$.

Proof

We may assume that ${\mathbb {E}}|\mu _0^{(\varepsilon )}|_{L_p}^p<\infty$. Define

$$\begin{aligned}{} & {} Q_p(b, \sigma , \rho , \beta ,\mu , k_{\varepsilon }) =p\big (|\mu ^{(\varepsilon )}|^{p-2}\mu ^{(\varepsilon )},({\tilde{{\mathcal {L}}}}^*\mu )^{(\varepsilon )}\big )\nonumber \\{} & {} \quad +\tfrac{p(p-1)}{2}\sum _k\big (|\mu ^{(\varepsilon )}|^{p-2}, |({\mathcal {M}}^{k*}\mu )^{(\varepsilon )}|^2 \big ), \nonumber \\{} & {} \quad {\mathcal {Q}}_p^{(0)}(\eta ({\mathfrak {z}}_0),\mu , k_{\varepsilon }) =p\big (|\mu ^{{(\varepsilon )}}|^{p-2}\mu ^{{(\varepsilon )}},(J^{\eta ({\mathfrak {z}}_0)*}\mu )^{{(\varepsilon )}}\big ),\nonumber \\{} & {} \quad {\mathcal {Q}}_p^{(1)}(\xi ({\mathfrak {z}}_1), \mu , k_{\varepsilon }) = p(|\mu ^{{(\varepsilon )}}|^{p-2}\mu ^{{(\varepsilon )}}, (J^{\xi ({\mathfrak {z}}_1)*}\mu )^{{(\varepsilon )}}),\nonumber \\{} & {} {\mathcal {R}}_p(\xi ({\mathfrak {z}}_1),\mu ,k_{\varepsilon })=|\mu ^{{(\varepsilon )}} + (I^{\xi ({\mathfrak {z}}_1)*}\mu )^{{(\varepsilon )}}|^p_{L_p} \nonumber \\{} & {} \quad - |\mu ^{{(\varepsilon )}}|^p_{L_p} - p(|\mu ^{{(\varepsilon )}}|^{p-2}\mu ^{{(\varepsilon )}},(I^{\xi ({\mathfrak {z}}_1)*}\mu )^{{(\varepsilon )}}), \end{aligned}$$

(5.29)

for $\mu \in {\mathbb {M}}$, $\beta \in {\mathbb {R}}^{d'}$, functions b, $\sigma$ and $\rho$ on ${\mathbb {R}}^d$, with values in ${\mathbb {R}}^d$, ${\mathbb {R}}^{d\times d_1}$ and ${\mathbb {R}}^{d\times d'}$, respectively, and ${\mathbb {R}}^d$-valued functions $\eta ({\mathfrak {z}}_0)$ and $\xi ({\mathfrak {z}}_1)$ on ${\mathbb {R}}^d$ for each ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$, $i=0,1$, where

$$\begin{aligned} {\tilde{{\mathcal {L}}}}=\tfrac{1}{2}(\sigma ^{il}\sigma ^{jl}+\rho ^{ik}\rho ^{jk})D_{ij}+ \beta ^l\rho ^{il}D_i+\beta ^lB^l, \quad \\ {\mathcal {M}}^k=\rho ^{ik}D_i+B^k, \quad k=1,2,\ldots ,d'. \end{aligned}$$

Recall that (f, g) denotes the integral of the product of Lebesgue measurable functions f and g over ${\mathbb {R}}^d$ against the Lebesgue measure on ${\mathbb {R}}^d$. By Lemma 5.2

$$\begin{aligned} d|\mu _t^{(\varepsilon )}|^p_{L_p}{} & {} ={\mathcal {Q}}_p(b_t, \sigma _t, \rho _t, \beta _t,\mu _t, k_{\varepsilon })\,dt +\int _{{\mathfrak {Z}}_0}{\mathcal {Q}}_p^{(0)}(\eta _t({\mathfrak {z}}),\mu _t, k_{\varepsilon }) \,\nu _0(d{\mathfrak {z}})\,dt \nonumber \\{} & {} \quad +\int _{{\mathfrak {Z}}_1}{\mathcal {Q}}_p^{(1)}(\xi _t({\mathfrak {z}}), \mu _t, k_{\varepsilon })\,\nu _1(d{\mathfrak {z}})\,dt\nonumber \\{} & {} \quad +\int _{{\mathfrak {Z}}_1}{\mathcal {R}}_p(\xi _{t}({\mathfrak {z}}), \mu _{t-}, k_{\varepsilon })\,N_1(d{\mathfrak {z}},dt) +d\zeta _1(t)+d\zeta _2(t), \end{aligned}$$

(5.30)

where $\beta _t=B_t(X_t)$ and

$$\begin{aligned} \zeta _1(t)= & {} p\int _0^t \big (|\mu ^{(\varepsilon )}_s|^{p-2} \mu ^{(\varepsilon )}_s,({\mathcal {M}}_s^{k *}\mu _s)^{(\varepsilon )}\big )\,dV^k_s,\nonumber \\ \zeta _2(t)= & {} p\int _0^t\int _{{\mathfrak {Z}}_1} \big (|\mu ^{(\varepsilon )}_{s}|^{p-2}\mu ^{(\varepsilon )}_{s},(I_s^{\xi *}\mu _{s})^{(\varepsilon )}\big )\,{\tilde{N}}_{1}(d{\mathfrak {z}},ds)\quad t\in [0,T] \end{aligned}$$

(5.31)

are local martingales under P. We write

$$\begin{aligned} \int _{{\mathfrak {Z}}_1}{\mathcal {R}}_p(\xi _{t}({\mathfrak {z}}_1), \mu _{t-}, k_{\varepsilon })\,N_1(d{\mathfrak {z}},dt) =\int _{{\mathfrak {Z}}_1}{\mathcal {R}}_p(\xi _{t}({\mathfrak {z}}_1), \mu _{t-}, k_{\varepsilon })\,\nu _1(d{\mathfrak {z}})dt+d\zeta _3(t)\nonumber \\ \end{aligned}$$

(5.32)

with

$$\begin{aligned} \zeta _3(t)=\int _0^t\int _{{\mathfrak {Z}}_1}{\mathcal {R}}_p(\xi _s({\mathfrak {z}}), \mu _{s-}, k_{\varepsilon })\,N_1(d{\mathfrak {z}},ds) -\int _0^t\int _{{\mathfrak {Z}}_1}{\mathcal {R}}_p(\xi _s({\mathfrak {z}}), \mu _{s-}, k_{\varepsilon })\,\nu _1(d{\mathfrak {z}})ds, \end{aligned}$$

which we can justify if we show

$$\begin{aligned} A:=\int _0^T \int _{{\mathfrak {Z}}_1} |{\mathcal {R}}_p(\xi _s({\mathfrak {z}}), \mu _s, k_{\varepsilon })|\, \nu _1(d{\mathfrak {z}})\,ds<\infty \,\text {(a.s.)}. \end{aligned}$$

(5.33)

To this end observe that by Taylor’s formula

$$\begin{aligned} 0\leqslant {\mathcal {R}}_p(\xi _t({\mathfrak {z}}), \mu _t, k_{\varepsilon }) \leqslant N\int _{{\mathbb {R}}^d} |\mu ^{(\varepsilon )}_t|^{p-2}| (I^{\xi ({\mathfrak {z}})*}_t\mu _t)^{(\varepsilon )}|^{2} +|(I^{\xi ({\mathfrak {z}})*}_t\mu _t)^{(\varepsilon )}|^{p}\,dx\nonumber \\ \end{aligned}$$

(5.34)

with a constant $N=N(d,p)$. Hence

$$\begin{aligned}{} & {} \int _{{\mathfrak {Z}}_1}{\mathcal {R}}_p(\xi _t({\mathfrak {z}}), \mu _t, k_{\varepsilon })\,\nu _1(d{\mathfrak {z}}) \leqslant N\int _{{\mathbb {R}}^d} |\mu ^{(\varepsilon )}_t|^{p-2}| (I^{\xi ({\mathfrak {z}})*}_t\mu _t)^{(\varepsilon )}|_{L_2({\mathfrak {Z}}_1)}^{2}\\{} & {} \quad +|(I^{\xi ({\mathfrak {z}})*}_t\mu _t)^{(\varepsilon )}|^{p}_{L_p({\mathfrak {Z}}_1)}\,dx \leqslant N'\big ( |\mu _t^{(\varepsilon )}|^p_{L_p}+A_1(t)+A_2(t)\big ) \end{aligned}$$

with

$$\begin{aligned} A_1(t)=\int _{{\mathbb {R}}^d}|(I^{\xi ({\mathfrak {z}}) *}_t\mu _t)^{(\varepsilon )}|^{p}_{L_2({\mathfrak {Z}}_1)}\,dx, \quad A_2(t)= \int _{{\mathbb {R}}^d}|(I^{\xi ({\mathfrak {z}}) *}_t\mu _t)^{(\varepsilon )}|^{p}_{L_p({\mathfrak {Z}}_1)}\,dx\nonumber \\ \end{aligned}$$

(5.35)

and constants N and $N'$ depending only on d and p. By Minkowski’s inequality

$$\begin{aligned} |\mu _t^{(\varepsilon )}|^p_{L_p}{} & {} =\int _{{\mathbb {R}}^d}\Big |\int _{{\mathbb {R}}^d}k_{\varepsilon }(x-y)\,\mu _t(dy)\Big |^p\,dx \leqslant \Big | \int _{{\mathbb {R}}^d}|k_{\varepsilon }|_{L_p}\,\mu _t(dy) \Big |^p \leqslant |k_{\varepsilon }|_{L_p}^p\,\mu _t^p(\textbf{1}), \nonumber \\ \end{aligned}$$

(5.36)

$$\begin{aligned} A_1(t){} & {} =\int _{{\mathbb {R}}^d}\Big |\int _{{\mathfrak {Z}}_1} \big | \int _{{\mathbb {R}}^d}(k_{\varepsilon }(x-y-\xi _t(y,{\mathfrak {z}}))-k_{\varepsilon }(x-y))\,\mu _t(dy)\big |^2\nu _1(d{\mathfrak {z}}) \Big |^{p/2}dx \nonumber \\{} & {} \leqslant \Big |\int _{{\mathfrak {Z}}_1} \Big | \int _{{\mathbb {R}}^d}(k_{\varepsilon }(\cdot -y-\xi _t(y,{\mathfrak {z}}))-k_{\varepsilon }(\cdot -y))\,\mu _t(dy) \Big |^{2}_{L_p} \nu _1(d{\mathfrak {z}})\Big |^{p/2} \nonumber \\{} & {} \leqslant \Big | \int _{{\mathfrak {Z}}_1} \Big | \int _{{\mathbb {R}}^d}|Dk_{\varepsilon }|_{L_p}|\xi _t(y,{\mathfrak {z}})|\,\mu _t(dy) \Big |^{2} \nu _1(d{\mathfrak {z}})\Big |^{p/2}\nonumber \\{} & {} \leqslant |Dk_{\varepsilon }|^p_{L_p} |{\bar{\xi }}|_{L_2({\mathfrak {Z}}_1)}^p \Big ( \int _{{\mathbb {R}}^d}(K_0+K_1|y|+K_1|Y_t|)\,\mu _t(dy) \Big )^p, \end{aligned}$$

(5.37)

and similarly, using Assumption 2.2,

$$\begin{aligned} A_2(t){} & {} = \int _{{\mathbb {R}}^d} \int _{{\mathfrak {Z}}_1} \Big | \int _{{\mathbb {R}}^d}(k_{\varepsilon }(x-y-\xi _{t}(y,{\mathfrak {z}}))-k_{\varepsilon }(x-y))\,\mu _t(dy) \Big |^{p} \nu _1(d{\mathfrak {z}})dx\nonumber \\{} & {} \leqslant \int _{{\mathfrak {Z}}_1} \Big |\int _{{\mathbb {R}}^d} \big |k_{\varepsilon }(\cdot -y-\xi _t(y,{\mathfrak {z}}))-k_{\varepsilon }(\cdot -y))\big |_{L_p}\mu _t(dy) \Big |^{p} \nu _1(d{\mathfrak {z}})\nonumber \\{} & {} \leqslant K^{p-2}_{\xi }|Dk_{\varepsilon }|^p_{L_p} |{\bar{\xi }}|^{2}_{L_2({\mathfrak {Z}}_1)} \Big ( \int _{{\mathbb {R}}^d}(K_0+K_1|y|+K_1|Y_t|)\,\mu _t(dy) \Big )^p. \end{aligned}$$

(5.38)

By (5.34)–(5.38) we have a constant $N=N(K_{\xi },p,d,\varepsilon ,|{\bar{\xi }}|_{L_2{({\mathfrak {Z}}_1)}})$ such that

$$\begin{aligned} A\leqslant N\int _0^T\mu _t^{p}(\textbf{1})\,dt +N\int _0^T \Big (\int _{{\mathbb {R}}^d}(K_0+K_1|y|+K_1|Y_t|)\,\mu _t(dy) \Big )^{p}dt<\infty \,\text {(a.s.)}. \end{aligned}$$

Next we claim that, with the operator $T^{\xi }$ defined in (4.22),

$$\begin{aligned} \zeta _2(t)+\zeta _3(t) =\int _0^t\int _{{\mathfrak {Z}}_1} |(T^{\xi *}_s\mu _s)^{(\varepsilon )}|^p_{L_p}-|\mu _s^{(\varepsilon )}|^p_{L_p} {{\tilde{N}}}_1(d{\mathfrak {z}},ds)=:\zeta (t)\quad \text {for }t\in [0,T]. \nonumber \\ \end{aligned}$$

(5.39)

To see that the stochastic integral $\zeta (t)$ is well-defined as an Itô integral note that by Lemma 4.5 and (5.36)

$$\begin{aligned}{} & {} \int _0^T\int _{{\mathfrak {Z}}_1}||(T_s^{\xi *}\mu _s)^{(\varepsilon )}|^p_{L_p} -|\mu _s^{(\varepsilon )}|^p_{L_p}|^2\,\nu _1(d{\mathfrak {z}})ds \leqslant N|{\bar{\xi }}|^2_{L_2({\mathfrak {Z}}_1)} \int _{0}^T|\mu _s^{(\varepsilon )}|^{2p}_{L_p}\,ds\nonumber \\{} & {} \quad \leqslant N|{\bar{\xi }}|^2_{L_2({\mathfrak {Z}}_1)}|k_{\varepsilon }|_{L_p}^{2p} \int _0^T\mu _s^{2p}(\textbf{1})\,ds<\infty \,\text {(a.s.)} \end{aligned}$$

(5.40)

with a constant $N=N(d,p,\lambda , K_{\xi })$. Since ${\mathfrak {Z}}_1$ is $\sigma$-finite, there is an increasing sequence $({\mathfrak {Z}}_{1n})_{n=1}^{\infty }$, ${\mathfrak {Z}}_{1n}\in {\mathcal {Z}}_1$, such that $\nu _1({\mathfrak {Z}}_{1n})<\infty$ for every n and $\cup _{n=1}^{\infty }{\mathfrak {Z}}_{1n}={\mathfrak {Z}}_1$. Then it is easy to see that

$$\begin{aligned}{} & {} {\bar{\zeta }}_{2n}(t)=p\int _0^t\int _{{\mathfrak {Z}}_1}{} \textbf{1}_{{\mathfrak {Z}}_{1n}}({\mathfrak {z}}) \big (|\mu ^{(\varepsilon )}_{s}|^{p-2}\mu ^{(\varepsilon )}_{s},(I_s^{\xi *}\mu _{s})^{(\varepsilon )}\big )\,N_{1}(d{\mathfrak {z}},ds),\\{} & {} {\hat{\zeta }}_{2n}(t)=p\int _0^t\int _{{\mathfrak {Z}}_1}{} \textbf{1}_{{\mathfrak {Z}}_{1n}}({\mathfrak {z}}) \big (|\mu ^{(\varepsilon )}_{s}|^{p-2}\mu ^{(\varepsilon )}_{s},(I_s^{\xi *}\mu _{s})^{(\varepsilon )}\big )\,\nu _1(d{\mathfrak {z}})ds,\\{} & {} {\bar{\zeta }}_{3n}(t)=\int _0^t\int _{{\mathfrak {Z}}_1} \textbf{1}_{{\mathfrak {Z}}_{1n}}({\mathfrak {z}}){\mathcal {R}}_p(\xi _s({\mathfrak {z}}), \mu _{s-}, k_{\varepsilon })\,N_1(d{\mathfrak {z}},ds),\\{} & {} {\hat{\zeta }}_{3n}(t)=\int _0^t\int _{{\mathfrak {Z}}_1}{} \textbf{1}_{{\mathfrak {Z}}_{1n}}({\mathfrak {z}}) {\mathcal {R}}_p(\xi _s({\mathfrak {z}}), \mu _{s-}, k_{\varepsilon })\,\nu _1(d{\mathfrak {z}})ds \end{aligned}$$

are well-defined, and

$$\begin{aligned} \zeta _2(t)= & {} \lim _{n\rightarrow \infty }({\bar{\zeta }}_{2n}(t)-{\hat{\zeta }}_{2n}(t)),\\ \zeta _3(t)= & {} \lim _{n\rightarrow \infty }{\bar{\zeta }}_{3n}(t)-\lim _{n\rightarrow \infty }{\hat{\zeta }}_{3n}(t), \end{aligned}$$

where the limits are understood in probability. Hence

$$\begin{aligned} \zeta _2(t)+\zeta _3(t)= & {} \lim _{n\rightarrow \infty }\Big ({\bar{\zeta }}_{2n}(t) +{\bar{\zeta }}_{3n}(t)-\big ({\hat{\zeta }}_{2n}(t)+{\hat{\zeta }}_{3n}(t)\big )\Big ) \\{} & {} =\lim _{n\rightarrow \infty }\Big (\int _0^t\int _{{\mathfrak {Z}}_1}{} \textbf{1}_{{\mathfrak {Z}}_{1n}}({\mathfrak {z}}) (|(T_s^{\xi *}\mu _s)^{(\varepsilon )}|^p_{L_p}-|\mu _s^{(\varepsilon )})|^p_{L_p}) {{\tilde{N}}}_1(d{\mathfrak {z}},ds)\Big )=\zeta (t), \end{aligned}$$

which completes the proof of (5.39). Consequently, from (5.30)–(5.32) we have

$$\begin{aligned} d|\mu _t^{(\varepsilon )}|^p_{L_p}&={\mathcal {Q}}_p(b_t, \sigma _t, \rho _t, \beta _t,\mu _t, k_{\varepsilon })\,dt +\int _{{\mathfrak {Z}}_0}{\mathcal {Q}}_p^{(0)}(\eta _t({\mathfrak {z}}_0),\mu _t, k_{\varepsilon }) \,\nu _0(d{\mathfrak {z}})\,dt\nonumber \\&\quad +\int _{{\mathfrak {Z}}_1}{\mathcal {Q}}_p^{(1)}(\xi _t({\mathfrak {z}}_1), \mu _t, k_{\varepsilon })+{\mathcal {R}}_p(\xi _{t}({\mathfrak {z}}_1), \mu _{t}, k_{\varepsilon })\,\nu _1(d{\mathfrak {z}})\,dt +d\zeta _1(t)+d\zeta (t). \end{aligned}$$

(5.41)

By Lemmas 4.1 and 4.2 we have

$$\begin{aligned} Q(b_s, \sigma _s, \rho _s, \beta _s,\mu _s, k_{\varepsilon }) \leqslant N(L^2+K^2)|\mu _s^{(\varepsilon )}|_{L_p}^p \end{aligned}$$

(5.42)

with a constant $N=N(d,p)$, and by Lemma 4.3 and Corollary 4.4, using that ${\bar{\xi }}\leqslant K_{\xi }$ and ${\bar{\eta }}\leqslant K_{\eta }$, we have

$$\begin{aligned}{} & {} {\mathcal {Q}}^{(0)}(\eta _s({\mathfrak {z}}), \mu _s, k_{\varepsilon }) \leqslant N{{\bar{\eta }}}^2({\mathfrak {z}})|\mu _s^{(\varepsilon )}|_{L_p}^p, \quad ({\mathcal {Q}}^{(1)}+{\mathcal {R}}_p)(\xi _s({\mathfrak {z}}), \mu _s, k_{\varepsilon }) \nonumber \\{} & {} \quad \leqslant N{{\bar{\xi }}}^2({\mathfrak {z}})|\mu _s^{(\varepsilon )}|_{L_p}^p \end{aligned}$$

(5.43)

with a constant $N=N(K_{\xi },K_{\eta },d,p,\lambda )$. Thus from (5.41) for $c^{\varepsilon }_t:=|\mu ^{(\varepsilon )}_t|^p_{L_p}$ we obtain that almost surely

$$\begin{aligned} c_t^{\varepsilon } \leqslant |\mu ^{(\varepsilon )}_0|^p_{L_p}+N\int _0^tc_s^\varepsilon \,ds+m^{\varepsilon }_t \quad \text {for all }t\in [0,T] \end{aligned}$$

(5.44)

with a constant $N=N(T,p,d,K,K_{\xi }, K_{\eta }, L,\lambda ,|{\bar{\xi }}|_{L_2},|{\bar{\eta }}|_{L_2})$ and the local martingale $m^{\varepsilon }=\zeta _1+\zeta$. For integers $n\geqslant 1$ set $\tau _n={\bar{\tau }}_n\wedge {\tilde{\tau }}_n$, where $({\tilde{\tau }}_n)_{n=1}^{\infty }$ is a localising sequence of stopping times for $m^{\varepsilon }$ and

$$\begin{aligned} {\bar{\tau }}_n={\bar{\tau }}_n(\varepsilon )=\inf \Big \{t\in [0,T]:\int _0^tc_s^\varepsilon \,ds\geqslant n\Big \}. \end{aligned}$$

Then from (5.44) we get

$$\begin{aligned} {\mathbb {E}}c^{\varepsilon }_{t\wedge \tau _n} \leqslant {\mathbb {E}}|\mu ^{(\varepsilon )}_0|_{L_p}^p+N\int _0^t{\mathbb {E}}c^{\varepsilon }_{s\wedge \tau _n}\,ds<\infty \quad \text {for }t\in [0,T]\text { and integers }n\geqslant 1. \end{aligned}$$

Hence by Gronwall’s lemma

$$\begin{aligned} {\mathbb {E}}c^{\varepsilon }_{t\wedge \tau _n}\leqslant N{\mathbb {E}}|\mu ^{(\varepsilon )}_0|_{L_p}^p \quad \text {for }t\in [0,T]\text { and integers }n\geqslant 1 \end{aligned}$$

with a constant $N=N(T,p,d,K,,K_{\xi }, K_{\eta }, L,\lambda ,|{\bar{\xi }}|_{L_2},|{\bar{\eta }}|_{L_2})$. Letting here $n\rightarrow \infty$, by Fatou’s lemma we obtain

$$\begin{aligned} \sup _{t\in [0,T]}{\mathbb {E}}|\mu ^{(\varepsilon )}_t|^p_{L_p}\leqslant N{\mathbb {E}}|\mu ^{(\varepsilon )}_0|^p_{L_p}. \end{aligned}$$

(5.45)

Hence we follow a standard way to prove (5.28). Clearly, from (5.44), taking into account (5.45), we have a constant $N=N(T,p,d,K,K_{\xi }, K_{\eta }, L,\lambda ,|{\bar{\xi }}|_{L_2},|{\bar{\eta }}|_{L_2})$ such that

$$\begin{aligned} {\mathbb {E}}\sup _{t\leqslant T}c^{\varepsilon }_{t\wedge \tau }\leqslant N{\mathbb {E}}|\mu _0^{(\varepsilon )}|^p_{L_p} + {\mathbb {E}}\sup _{t\leqslant T}|\zeta _1(t\wedge \tau )|+{\mathbb {E}}\sup _{t\leqslant T}|\zeta (t\wedge \tau )| \end{aligned}$$

(5.46)

for every stopping time $\tau$. By estimates in Lemmas 4.1 and 4.5 for the Doob-Meyer processes $\langle \zeta _1\rangle$ and $\langle \zeta \rangle$ of $\zeta _1$ and $\zeta$ we have

$$\begin{aligned} \langle \zeta _1\rangle (t)&=p^2\int _0^t \big |(|\mu ^{(\varepsilon )}_s|^{p-2} \mu ^{(\varepsilon )}_s,({\mathcal {M}}_s^{k*}\mu _s)^{(\varepsilon )}\big )|^2\,ds \leqslant N_1\int _0^{t} |\mu _s^{(\varepsilon )}|_{L_p}^{2p}\,ds<\infty , \nonumber \\ \langle \zeta \rangle (t)&=\int _0^t\int _{{\mathfrak {Z}}_1}||(T_s^{\xi *}\mu _s)^{(\varepsilon )}|^p_{L_p} -|\mu _s^{(\varepsilon )}|^p_{L_p}|^2\nu _1(d{\mathfrak {z}})ds \leqslant N_2\int _0^{t} |\mu _s^{(\varepsilon )}|_{L_p}^{2p}\,ds<\infty \end{aligned}$$

(5.47)

almost surely for all $t\in [0,T]$, with constants $N_1=N_1(d,p,L)$ and $N_2=N_2(d,p,\lambda , K_\xi , |{\bar{\xi }}|_{L_2({\mathfrak {Z}}_1)})$. Using the Davis inequality, by (5.47) and (5.45) we get

$$\begin{aligned}{} & {} {\mathbb {E}}\sup _{t\leqslant T}|\zeta _1(t\wedge \tau )|+{\mathbb {E}}\sup _{t\leqslant T}|\zeta (t\wedge \tau )| \leqslant 3{\mathbb {E}}\langle \zeta _1\rangle ^{1/2}(t\wedge \tau )+3{\mathbb {E}}\langle \zeta \rangle ^{1/2}(t\wedge \tau ) \nonumber \\{} & {} \leqslant N'{\mathbb {E}}\Big (\int _0^{T} |\mu _{s\wedge \tau }^{(\varepsilon )}|_{L_p}^{2p}\,ds \Big )^{1/2} \leqslant N'{\mathbb {E}}\Big (\sup _{t\leqslant T}|\mu _{s}^{(\varepsilon )}|_{L_p}^{p} \int _0^{T} |\mu _{s\wedge \tau }^{(\varepsilon )}|_{L_p}^{p}\,ds \Big )^{1/2}\nonumber \\{} & {} \quad \leqslant \frac{1}{2}{\mathbb {E}}\sup _{t\leqslant T}|\mu _{t\wedge \tau }^{(\varepsilon )}|_{L_p}^p + N^{''}{\mathbb {E}}\int _0^T |\mu _{s}^{(\varepsilon )}|_{L_p}^{p}\,ds \leqslant \tfrac{1}{2}{\mathbb {E}}\sup _{t\leqslant T}|\mu _{t\wedge \tau }^{(\varepsilon )}|_{L_p}^p +N^{'''}{\mathbb {E}}|\mu _0^{(\varepsilon )}|^p_{L_p},\nonumber \\ \end{aligned}$$

(5.48)

which by (5.46) gives

$$\begin{aligned} {\mathbb {E}}\sup _{t\leqslant T}c^{\varepsilon }_{t\wedge \tau }\leqslant N{\mathbb {E}}|\mu _0^{(\varepsilon )}|^p_{L_p} + \tfrac{1}{2}{\mathbb {E}}\sup _{t\leqslant T}c^{\varepsilon }_{t\wedge \tau } \end{aligned}$$

with constants $N,N',N'',N'''$ depending on T, p, d, K, $K_{\xi }$, $K_\eta$, L, $\lambda$, $|{\bar{\xi }}|_{L_2}$ and $|{\bar{\eta }}|_{L_2}$ Substituting here the stopping time

$$\begin{aligned} \rho _n=\inf \{t\in [0,T]: \langle \zeta _1\rangle (t)+ \langle \zeta \rangle (t)\geqslant n\} \end{aligned}$$

in place of $\tau$, from (5.46) by virtue of the Davis inequality we have

$$\begin{aligned} {\mathbb {E}}\sup _{t\leqslant T}c^{\varepsilon }_{t\wedge \rho _n}\leqslant N{\mathbb {E}}|\mu _0^{(\varepsilon )}|^p_{L_p} + \tfrac{1}{2}{\mathbb {E}}\sup _{t\leqslant T}c^{\varepsilon }_{t\wedge \rho _n}<\infty \end{aligned}$$

for every integer $n\geqslant 1$. Hence

$$\begin{aligned} {\mathbb {E}}\sup _{t\leqslant T\wedge \rho _n}|\mu _t^{(\varepsilon )}|_{L_p}^p \leqslant 2N{\mathbb {E}}|\mu _0^{(\varepsilon )}|^p_{L_p}, \end{aligned}$$

and letting here $n\rightarrow \infty$ by Fatou’s lemma we finish the proof of (5.28). $\square$

Lemma 5.5

Let Assumptions 2.1, 2.2 and 2.4 hold. Let $(u _t)_{t\in [0,T]}$ be an $L_p$-solution to (3.9) for an even integer $p\geqslant 2$ such that $\mathrm{ess\,sup}_{t\in [0,T]}|u_t|_{L_1}<\infty$ (a.s.). If $K_1\ne 0$ in Assumption 2.1, then assume additionally (5.8). Then we have

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|u_t|^p_{L_p}\leqslant N{\mathbb {E}}|u_0|^p_{L_p} \end{aligned}$$

(5.49)

with a constant $N=N(p,d,T,K,K_{\xi }, K_{\eta }, L, \lambda , |{\bar{\xi }}|_{L_2}, |{\bar{\eta }}|_{L_2})$.

Proof

We may assume ${\mathbb {E}}|u_0|^p_{L_p}<\infty$. By Lemma 5.3 for every $\varepsilon >0$ equation (5.27) holds almost surely for all $t\in [0,T]$. Hence following the proof of Lemma 5.4 with $u^{(\varepsilon )}_t(x)$, $u_t(x)dx$, $u_{t-}(x)dx$, $|u_t(x)|dx$ in place of $\mu ^{(\varepsilon )}_t(x)$, $\mu _t(dx)$, $\mu _{t-}(dx)dx$ and $|\mu _t|(dx)$, respectively, and taking into account that almost surely $u_t=u_{t-}$ for all but countable many $t\in [0,T]$, we obtain the counterpart of (5.44),

$$\begin{aligned} |u^{(\varepsilon )}_t|^p_{L_p}&\leqslant |u^{(\varepsilon )}_0|^p_{L_p} +N\int _0^t||u_s|^{(\varepsilon )}|^p_{L_p}\,ds+m^{\varepsilon }_t \nonumber \\&\leqslant |u_0|^p_{L_p} +N\int _0^t|u_s|^p_{L_p}\,ds+m^{\varepsilon }_t \quad \text {almost surely for all }t\in [0,T] \end{aligned}$$

(5.50)

with a constant $N=N(T,p,d,K,K_{\xi }, K_{\eta }, L,\lambda ,|{\bar{\xi }}|_{L_2},|{\bar{\eta }}|_{L_2})$ and a (cadlag) local martingale $m^{\varepsilon }_t=\zeta _1^{\varepsilon }(t)+\zeta ^{\varepsilon }(t)$, $t\in [0,T]$, where

$$\begin{aligned}{} & {} \zeta _1^{\varepsilon }(t)=p\int _0^t \big (|u^{(\varepsilon )}_s|^{p-2} u^{(\varepsilon )}_s,({\mathcal {M}}^{k*}_su_s)^{\varepsilon }\big )\,dV^k_s, \\{} & {} \zeta ^{\varepsilon }(t):=\int _0^t\int _{{\mathfrak {Z}}_1} |(T^{\xi *}u_s)^{(\varepsilon )}|^p_{L_p}-|u_s^{(\varepsilon )}|^p_{L_p} {{\tilde{N}}}_1(d{\mathfrak {z}},ds). \end{aligned}$$

Since $(u_t)_{t\in [0,T]}$ is a weakly cadlag ${\mathcal {F}}_t$-adapted process, we have $\sup _{t\in [0,T]}|u_t|_{L_p}<\infty$ (a.s.), and hence

$$\begin{aligned} \int _0^t|u_s|^r_{L_p}\,ds, \quad t\in [0,T] \end{aligned}$$

is a continuous ${\mathcal {F}}_t$-adapted process for every $r>0$. For $\varepsilon >0$ and integers $n\geqslant 1$, $k\geqslant 1$ define the stopping times $\tau ^{\varepsilon }_{n,k}:={\bar{\tau }}_{n}\wedge {\tilde{\tau }}^{\varepsilon }_{k}$, where

$$\begin{aligned} {\bar{\tau }}_n:=\inf \Big \{t\in [0,T]:\int _0^t|u_s|^p_{L_p}\,ds\geqslant n\Big \} \end{aligned}$$

for integers $n\geqslant 1$, and $(\tilde{\tau }_k^{\varepsilon })_{k=1}^{\infty }$ is a localizing sequence for the local martingale $m^{\varepsilon }$. Thus from (5.50) for $c^{\varepsilon }_t:=|u^{\varepsilon }_t|_{L_p}^p$ and $c_t:=|u_t|_{L_p}^p$ we get

$$\begin{aligned} {\mathbb {E}}c^{\varepsilon }_{t\wedge \tau ^{\varepsilon }_{n,k}} \leqslant {\mathbb {E}}c_0+N{\mathbb {E}}\int _0^tc_{s\wedge {\bar{\tau }}_n}\,ds<\infty \end{aligned}$$

for every $t\in [0,T]$. Letting here first $k\rightarrow \infty$ and then $\varepsilon \rightarrow 0$ by Fatou’s lemma we obtain

$$\begin{aligned} {\mathbb {E}}c_{t\wedge {\bar{\tau }}_n} \leqslant {\mathbb {E}}c_0+N{\mathbb {E}}\int _0^tc_{s\wedge {\bar{\tau }}_n}\,ds<\infty , \quad t\in [0,T], \end{aligned}$$

which by Gronwall’s lemma gives

$$\begin{aligned} {\mathbb {E}}c_{t\wedge {\bar{\tau }}_n} \leqslant e^{NT}{\mathbb {E}}|u_0|^p_{L_p}\quad \text {for }t\in [0,T]. \end{aligned}$$

Letting now $n\rightarrow \infty$ by Fatou’s lemma we have

$$\begin{aligned} \sup _{t\in [0,T]}{\mathbb {E}}|u_t|_{L_p}^p\leqslant e^{NT}{\mathbb {E}}|u_0|^p_{L_p}. \end{aligned}$$

(5.51)

Hence we are going to prove (5.49) in an already familiar way. Analogously to (5.46), due to Lemmas 4.1 and 4.5, for the Doob-Meyer processes of $\zeta _1^{\varepsilon }$ and $\zeta ^{\varepsilon }$ we have with constants $N_1=N_1(d,p,L)$ and $N_2=N_2(d,p,\lambda , K_\xi , |{\bar{\xi }}|_{L_2({\mathfrak {Z}}_1)})$,

$$\begin{aligned} \langle \zeta ^{\varepsilon }_1\rangle (t)&=p^2\int _0^t \big |(|u^{(\varepsilon )}_s|^{p-2} u^{(\varepsilon )}_s,({\mathcal {M}}_s^{k*}u_s)^{(\varepsilon )}\big )|^2\,ds \nonumber \\&\leqslant N_1\int _0^{t} ||u_s|^{(\varepsilon )}|_{L_p}^{2p}\,ds \leqslant N_1\int _0^{t} |u_s|_{L_p}^{2p}\,ds, \nonumber \\ \langle \zeta ^{\varepsilon }\rangle (t)&=\int _0^t\int _{{\mathfrak {Z}}_1}||(T_s^{\xi *}u_s)^{(\varepsilon )}|^p_{L_p} -|u_s^{(\varepsilon )}|^p_{L_p}|^2\nu _1({\mathfrak {z}})ds \nonumber \\&\leqslant N_2\int _0^{t} ||u_s|^{(\varepsilon )}|_{L_p}^{2p}\,ds \leqslant N_2\int _0^{t} |u_s|_{L_p}^{2p}\,ds. \end{aligned}$$

(5.52)

We define the stopping time $\rho _{n,k}^{\varepsilon }={\tilde{\tau }}^{\varepsilon }_k\wedge \rho _n$, where

$$\begin{aligned} \rho _n=\inf \Big \{t\in [0,T]:\int _0^t|u_s|^{2p}_{L_p}\,ds\geqslant n\Big \} \quad \text {for every integer }n\geqslant 1, \end{aligned}$$

and $({\tilde{\tau }}^{\varepsilon }_k)_{k=1}^{\infty }$ denotes, as before, a localizing sequence of stopping times for $m^{\varepsilon }$. Notice that from (5.50), due to (5.51) and (5.52), by using the Davis inequality we have

$$\begin{aligned} {\mathbb {E}}\sup _{t\leqslant T}c^{\varepsilon }_{t\wedge \rho ^{\varepsilon }_{n,k}}&\leqslant N'{\mathbb {E}}|u_0|^p_{L_p}+ {\mathbb {E}}\sup _{t\leqslant T}|\zeta _1(t\wedge \rho ^{\varepsilon }_{n,k})| +{\mathbb {E}}\sup _{t\leqslant T}|\zeta (t\wedge \rho ^{\varepsilon }_{n,k})| \\&\leqslant N{\mathbb {E}}|u_0|^p_{L_p} +N{\mathbb {E}}\left( \int _0^{T\wedge \rho _n}|u_t|^{2p}_{L_p}\,dt \right) ^{1/2}<\infty , \end{aligned}$$

where $N'$ and N are constants, depending only on p, d, T, K, $K_{\xi }$, $K_{\eta }$ L $\lambda$, $|{\bar{\xi }}|_{L_2}$ and $|{\bar{\eta }}|_{L_2}$. Letting here first $k\rightarrow \infty$ and then $\varepsilon \rightarrow 0$ by Fatou’s lemma we obtain

$$\begin{aligned} {\mathbb {E}}\sup _{t\leqslant T}c_{t\wedge \rho _{n}} \leqslant N{\mathbb {E}}|u_0|^p_{L_p} +N{\mathbb {E}}\left( \int _0^{T\wedge \rho _n}|u_t|^{2p}_{L_p}\,dt \right) ^{1/2}<\infty \quad \text {for every }n.\nonumber \\ \end{aligned}$$

(5.53)

Hence, in the same standard way as before, by Young’s inequality and (5.51) we have

$$\begin{aligned} {\mathbb {E}}\sup _{t\leqslant T}c_{t\wedge \rho _{n}}&\leqslant N{\mathbb {E}}|u_0|^p_{L_p} +N{\mathbb {E}}\Big ( \sup _{t\leqslant T}c_{t\wedge \rho _{n}}\int _0^{T}|u_t|^{p}_{L_p}\,dt \Big )^{1/2} \\&\leqslant N{\mathbb {E}}|u_0|^p_{L_p} +\tfrac{1}{2}{\mathbb {E}}\sup _{t\leqslant T}c_{t\wedge \rho _{n}} +N^2{\mathbb {E}}\int _0^{T}|u_t|^{p}_{L_p}\,dt \\&\leqslant N'{\mathbb {E}}|u_0|^p_{L_p} +\tfrac{1}{2}{\mathbb {E}}\sup _{t\leqslant T}c_{t\wedge \rho _{n}}<\infty , \end{aligned}$$

with a constant $N'=N'(T,p,d,K,K_{\xi }, K_{\eta }, L,\lambda ,|{\bar{\xi }}|_{L_2},|{\bar{\eta }}|_{L_2})$, which gives

$$\begin{aligned} {\mathbb {E}}\sup _{t\leqslant T}c_{t\wedge \rho _{n}}\leqslant 2N'{\mathbb {E}}|u_0|^p_{L_p}. \end{aligned}$$

Letting here $n\rightarrow \infty$ by Fatou’s lemma we finish the proof of (5.49). $\square$

To formulate the next lemma let $(S,{\mathcal {S}})$ denote a measurable space, and let ${\mathcal {H}}\subset {\mathcal {F}}\otimes {\mathcal {S}}$ be a $\sigma$-algebra.

Lemma 5.6

Let $\mu =(\mu _s)_{s\in S}$ be an ${\mathbb {M}}$-valued function on $\Omega \times S$ such that $\mu _s(\varphi )$ is an ${\mathcal {H}}$-measurable random variable for every bounded Borel function $\varphi$ on ${\mathbb {R}}^d$, and ${\mathbb {E}}\mu _s(\textbf{1})<\infty$ for every $s\in S$. Let $p>1$ and assume that for a positive sequence $\varepsilon _n\rightarrow 0$ we have

$$\begin{aligned} \limsup _{\varepsilon _n\rightarrow 0}{\mathbb {E}}|\mu _s^{(\varepsilon _n)}|^p_{L_p}=:N_s^p<\infty \quad \text {for every }s\in S. \end{aligned}$$

Then for every $s\in S$ the density $d\mu _s/dx$ exists almost surely, and there is an $L_p({\mathbb {R}}^d)$-valued ${\mathcal {H}}$-measurable mapping u on $\Omega \times S$ such that for each s we have $u_s=d\mu _s/dx$ (a.s.). Moreover, $\lim _{n\rightarrow \infty }|\mu _s^{(\varepsilon _n)}-u_s|_{L_p}=0$ (a.s.) and ${\mathbb {E}}|u_s|_{L_p}^p\leqslant N_s^p$ for each $s\in S$.

Proof

Fix $s\in S$. Since $(\mu _s^{(\varepsilon _n)})_{n=1}^{\infty }$ is a bounded sequence in ${\mathbb {L}}_p:=L_p((\Omega ,{\mathcal {F}},P), L_p({\mathbb {R}}^d))$ from any subsequence of it one can choose a subsequence, $\mu _s^{(\varepsilon _{n'})}$, which converges weakly in ${\mathbb {L}}_p$ to some ${{\bar{u}}}_s\in {\mathbb {L}}_p$. Thus for every $\varphi \in C^{\infty }_0({\mathbb {R}}^d)$ and $G\in {\mathcal {F}}$ we have

$$\begin{aligned} {\mathbb {E}}\int _{{\mathbb {R}}^d}\mu _s^{(\varepsilon _{n'})}(x)\textbf{1}_G\varphi (x)\,dx\rightarrow {\mathbb {E}}\int _{{\mathbb {R}}^d}{{\bar{u}}}_s(x)\textbf{1}_G\varphi (x)\,dx \quad \text {as }n'\rightarrow \infty . \end{aligned}$$

On the other hand, since

$$\begin{aligned} {\mathbb {E}}\int _{{\mathbb {R}}^d}\int _{{\mathbb {R}}^d}k_{\varepsilon _n}(x-y)\textbf{1}_G|\varphi (x)|\,\mu _s(dy)dx \leqslant |\mu _s^{(\varepsilon _{n})}|_{{\mathbb {L}}_p}|\varphi |_{L_q}<\infty \quad \text {with }q=p/(p-1), \end{aligned}$$

we can use Fubini’s theorem, and then, due to ${\mathbb {E}}\mu _s(\textbf{1})<\infty$, we can use Lebesgue’s theorem on dominated convergence to get

$$\begin{aligned} {\mathbb {E}}\int _{{\mathbb {R}}^d}\mu ^{(\varepsilon _{n'})}_s(x)\textbf{1}_G\varphi (x)\,dx ={\mathbb {E}}\int _{{\mathbb {R}}^d}{} \textbf{1}_G\varphi ^{(\varepsilon _{n'})}(x)\,\mu _s(dx) \rightarrow {\mathbb {E}}\int _{{\mathbb {R}}^d}{} \textbf{1}_G\varphi (x)\,\mu _s(dx). \end{aligned}$$

Consequently,

$$\begin{aligned} {\mathbb {E}}\textbf{1}_G\int _{{\mathbb {R}}^d}\varphi (x)\,\mu _s(dx) ={\mathbb {E}}\textbf{1}_G\int _{{\mathbb {R}}^d}\varphi (x){{\bar{u}}}_s(x)\,dx \quad \text {for any }G\in {\mathcal {F}}\text { and }\varphi \in C_0^{\infty },\nonumber \\ \end{aligned}$$

(5.54)

which implies that $d\mu _s/dx$ almost surely exists in $L_p$ and equals ${{\bar{u}}}_s$. Notice, that ${{\bar{u}}}_s$, as an element of ${\mathbb {L}}_p$, is independent of the chosen subsequences, i.e., if ${\tilde{u}}_s$ is the weak limit in ${\mathbb {L}}_p$ of some subsequence of a subsequence of $\mu _s^{(\varepsilon _n)}$, then by (5.54) we have

$$\begin{aligned} {\mathbb {E}}\textbf{1}_G\int _{{\mathbb {R}}^d}\varphi (x){{\bar{u}}}_s(x)\,dx ={\mathbb {E}}\textbf{1}_G\int _{{\mathbb {R}}^d}\varphi (x){{\tilde{u}}}_s(x)\,dx \quad \text {for any }G\in {\mathcal {F}}\text { and }\varphi \in C_0^{\infty }, \end{aligned}$$

which means ${{\bar{u}}}_s={{\tilde{u}}}_s$ in ${\mathbb {L}}_p$. Consequently, the whole sequence $\mu ^{(\varepsilon _n)}_s$ converges weakly to ${\bar{u}}_s$ in ${\mathbb {L}}_p$ for every s, and for each s almost surely ${\bar{u}}_s=d\mu _s/dx\in L_p$. Hence $\mu _s^{(\varepsilon _n)}={\bar{u}}_s^{(\varepsilon _n)}\in L_p$ (a.s.), and thus by a well-known property of mollifications, $\lim _{n\rightarrow \infty }|\mu ^{(\varepsilon _n)}_s-{{\bar{u}}}_s|_{L_p}=0$ (a.s.). Set

$$\begin{aligned} A:=\{(\omega ,s)\in \Omega \times S: \mu _s^{(\varepsilon _n)}\text { is convergent in }L_p\text { as }n\rightarrow \infty \}, \end{aligned}$$

and let $u_s$ denote the limit of $\textbf{1}_A\mu _s^{(\varepsilon _n)}$ in $L_p$. Then, since $(\mu _s^{(\varepsilon _n)})_{s\in S}$ is an $L_p$-valued ${\mathcal {H}}$-measurable function of $(\omega ,s)$ for every n, the function $u=(u_s)_{s\in S}$ is also an $L_p$-valued ${\mathcal {H}}$-measurable function, and clearly, $u_s=d\mu _s/dx$ (a.s.) and ${\mathbb {E}}|u_s|^p_{L_p}\leqslant N_s^p$ for each s. $\square$

Lemma 5.7

Let Assumptions 2.1, 2.2 and 2.4 hold. Let $\mu =(\mu _t)_{t\in [0,T]}$ be a measure-valued solution to (3.8). If $K_1\ne 0$ in Assumption 2.1, then assume additionally (5.8). Assume $u_0= d\mu _0/dx$ exists almost surely and ${\mathbb {E}}|u_0|^p_{L_p}<\infty$ for some even $p\geqslant 2$. Then the following statements hold.

• (i) For each $t\in [0,T]$ the density $d\mu _t/dx$ exists almost surely, and there is an $L_p$-valued ${\mathcal {F}}_t$-adapted weakly cadlag process $(u_t)_{t\in [0,T]}$ such that almost surely $u_t=d\mu _t/dx$ for every $t\in [0,T]$ and ${\mathbb {E}}\sup _{t\in [0,T]}|u_t|^p_{L_p}<\infty$.

• (ii) If $\mu '=(\mu '_t)_{t\in [0,T]}$ satisfies the same conditions (with the same even integer p) as $\mu$, then for $u_t=d\mu _t/dx$ and $u'_t=d\mu '_t/dx$ we have

  $$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|u_t-u'_t|^p_{L_p} \leqslant N{\mathbb {E}}|u_0-u_0'|^p_{L_p} \quad \text {for }t\in [0,T], \end{aligned}$$

  (5.55)

  with a constant N depending only on d, p, K, $K_{\xi }$, $K_{\eta }$, L, $\lambda$, T, $|{\bar{\eta }}|_{L_2({\mathfrak {Z}}_1)}$ and $|{\bar{\xi }}|_{L_2({\mathfrak {Z}}_0)}$.

Proof

By Lemma 5.4 we have

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|\mu _t^{(\varepsilon )}|^p_{L_p} \leqslant N{\mathbb {E}}|\mu _0^{(\varepsilon )}|^p_{L_p}<\infty \quad \text {for every }t\in [0,T]\text { and }\varepsilon >0 \end{aligned}$$

with a constant $N=N(d,p,K, T, K_{\xi },K_{\eta }, L,\lambda ,|{\bar{\eta }}|_{L_2({\mathfrak {Z}}_1)},|{\bar{\xi }}|_{L_2({\mathfrak {Z}}_0)})$. Moreover, by Lemma 5.6, there is an $L_p$-valued ${\mathcal {F}}_t$-adapted ${\mathcal {F}}\otimes {\mathcal {B}}([0,T])$-measurable process $({\bar{u}}_t)_{t\in [0,T]}$ such that ${{\bar{u}}}_t=d\mu _t/dx$ (a.s.) for every $t\in [0,T]$. To prove (i) let A be a countable dense subset of [0, T], such that $T\in A$. Then

$$\begin{aligned}{} & {} {\mathbb {E}}\sup _{t\in A}|{{\bar{u}}}_t|_{L_p}^p ={\mathbb {E}}\sup _{t\in A} \liminf _{n\rightarrow \infty }|\mu _t^{(\varepsilon _n)}|_{L_p}^p \leqslant {\mathbb {E}}\liminf _{n\rightarrow \infty }\sup _{t\in A}|\mu _t^{(\varepsilon _n)}|_{L_p}^p \nonumber \\{} & {} \quad \leqslant \liminf _{n\rightarrow \infty } {\mathbb {E}}\sup _{t\in A}|\mu _t^{(\varepsilon _n)}|_{L_p}^p \leqslant N{\mathbb {E}}|d\mu _0/dx|^p_{L_p}<\infty \end{aligned}$$

(5.56)

for a sequence $\varepsilon _n\downarrow 0$, and there is a set $\Omega '\in {\mathcal {F}}_0$ of full probability such that

$$\begin{aligned} \sup _{t\in A}|{{\bar{u}}}_t(\omega )|_{L_p}<\infty , \quad d\mu _t/dx={\bar{u}}_t \quad \text {for every }\omega \in \Omega '\text { and }t\in A, \end{aligned}$$

and $\mu _t(\varphi )$ is a cadlag function in $t\in [0,T]$ for $\omega \in \Omega '$ and $\varphi \in C^{\infty }_0({\mathbb {R}}^d)$. Hence, if $t\in [0,T]$ and $\omega \in \Omega '$, then there is a sequence $t_n=t_n(\omega )\in A$ such that $t_n\downarrow t$ and ${\bar{u}}_{t_n}(\omega )$ converges weakly in $L_p$ to an element, denoted by $u_t(\omega )$. Note that since ${{\bar{u}}}_{t_n}(\omega )$ is dx-everywhere nonnegative for every n, the function $u_t(\omega )$ is also dx-almost everywhere nonnegative. Moreover, by property of a weak limit we have

$$\begin{aligned} |u_t(\omega )|_{L_p}\leqslant \liminf _{n\rightarrow \infty }|{\bar{u}}_{t_n}(\omega )|_{L_p} \leqslant \sup _{s\in A}|{{\bar{u}}}_{s}(\omega )|_{L_p}, \end{aligned}$$

which gives

$$\begin{aligned} \sup _{t\in [0,T]}|u_t(\omega )|_{L_p} \leqslant \sup _{s\in A}|{\bar{u}}_{s}(\omega )|_{L_p}<\infty \quad \text {for }\omega \in \Omega '. \end{aligned}$$

(5.57)

Notice that

$$\begin{aligned} (u_t(\omega ),\varphi ) =\lim _{n\rightarrow \infty }\mu _{t_n}(\omega ,\varphi )=\mu _{t}(\omega ,\varphi ) \quad \text {for }\omega \in \Omega ', t\in [0,T]\text { and }\varphi \in C_0^{\infty },\nonumber \\ \end{aligned}$$

(5.58)

which shows that $u_t(\omega )$ does not depend on the sequence $t_n$. In particular, for $\omega \in \Omega '$ we have ${\bar{u}}_t(\omega )=u_t(\omega )$ for $t\in A$. Moreover, it shows that $(u_t(\omega ),\varphi )$ is a cadlag function of $t\in [0,T]$ for every $\varphi \in C_0^{\infty }$. Hence, due to (5.57), since $C_0^{\infty }$ is dense in $L_q$, it follows that $u_t(\omega )$ is a weakly cadlag $L_p$-valued function of $t\in [0,T]$ for each $\omega \in \Omega '$. Moreover, from (5.58), by the monotone class lemma it follows that $u_t=d\mu _t/dx$ for every $\omega \in \Omega '$ and $t\in [0,T]$. Define $u_t(\omega )=0$ for $\omega \notin \Omega '$ and $t\in [0,T]$. Then $(u_t)_{t\in [0,T]}$ is an $L_p$-valued weakly cadlag function in $t\in [0,T]$ for every $\omega \in \Omega$, and since due to (5.58) almost surely $(u_t,\varphi )=\mu _t(\varphi )$ for $\varphi \in C_0^{\infty }$, it follows that $u_t$ is an ${\mathcal {F}}_t$-measurable $L_p$-valued random variable for every $t\in [0,T]$. Moreover, by virtue of (5.56) and (5.57) we have ${\mathbb {E}}\sup _{t\in [0,T]}|u_t|^p_{L_p}<\infty$. To prove (ii), notice that by (i) the process ${{\bar{u}}}_t:=u_t-u'_t$, $t\in [0,T]$, is an $L_p$-solution to Eq. (3.9) such that $\mathrm{ess\,sup}_{t\in [0,T]}|{{\bar{u}}}_t|_{L_1}<\infty$ (a.s.). Thus we have (5.55) by Lemma 5.5. $\square$

Definition 5.1

Let $p>1$ and let $\psi$ be an $L_p$-valued ${\mathcal {F}}_0$-measurable random variable. Then we say that an $L_p$-valued ${\mathcal {F}}_t$-optional process $v=(v_t)_{t\in [0,T]}$ is a ${\mathbb {V}}_p$-solution to (3.9) with initial value $\psi$ if for each $\varphi \in C_0^{\infty }$

$$\begin{aligned} \begin{aligned} (v_t,\varphi )&=(\psi ,\varphi ) + \int _0^t(v_{s},{\tilde{{\mathcal {L}}}}_s\varphi )\,ds + \int _0^t(v_{s},{\mathcal {M}}_s^k\varphi )\,dV^k_s + \int _0^t\int _{{\mathfrak {Z}}_0}(v_{s},J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})ds\\&\quad + \int _0^t\int _{{\mathfrak {Z}}_1}(v_{s},J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})ds +\int _0^t\int _{{\mathfrak {Z}}_1}(v_{s},I_s^{\xi }\varphi )\,{{\tilde{N}}}_1(d{\mathfrak {z}},ds) \end{aligned} \end{aligned}$$

(5.59)

for $P\otimes dt$-a.e. $(\omega ,t)\in \Omega \times [0,T]$.

Lemma 5.8

Let Assumption 2.1 (ii) holds. Let $(v_t)_{t\in [0,T]}$ be a ${\mathbb {V}}_p$-solution for a $p>1$ such that $\mathrm{ess\,sup}_{t\in [0,T]}|v_t|_{L_p}<\infty$ (a.s.), and there is an $L_p$-valued random variable g such that for each $\varphi \in C_0^{\infty }$ equation (5.59) for $t:=T$ holds almost surely with g in place of $v_T$. Then there exists an $L_p$-solution $u=(u_t)_{t\in [0,T]}$ to equation (3.9) such that $u_0=\psi$ and $u=v$, $P\otimes dt$-almost everywhere.

Proof

Let $\Phi \in C_0^\infty$ be a countable dense set in $L_q$ for $q=p/(p-1)$. Then there is a set $\Omega '\in \Omega$ of full probability and for every $\omega \in \Omega '$ there is a set ${\mathbb {T}}_\omega \subset [0,T]$ of full Lebesgue measure in [0, T], such that $\sup _{t\in {\mathbb {T}}_{\omega }}|v_t(\omega )|_{L_p}<\infty$ for $\omega \in \Omega '$, and for all $\varphi \in \Phi$ equation (5.59) holds for all $\omega \in \Omega '$ and $t\in {\mathbb {T}}_\omega$. We may also assume that for each $\varphi \in \Phi$ and $\omega \in \Omega '$ equation (3.6) holds for $t=T$ with g in place of $v_T$. Since the right-hand side of equation (5.59), which we denote by $F_t(\varphi )$ for short, is almost surely a cadlag function of t, we may assume, that for $\omega \in \Omega '$ it is cadlag for all $\varphi \in \Phi$. Since ${\mathbb {T}}_{\omega }$ is dense in [0, T] and $\sup _{t\in {\mathbb {T}}_{\omega }}|v_t(\omega )|_{L_p}<\infty$ for $\omega \in \Omega '$, for each $\omega \in \Omega '$ and $t\in [0,T)$ we have a sequence $t_n=t_n(\omega )\in {\mathbb {T}}_{\omega }$ such that $t_n\downarrow t$ and $v_{t_n}\rightarrow {{\bar{v}}}_t$ weakly in $L_p$ for some element ${{\bar{v}}}_t={{\bar{v}}}_t(\omega )\in L_p$. Hence

$$\begin{aligned} ({\bar{v}}_t(\omega ),\varphi )=\lim _{n\rightarrow \infty }(v_{t_n}(\omega ),\varphi ) =\lim _{n\rightarrow \infty }F_{t_n(\omega )}(\omega ,\varphi ) =F_{t}(\omega ,\varphi )\quad \text {for all }\varphi \in \Phi ,\nonumber \\ \end{aligned}$$

(5.60)

which implies that for every sequence $t_n=t_n(\omega )\in {\mathbb {T}}_{\omega }$ such that $t_n\downarrow t$ the sequence $v_{t_n(\omega )}(\omega )$ converges weakly to ${\bar{v}}_t(\omega )$ in $L_p$. In particular, ${{\bar{v}}}_t(\omega )=v_t(\omega )$ for $\omega \in \Omega '$ and $t\in {\mathbb {T}}_{\omega }$. For $\omega \in \Omega '$ we define $u_t(\omega ):={{\bar{v}}}_t(\omega )$ for $t\in [0,T)$ and $u_T(\omega ):=g(\omega )$, and for $\omega \in \Omega \setminus \Omega '$ we set $u_t(\omega )=0$ for all $t\in [0,T]$. Then due to (5.60) and that almost surely $(u_T,\varphi )=F_T(\varphi )$ for all $\varphi \in \Phi$, the process $u=(u_t)_{t\in [0,T]}$ is an $L_p$-valued ${\mathcal {F}}_t$-adapted weakly cadlag process such that almost surely (5.59) holds for all $\varphi \in C_0^{\infty }$. Clearly, $u=v$ $P\otimes dt$ (a.e.). Thus we also have that almost surely

$$\begin{aligned} \int _0^t\int _{{\mathfrak {Z}}_1} (u_{s-},I_s^\xi \varphi )\,{\tilde{N}}_{1}(d{\mathfrak {z}},ds) = \int _0^t\int _{{\mathfrak {Z}}_1} (u_{s},I_s^\xi \varphi )\,{\tilde{N}}_{1}(d{\mathfrak {z}},ds) \end{aligned}$$

for all $t\in [0,T]$ and hence u satisfies (5.59), with $u_s$ replaced by $u_{s-}$ in the last term on the right-hand side, almost surely for all $\varphi \in C_0^\infty$ for all $t\in [0,T]$, i.e., u is an $L_p$ solution to (3.9). $\square$

6 Solvability of the filtering equations in $L_p$-spaces

To show the solvability of the linear filtering equation (3.9), the Zakai equation, with any ${\mathcal {F}}_0$-measurable $L_p$-valued initial condition, we want to apply the existence and uniqueness theorem for stochastic integro-differential equations proved in [8]. With this purpose in mind first we assume that the coefficients $\sigma$, b, $\rho$, B, $\xi$, $\eta$ are smooth in $x\in {\mathbb {R}}^d$, and under this additional assumption we are going to determine the form of the “adjoint" operators ${\tilde{{\mathcal {L}}}}^{*}$, ${\mathcal {M}}^{k*}$, $J^{\eta *}$, $J^{\xi *}$ and $I^{\xi *}$ as operators acting directly on $C^{\infty }_0$ such that

$$\begin{aligned} \int _{{\mathbb {R}}^d} A^{*}\varphi (x)\phi (x)\,dx= \int _{{\mathbb {R}}^d} \varphi (x)A\phi (x)\,dx\quad \text {for all }\varphi , \phi \in C_0^{\infty }, \end{aligned}$$

for ${\tilde{{\mathcal {L}}}}$, ${\mathcal {M}}$, $J^{\xi }$, $J^{\eta }$ and $I^{\xi }$ in place of A. The form of ${\tilde{{\mathcal {L}}}}^{*}$ and ${\mathcal {M}}^{k*}$ is immediately obvious by integrating by parts. To find the form of the other operators (defined in (4.22)), let $\zeta :{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$ such that

$$\begin{aligned} \tau (x):=\tau ^{\zeta }(x):=x+\zeta (x),\quad x\in {\mathbb {R}}^d, \end{aligned}$$

is a $C^1$-diffeomorphism on ${\mathbb {R}}^d$. Then observe that for $\varphi$, $\phi \in C^{\infty }_0$ we have

$$\begin{aligned} (\phi , T^{\zeta }\varphi ) = \int _{{\mathbb {R}}^d} \phi (\tau ^{ -1}(x))|\det D\tau ^{-1}(x)|\varphi (x)\,dx =(|\det D\tau ^{-1}|\, T^{\zeta ^{*}}\phi ,\varphi ) \end{aligned}$$

with

$$\begin{aligned} \zeta ^{*}(x):=-x+\tau ^{-1}(x)=-\zeta (\tau ^{-1}(x)), \quad T^{\zeta ^{*}}\phi (x)=\phi (x+\zeta ^{*}(x)). \end{aligned}$$

(6.1)

Similarly,

$$\begin{aligned} (\phi , I^{\zeta }\varphi )&= \int _{{\mathbb {R}}^d} \big ( \phi (\tau ^{ -1}(x))|\det D\tau ^{-1}(x)|-\phi (x) \big )\varphi (x)\,dx \\&=\int _{{\mathbb {R}}^d} \big ( \phi (\tau ^{-1}(x))|\det D\tau ^{-1}(x)|-\phi (\tau ^{-1}(x))+\phi (\tau ^{-1}(x))-\phi (x) \big ) \varphi (x)\,dx \\&=({\mathfrak {c}}T^{\zeta ^{*}}\phi ,\varphi )+(I^{\zeta ^{*}}\phi , \varphi ), \end{aligned}$$

where

$$\begin{aligned} {\mathfrak {c}}(x)=|\det D\tau ^{-1}(x)|-1, \end{aligned}$$

and

$$\begin{aligned} (\phi ,J^{\zeta }\varphi )&= (\phi ,I^\zeta \varphi ) - (\phi ,\zeta ^iD_i\varphi ) \\&= (I^{\zeta ^{*}}\phi , \varphi ) + ({\mathfrak {c}}T^{\zeta ^{*}}\phi ,\varphi ) + (\zeta ^iD_i\phi ,\varphi ) + ((D_i\zeta ^i)\phi ,\varphi ) \\&= (J^{\zeta ^{*}}\phi , \varphi ) + ({\mathfrak {c}}I^{\zeta ^{*}}\phi ,\varphi ) + (({\mathfrak {c}}+ D_i\zeta ^i)\phi ,\varphi ) + (({\zeta ^{*}}^i+\zeta ^i)D_i\phi ,\varphi ) \\&=(J^{\zeta ^{*}}\phi , \varphi ) + ({\mathfrak {c}}I^{\zeta ^{*}}\phi ,\varphi ) + (({\bar{{\mathfrak {c}}}} +D_i{\zeta ^{*}}^i +D_i\zeta ^i)\phi ,\varphi ) + (({\zeta ^{*}}^i+\zeta ^i)D_i\phi ,\varphi ), \end{aligned}$$

where

$$\begin{aligned} {\bar{{\mathfrak {c}}}}=|\det D\tau ^{-1}(x)|-1-D_i{\zeta ^{*}} ^i. \end{aligned}$$

Consequently, $T^{\zeta *}$, $I^{\zeta *}$ and $J^{\zeta *}$, the formal adjoint of $T^{\zeta }$, $I^{\zeta }$ and $J^{\zeta }$, can be written in the form

$$\begin{aligned}{} & {} T^{\zeta *}=|\det D\tau ^{-1}|\, T^{\zeta ^{*}}, I^{\zeta *}=I^{\zeta ^{*}}+{\mathfrak {c}}T^{\zeta ^{*}}, \nonumber \\{} & {} J^{\zeta *}=J^{\zeta ^{*}}+{\mathfrak {c}}I^{\zeta ^{*}} +(\zeta ^{*i}+\zeta ^i)D_i +{\bar{{\mathfrak {c}}}} +D_i(\zeta ^{*i}+\zeta ^i). \end{aligned}$$

(6.2)

Lemma 6.1

Let $\zeta$ be an ${\mathbb {R}}^d$-valued function on ${\mathbb {R}}^d$ such that for an integer $m\geqslant 1$ it is continuously differentiable up to order m, and

$$\begin{aligned} \inf _{x\in {\mathbb {R}}^d}|\det (I+D\zeta (x))|=:\lambda >0, \quad \max _{1\leqslant k\leqslant m}\sup _{x\in {\mathbb {R}}^d}|D^k\zeta (x)|=:M_m<\infty . \end{aligned}$$

(6.3)

Then the following statements hold.

• (i) The function $\tau =x+\zeta (x)$, $x\in {\mathbb {R}}^d$, is a $C^m$-diffeomorphism, such that

  $$\begin{aligned} \inf _{x\in {\mathbb {R}}^d}|\det D\tau ^{-1}(x)|\geqslant \lambda ', \quad \max _{1\leqslant k\leqslant m}\sup _{x\in {\mathbb {R}}^d}|D^k\tau ^{-1}|\leqslant M'_{m}<\infty , \end{aligned}$$

  (6.4)

  with constants $\lambda '=\lambda '(d, M_1)>0$ and $M'_m=M'_m(d,\lambda ,M_m)$.

• (ii) The function $\zeta ^{*}(x)=-x+\tau ^{-1}(x)$, $x\in {\mathbb {R}}^d$, is continuously differentiable up to order m, such that

  $$\begin{aligned} \sup _{{\mathbb {R}}^d}|\zeta ^{*}|&=\sup _{{\mathbb {R}}^d}|\zeta |, \end{aligned}$$

  (6.5)
  $$\begin{aligned} \sup _{{\mathbb {R}}^d}|D^k\zeta ^{*}|&\leqslant M^{*}_m\max _{1\leqslant j\leqslant k}\sup _{{\mathbb {R}}^d}|D^j\zeta |, \quad \text {for }k=1,2,\ldots ,m, \end{aligned}$$

  (6.6)
  $$\begin{aligned} \inf _{\theta \in [0,1]}\inf _{{\mathbb {R}}^d}|\det (I+\theta D\zeta ^{*})|&\geqslant \lambda '\inf _{\theta \in [0,1]}\inf _{{\mathbb {R}}^d}|\det (I+\theta D\zeta )|, \end{aligned}$$

  (6.7)

  with a constant $M_m^{*}=M_m^{*}(d,\lambda ,M_m)$ and with $\lambda '$ from (6.4).

• (iii) For the functions ${\mathfrak {c}}=\det (I+D\zeta ^{*})-1$, ${\bar{{\mathfrak {c}}}}={\mathfrak {c}}-D_i\zeta ^{*i}$ and $\zeta +\zeta ^{*}$ we have

  $$\begin{aligned}{} & {} \sup _{x\in {\mathbb {R}}^d}|D^k{\mathfrak {c}}(x)| \leqslant N\max _{1\leqslant j\leqslant k+1}\sup _{{\mathbb {R}}^d}|D^j\zeta |, \quad \sup _{x\in {\mathbb {R}}^d}|D^k{\bar{{\mathfrak {c}}}}(x)| \leqslant N\max _{1\leqslant j\leqslant k+1}\sup _{{\mathbb {R}}^d}|D^j\zeta |^2 \nonumber \\ \end{aligned}$$

  (6.8)
  $$\begin{aligned}{} & {} \sup _{{\mathbb {R}}^d}|D^{k}(\zeta +\zeta ^{*})| \leqslant N \max _{1\leqslant j\leqslant k+1}\sup _{{\mathbb {R}}^d}|D^j\zeta |^2. \nonumber \\ \end{aligned}$$

  (6.9)

  for $1\leqslant k\leqslant m-1$ with a constant $N=N(d,\lambda , m, M_m)$.

Proof

To prove (i) note that (6.3) implies that $\tau$ is a $C^m$-diffeomorphism and the estimates in (6.4) are proved in [3] (see Lemma 3.3 therein). By (i) the mapping $\tau$ is a $C^m$-diffeomorphism. From $\tau (x)=x+\zeta (x)$, by substituting $\tau ^{-1}(x)$ in place of x we obtain $\zeta ^{*}(x)=-\zeta (\tau ^{-1}(x))$. Hence (6.5) follows immediately, and due to the second estimates in (6.4), the estimate in (6.6) also follows. Notice that

$$\begin{aligned} x+\theta \zeta ^{*}(x)=\tau ^{-1}(x)+\zeta (\tau ^{-1}(x))-\theta \zeta (\tau ^{-1}(x)) =\tau ^{-1}(x)+(1-\theta )\zeta (\tau ^{-1}(x)). \end{aligned}$$

Hence, by the first inequality in (6.4),

$$\begin{aligned} |\det (I+\theta D\zeta ^{*})|&=|\det (I+(1-\theta )D\zeta (\tau ^{-1}))||\det D\tau ^{-1}| \\&\geqslant \lambda ' |\det (I+(1-\theta )D\zeta (\tau ^{-1}))|, \end{aligned}$$

which implies (6.7). To prove the inequalities in (6.8) notice that for the function $F(A)=\det A$, considered as the function of the entries $A^{ij}$ of $d\times d$ real matrices A, we have

$$\begin{aligned} \frac{\partial }{\partial A^{ij}}\det A\big |_{A=I}=\delta _{ij}, \quad i,j=1,2,\ldots ,d. \end{aligned}$$

Thus

$$\begin{aligned} \frac{\partial }{\partial \theta }\det (I+\theta D\zeta ^{*})\big |_{\theta =0} =\delta _{ij}D_j\zeta ^{*i}=D_i\zeta ^{*i}, \end{aligned}$$

and by Taylor’s formula we get

$$\begin{aligned} {\mathfrak {c}}=\det (I+D\zeta ^{*})-\det I =\int _0^1 \tfrac{\partial }{\partial A^{ij}}F(I+\theta D\zeta ^{*})\,d\theta D_i\zeta ^{*j} \end{aligned}$$

and

$$\begin{aligned} {\bar{{\mathfrak {c}}}}= & {} \det (I+D\zeta ^{*})-\det I-D_i\zeta ^{*i} \\= & {} \int _0^1(1-\theta )\tfrac{\partial ^2}{\partial A^{ij}\partial A^{kl}} F(I+\theta D\zeta ^{*})\,d\theta D_i\zeta ^{*j}D_k\zeta ^{*l}. \end{aligned}$$

Hence using the estimates in (6.6) we get (6.8). Note that

$$\begin{aligned} \zeta +\zeta ^{*}=\zeta (\tau ^{-1}-\theta \zeta ^{*})\big |^{\theta =1}_{\theta =0} =\zeta ^{*i}\int _0^1(D_i\zeta )(\tau ^{-1}-\theta \zeta ^{*})\,d\theta . \end{aligned}$$

Hence by the second estimate in (6.3) and (6.6) we obtain (6.9). $\square$

In this section for $\varepsilon >0$ and functions v on ${\mathbb {R}}^d$ we use the notation $v^{(\varepsilon )}$ for the convolution of v with $\kappa _{\varepsilon }(\cdot )=\varepsilon ^{-d}\kappa (\cdot /\varepsilon )$, where $\kappa$ is a fixed nonnegative $C_0^{\infty }$ function of unit integral such that $\kappa (x)=0$ for $|x|\geqslant 1$ and $\kappa (-x)=\kappa (x)$ for $x\in {\mathbb {R}}^d$.

Lemma 6.2

Let $\tau$ be an ${\mathbb {R}}^d$-valued function on ${\mathbb {R}}^d$ with uniformly continuous derivative $D\tau$ on ${\mathbb {R}}^d$ such that with positive constants $\lambda$ and K

$$\begin{aligned} \lambda \leqslant |\det D\tau | \quad \text {and}\quad |D\tau |\leqslant M\quad \text {on }{\mathbb {R}}^d. \end{aligned}$$

Then

$$\begin{aligned} \tfrac{1}{2}\lambda \leqslant |\det D\tau ^{(\varepsilon )}| \quad \text {on }{\mathbb {R}}^d \end{aligned}$$

for $\varepsilon \in (0,\varepsilon _0)$ for $\varepsilon _0>0$ satisfying $\delta (\varepsilon _0)\leqslant \lambda /(2d!dM^{d-1})$, where $\delta =\delta (\varepsilon )$ is the modulus of continuity of $D\tau$.

Proof

Clearly,

$$\begin{aligned} \sup _{x\in {\mathbb {R}}^d}|D_{j}\tau ^i-D_{j}\tau ^{i(\varepsilon )}|\leqslant \delta (\varepsilon ) \quad \text {for }\varepsilon >0, i,j=1,2,\ldots ,d. \end{aligned}$$

Hence

$$\begin{aligned}{} & {} \sup _{x\in {\mathbb {R}}^d}|\Pi _{i=1}^dD_{j_i}\tau ^i-\Pi _{i=1}^dD_{j_i}\tau ^{i(\varepsilon )}|\\{} & {} \leqslant \sum _{i=1}^dM^{d-1}\sup _{{\mathbb {R}}^d}|D_{j_i}\tau ^i-D_{j_i}\tau ^{i(\varepsilon )}| \leqslant dM^{d-1}\delta (\varepsilon ) \quad \text {for }\varepsilon >0, \end{aligned}$$

for every permutation $(j_1,\ldots ,j_d)$ of $1,2,\ldots ,d$. Therefore

$$\begin{aligned} \sup _{x\in {\mathbb {R}}^d}|\det D\tau -\det D\tau ^{(\varepsilon )}| \leqslant d!\,dM^{d-1}\delta (\varepsilon ) \quad \text {for }\varepsilon >0. \end{aligned}$$

Consequently, choosing $\varepsilon _0>0$ such that $\delta (\varepsilon _0)\leqslant \lambda /(2d!dM^{d-1})$, for $\varepsilon \in (0,\varepsilon _0)$ we have

$$\begin{aligned} |\det D\tau ^{(\varepsilon )}| \geqslant |\det D\tau |-|\det D\tau -\det D\tau ^{(\varepsilon )}| \geqslant \lambda /2 \quad \text {on }{\mathbb {R}}^d. \end{aligned}$$

$\square$

Corollary 6.3

Let $\zeta$ be an ${\mathbb {R}}^d$-valued function on ${\mathbb {R}}^d$ such that $D\zeta$ is a uniformly continuous function on ${\mathbb {R}}^d$ and

$$\begin{aligned} 0<\lambda \leqslant \inf _{{\mathbb {R}}^d}\det (I+D\zeta ), \quad \sup _{{\mathbb {R}}^d}|D\zeta |\leqslant M<\infty \end{aligned}$$

(6.10)

with some positive constants $\lambda$ and M. Let $\varepsilon _0>0$ such that $\delta (\varepsilon _0) \leqslant \lambda /(2d!dM^{d-1})$. Then for every $\varepsilon \in (0,\varepsilon _0)$ the first inequality in (6.10) holds for $\zeta ^{(\varepsilon )}$ in place of $\zeta$ with $\lambda /2$ in place of $\lambda$. Moreover, $\sup _{{\mathbb {R}}^d}|D^k\zeta ^{(\varepsilon )}|\leqslant M_k$ for every integer k with a constant $M_k=M_k(d, M,\varepsilon )$, where $M_1=M$. Hence Lemma 6.1 holds with $\zeta ^{(\varepsilon )}$ in place of $\zeta$, for $\varepsilon \in (0,\varepsilon _0)$ for every integer $m\geqslant 1$.

Consider for $\varepsilon \in (0,1)$ the equation

$$\begin{aligned} du_t^{\varepsilon }=&{\tilde{{\mathcal {L}}}}_t^{\varepsilon *}u_t^{\varepsilon }\,dt +{\mathcal {M}}_t^{\varepsilon k*}u_t^{\varepsilon }\,dV^k_t +\int _{{\mathfrak {Z}}_0}J_t^{\eta ^{\varepsilon }*}u_t^{\varepsilon }\,\nu _0(d{\mathfrak {z}})dt \nonumber \\&+\int _{{\mathfrak {Z}}_1}J_t^{\xi ^{\varepsilon }*}u_t^{\varepsilon }\,\nu _1(d{\mathfrak {z}})dt +\int _{{\mathfrak {Z}}_1}I_t^{\xi ^{\varepsilon }*} u_t^{\varepsilon }\,{\tilde{N}}_1(d{\mathfrak {z}},dt), \quad \text {with }u_0^{\varepsilon }=\psi ^{(\varepsilon )}, \end{aligned}$$

(6.11)

where

$$\begin{aligned}{} & {} {\mathcal {M}}_t^{\varepsilon k} = \rho ^{(\varepsilon ) ik}_t D_i + B_t^{(\varepsilon )k},\quad k=1,\dots ,d', \\{} & {} {\tilde{{\mathcal {L}}}}_t^{\varepsilon }=a_t^{\varepsilon , ij}D_{ij} +b_t^{(\varepsilon )i}D_i+ \beta ^k_t{\mathcal {M}}^{\varepsilon k}_t, \quad \beta _t=B(t,X_t,Y_t), \\{} & {} a_t^{\varepsilon , ij}:=\tfrac{1}{2}\sum _k (\sigma _t^{(\varepsilon )ik}\sigma _t^{(\varepsilon )jk} +\rho _t^{(\varepsilon )ik}\rho _t^{(\varepsilon )jk}), \quad i,j=1,2,\ldots ,d, \end{aligned}$$

the operators $J_t^{\eta ^{\varepsilon }}$ and $J_t^{\xi ^{\varepsilon }}$ are defined as $J^{\xi }_t$ in (3.1) with $\eta ^{(\varepsilon )}_t$ and $\xi ^{(\varepsilon )}_t$ in place of $\xi _t$, and the operator $I_t^{\xi ^{\varepsilon }}$ is defined as $I_t^{\xi }$ in (3.1) with $\xi _t^{(\varepsilon )}$ in place of $\xi _t$. (Remember that $v^{(\varepsilon )}$ denotes the convolution of functions v in $x\in {\mathbb {R}}^d$, with the kernel $\kappa _{\varepsilon }$ described above.) We define the $L_p$-solution $(u^{\varepsilon }_t)_{t\in [0,T]}$ to (6.11) in the sense of Definition 3.2. Define now for each $\omega \in \Omega$, $t\geqslant 0$ and ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$ the functions

$$\begin{aligned} \tau ^{\eta _t^{\varepsilon }}(x)=x + \eta _t^{(\varepsilon )}(x), \quad \tau ^{\xi _t^{\varepsilon }}(x)=x + \xi _t^{(\varepsilon )}(x), \quad x\in {\mathbb {R}}^d, \end{aligned}$$

(6.12)

where, and later on, we suppress the variables ${\mathfrak {z}}_i$, $i=0,1$.

We recall that for $p\geqslant 1$, ${\mathbb {L}}_p$ denotes the space of $L_p$-valued ${\mathcal {F}}_0$-measurable random variables Z such that ${\mathbb {E}}|Z|_{L_p}^p<\infty$, as well as that for $p,q\geqslant 1$ the notation ${\mathbb {L}}_{p,q}$ stands for the space of $L_p$-valued ${\mathcal {F}}_t$-optional processes $v=(v_t)_{t\in [0,T]}$ such that

$$\begin{aligned} |v|_{{\mathbb {L}}_{p,q}}^p:={\mathbb {E}}\left( \int _0^T|v_t|^q_{L_p}\,dt\right) ^{p/q}<\infty . \end{aligned}$$

Let ${\mathbb {B}}_0$ denote the set of ${\mathcal {F}}_0\otimes {\mathcal {B}}({\mathbb {R}}^d)$-measurable real-valued bounded functions $\psi$ on $\Omega \times {\mathbb {R}}^d$ such that and $\psi (x)=0$ for $|x|\geqslant R$ for some constant $R>0$ depending on $\psi$. It is easy to see that ${\mathbb {B}}_0$ is a dense subspace of ${\mathbb {L}}_p$ for every $p\in [1,\infty )$.

Lemma 6.4

Let Assumptions 2.1, 2.2 and 2.4 hold with $K_1=0$. Assume that the following “support condition" holds: There is some $R>0$ such that

$$\begin{aligned} \big (b_t(x),B_t(x),\sigma _t(x),\rho _t(x), \eta _t(x,{\mathfrak {z}}_0),\xi _t(x,{\mathfrak {z}}_1)\big )=0 \end{aligned}$$

(6.13)

for $\omega \in \Omega$, $t\geqslant 0$, ${\mathfrak {z}}_0\in {\mathfrak {Z}}_0$, ${\mathfrak {z}}_1\in {\mathfrak {Z}}_1$ and $x\in {\mathbb {R}}^{d}$ such that $|x|\geqslant R$. Let $\psi \in {\mathbb {B}}_0$ such that $\psi (x)=0$ if $|x|\geqslant R$. Then there exists an $\varepsilon _0>0$ and a constant ${{\bar{R}}}={\bar{R}}(R,K,K_0,K_\xi ,K_\eta )$ such that the following statements hold for all $m\geqslant 1$ and even integers $p \geqslant 2$.

• (i) For every $\varepsilon \in (0,\varepsilon _0)$ there is an $L_p$-solution $u^\varepsilon = (u^\varepsilon _t)_{t\in [0,T]}$ to (6.11), which is a $W^m_p$-valued weakly cadlag process. Moreover, it satisfies

  $$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|u^\varepsilon _t|_{W^m_p}^p<\infty \quad \text {and}\quad u^\varepsilon _t(x) = 0, \quad \text {for }dx\text {-a.e. }x\in \{x\in {\mathbb {R}}^d:|x|\geqslant {{\bar{R}}}\},\nonumber \\ \end{aligned}$$

  (6.14)

  almost surely for all $t\in [0,T]$.

• (ii) There exists a unique $L_p$-solution $u=(u_t)_{t\in [0,T]}$ to equation (3.9) with initial condition $u_0=\psi$, such that almost surely $u_t(x)=0$ for dx-almost every $x\in \{x\in {\mathbb {R}}^d:|x|\geqslant {{\bar{R}}}\}$ for every $t\in [0,T]$ and

  $$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|u_t|_{L_p}^p\leqslant N{\mathbb {E}}|\psi |_{L_p}^p \end{aligned}$$

  (6.15)

  with a constant $N=N(d,p,T,K, K_{\xi }, K_{\eta }, L,\lambda ,|\bar{\xi }|_{L_2},|\bar{\eta }|_{L_2})$.

• (iii) If $(\varepsilon _n)_{n=1}^\infty \subset (0,\varepsilon _0)$ such that $\varepsilon \rightarrow 0$ then we have

  $$\begin{aligned} u^{\varepsilon _n}\rightarrow u \quad \text {weakly in }{\mathbb {L}}_{p,q}, \text { for every integer }q\geqslant 1. \end{aligned}$$

Proof

We may assume ${\mathbb {E}}|\psi |^p_{L_p}<\infty$. To prove (i), we look for a $W^m_p$-valued weakly cadlag ${\mathcal {F}}_t$-adapted process $(u^{\varepsilon }_t)_{t\in [0,T]}$ such that for each $\varphi \in C_0^{\infty }$ almost surely

$$\begin{aligned} (u_t^{\varepsilon },\varphi )=&(\psi ^{(\varepsilon )},\varphi )+ \int _0^t({\tilde{{\mathcal {L}}}}_s^{\varepsilon *}u_s^{\varepsilon },\varphi )\,ds +\int _0^t({\mathcal {M}}_s^{\varepsilon k*}u_s^{\varepsilon },\varphi )\,dV^k_s \nonumber \\&\quad +\int _0^t\int _{{\mathfrak {Z}}_0} (J_s^{\eta ^{\varepsilon }*}u_s^{\varepsilon },\varphi )\,\nu _0(d{\mathfrak {z}})\,ds \nonumber \\&+\int _0^t\int _{{\mathfrak {Z}}_1} (J_s^{\xi ^{\varepsilon }*}u_s^{\varepsilon },\varphi )\,\nu _1(d{\mathfrak {z}})\,ds +\int _0^t\int _{{\mathfrak {Z}}_1} (I_s^{\xi ^{\varepsilon }*}u_s^{\varepsilon },\varphi )\,{\tilde{N}}_1(d{\mathfrak {z}},ds), \nonumber \\ \end{aligned}$$

(6.16)

holds for all $t\in [0,T]$, where by virtue of (6.2)

$$\begin{aligned} I_s^{\xi ^{\varepsilon }*}{} & {} =I^{\xi _s^{\varepsilon *}} +{\mathfrak {c}}^{\xi _s^{\varepsilon }}T^{\xi _s^{\varepsilon *}}, \quad J^{\xi ^{\varepsilon }*}_s=J^{\xi ^{\varepsilon *}}_s +{\mathfrak {c}}^{\xi _s^{\varepsilon }}I^{\xi _s^{\varepsilon *}} +(\xi _s^{\varepsilon *i}\nonumber \\{} & {} \quad +\xi _s^{(\varepsilon ) i})D_i +{\bar{{\mathfrak {c}}}}^{\xi _t^{\varepsilon }} +D_i(\xi _s^{\varepsilon *i}+\xi _s^{(\varepsilon ) i}),\nonumber \\ J^{\eta ^{\varepsilon }*}_s{} & {} =J^{\eta ^{\varepsilon *}}_s +{\mathfrak {c}}^{\eta _t^{\varepsilon }}I^{\eta _s^{\varepsilon *}} +(\eta _s^{\varepsilon *i}+\eta _s^{(\varepsilon ) i})D_i +{\bar{{\mathfrak {c}}}}^{\eta _s^{\varepsilon }} +D_i(\eta _s^{\varepsilon *i} +\eta _s^{(\varepsilon ) i}),\nonumber \\ \end{aligned}$$

(6.17)

with the functions

$$\begin{aligned} \eta _t^{\varepsilon *}(x)&=-x + (\tau ^{\eta ^{\varepsilon }_t})^{-1}(x), \quad \xi _t^{\varepsilon *}(x)=-x + (\tau ^{\xi _t^{\varepsilon }})^{-1}(x), \nonumber \\ {\mathfrak {c}}^{\xi _t^{\varepsilon }}(x)&=|\det D(\tau ^{\xi _t^{\varepsilon }})^{-1}(x)|-1, \quad {\mathfrak {c}}^{\eta _t^{\varepsilon }}(x)=|\det D(\tau ^{\eta _t^{\varepsilon }})^{-1}(x)|-1,\nonumber \\ {\bar{{\mathfrak {c}}}}^{\xi _t^{\varepsilon }}(x)&=|\det D(\tau ^{\xi _t^{\varepsilon }})^{-1}(x)|-1 -D_i\xi _t^{\varepsilon *i}(x), \nonumber \\ \quad {\bar{{\mathfrak {c}}}}^{\eta _t^{\varepsilon }}(x)&=|\det D(\tau ^{\eta _t^{\varepsilon }})^{-1}(x)| -1-D_i\eta _t^{\varepsilon *i}(x) \quad x\in {\mathbb {R}}^d, \end{aligned}$$

(6.18)

and clearly,

$$\begin{aligned} {\mathcal {M}}_s^{\varepsilon * k}\phi= & {} -D_i(\rho ^{(\varepsilon ) ik}_s \phi ) + B_s^{(\varepsilon )k}\phi , \\ {\tilde{{\mathcal {L}}}}_s^{\varepsilon *}\phi= & {} D_{ij}(a_s^{\varepsilon , ij}\phi ) -D_i(b^{(\varepsilon )i}_s\phi ) -\beta ^k_sD_i(\rho _s^{(\varepsilon )ik}\phi ) +\beta ^k_sB^{(\varepsilon )k}_s\phi \quad \text {for }\phi \in W^m_p. \end{aligned}$$

Note that by Assumption 2.1(i) together with Assumption 2.4(i) & (ii), for each $\omega \in \Omega$, $t\in [0,T]$ and ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$, $i=0,1$, the mappings

$$\begin{aligned} \tau ^\eta (x)=x+\eta _t(x,{\mathfrak {z}}_0),\quad \text {and}\quad \tau ^\xi (x) = x+\xi _t(x,{\mathfrak {z}}_1) \end{aligned}$$

are biLipschitz and continuously differentiable as functions of $x\in {\mathbb {R}}^d$. Hence, as biLipschitz functions admit Lipschitz continuous inverses, it is easy to see that for each $\omega \in \Omega$, $t\in [0,T]$ and ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$, $i=0,1$,

$$\begin{aligned} \lambda '\leqslant \inf _{x\in {\mathbb {R}}^d} |\det D\tau ^\eta (x)|,\quad \text {and}\quad \lambda '\leqslant \inf _{x\in {\mathbb {R}}^d} |\det D\tau ^\xi (x)| \end{aligned}$$

for some $\lambda '=\lambda '(d,\lambda ,L,K_\eta ,K_\xi )$. Due to Assumption 2.4 (i) by virtue of Corollary 6.3 there is $\varepsilon _0\in (0,1)$ such that for $\varepsilon \in (0,\varepsilon _0)$ the functions $\tau ^{\eta _t^{\varepsilon }}$ and $\tau ^{\xi _t^{\varepsilon }}$, defined in (6.12), are $C^{\infty }$-diffeomorphisms on ${\mathbb {R}}^d$ for all $\omega \in \Omega$, $t\in [0,T]$ and ${\mathfrak {z}}_i$, $i=0,1$. Moreover, the functions defined in (6.18) are infinitely differentiable functions in $x\in {\mathbb {R}}^d$, for all $t\in [0,T]$ and ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$, $i=0,1$.

Hence we can easily verify that for each $\varepsilon \in (0,\varepsilon _0)$ equation (6.11) satisfies the conditions of the existence and uniqueness theorem, Theorem 2.1 in [8]. Hence (6.11) has a unique $L_p$-solution $(u^{\varepsilon }_t)_{t\in [0,T]}$ which is weakly cadlag as $W^m_p$-valued process and satisfies the first equation in (6.14), for every $m\geqslant 1$. Due to the support condition (6.13) and that $|\xi |\leqslant K_0K_{\xi }$, $|\eta |\leqslant K_0K_{\eta }$, there is a constant ${{\bar{R}}}={{\bar{R}}}(R, K_0,K,K_{\xi }, K_{\eta })$ such that for $\varepsilon \in (0,\varepsilon _0)$ and $s\in [0,T]$ we have

$$\begin{aligned} \tilde{{\mathcal {L}}}^{\varepsilon }_s\varphi ={\mathcal {M}}_s^{\varepsilon k}\varphi =I_s^{\xi ^{\varepsilon }}\varphi =J_s^{\xi ^{\varepsilon }}\varphi =J_s^{\eta ^{\varepsilon }}\varphi =0, \quad k=1,2,\ldots , d', \end{aligned}$$

for all $\varphi \in C_0^{\infty }$ such that $\varphi (x)=0$ for $|x|\leqslant {{\bar{R}}}$. Thus from equation (6.16) we get that almost surely

$$\begin{aligned} (u_t^{\varepsilon },\varphi )=0 \quad \text {for all }\varphi \in C_0^{\infty }\text { such that }\varphi (x)=0\text { for }|x|\leqslant {{\bar{R}}} \end{aligned}$$

for all $t\in [0,T]$, which implies

$$\begin{aligned} u_t^{\varepsilon }=0 \quad \text {for }dx\text {-almost every }x\in \{x\in {\mathbb {R}}^d, |x|\geqslant {{\bar{R}}}\}\text { for all }t\in [0,T] \end{aligned}$$

(6.19)

for each $\varepsilon \in (0,\varepsilon _0)$. To prove (ii) and (iii), note first that

$$\begin{aligned} \sup _{t\in [0,T]}|u^{\varepsilon }_t|_{L_1} \leqslant {{{\bar{R}}}}^{d(p-1)/p} \sup _{t\in [0,T]}|u^{\varepsilon }_t|_{L_p}<\infty \,\mathrm {(a.s.)}. \end{aligned}$$

It is not difficult to see that $\sigma ^{(\varepsilon )}_t$, $\rho ^{(\varepsilon )}_t$, $b^{(\varepsilon )}_t$ and $B^{(\varepsilon )}_t$ are bounded and Lipschitz continuous in $x\in {\mathbb {R}}^d$, uniformly in $\omega \in \Omega$, $t\in [0,T]$ and $\varepsilon \in (0,\varepsilon _0)$. Moreover, for $\varepsilon \in (0,\varepsilon _0)$

$$\begin{aligned}{} & {} |\eta ^{(\varepsilon )}_t(x,{\mathfrak {z}}_0)| \leqslant K_0\bar{\xi }({\mathfrak {z}}_0), \quad |\xi ^{(\varepsilon )}_t(x,{\mathfrak {z}}_1)| \leqslant K_0\bar{\xi }({\mathfrak {z}}_1), \\{} & {} |\eta ^{(\varepsilon )}_t(x,{\mathfrak {z}}_0)-\eta ^{(\varepsilon )}_t(y,{\mathfrak {z}}_0)| \leqslant \bar{\eta }({\mathfrak {z}}_0)|x-y|, \quad |\xi ^{(\varepsilon )}_t(x,{\mathfrak {z}}_1)-\xi ^{(\varepsilon )}_t(y,{\mathfrak {z}}_1)| \leqslant \bar{\xi }({\mathfrak {z}}_1)|x-y|, \end{aligned}$$

for all $x,y\in {\mathbb {R}}^d$, $\omega \in \Omega$, $t\in [0,T]$, ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$, $i=0,1$. Hence by Lemma 5.5 for $\varepsilon \in (0,\varepsilon _0)$ we have

$$\begin{aligned} {\mathbb {E}}|u^{\varepsilon }_T|_{L_p}^p +{\mathbb {E}}\Big (\int _0^T|u_t^{\varepsilon }|_{L_p}^{q}\,dt \Big )^{p/q} \leqslant {\mathbb {E}}|u^{\varepsilon }_T|_{L_p}^p+T^{p/q}{\mathbb {E}}\sup _{t\in [0,T]} |u_t^{\varepsilon }|_{L_p}^p\leqslant N{\mathbb {E}}|\psi |_{L_p}^p \nonumber \\ \end{aligned}$$

(6.20)

for all $q\geqslant 1$ with a constant $N=N(d,p,T,K, K_{\xi }, K_{\eta },R,|{\bar{\eta }}|_{L_2},|{\bar{\xi }}|_{L_2})$. By virtue of (6.20) there exists a sequence $\varepsilon _n\downarrow 0$ such that $u^{\varepsilon _n}$ converges weakly in ${\mathbb {L}}_{p,q}$ to some ${{\bar{u}}}\in {\mathbb {L}}_{p,q}$ for every integer $q\geqslant 1$ and $u^{\varepsilon _n}_T$ converges weakly to some g in ${\mathbb {L}}_p({\mathcal {F}}_T)$, the space of $L_p$-valued ${\mathcal {F}}_T$-measurable random variables Z with the norm $({\mathbb {E}}|Z|^p_{L_p})^{1/p}<\infty$. From (6.20) we get

$$\begin{aligned} {\mathbb {E}}|g|_{L_p}^p+|{{\bar{u}}}|_{{\mathbb {L}}_{p,q}}^p\leqslant N{\mathbb {E}}|\psi |_{L_p}^p \quad \text {for every integer }q\geqslant 1 \end{aligned}$$

(6.21)

with the constant N in (6.20). Taking $\varepsilon _n$ in place of $\varepsilon$ in equation (6.16) then multiplying both sides of the equation with an ${\mathcal {F}}_t$-optional bounded process $\phi$ and integrating over $\Omega \times [0,T]$ against $P\otimes dt$ we obtain

$$\begin{aligned} F(u^{\varepsilon _n})=F(\psi ^{\varepsilon _n}) + \sum _{i=1}^5 F_i^{\varepsilon _n}(u^{\varepsilon _n}), \end{aligned}$$

(6.22)

where F and $F_i^{\varepsilon }$ are linear functionals over ${\mathbb {L}}_{p,q}$, defined by

$$\begin{aligned}{} & {} F(v):={\mathbb {E}}\int _0^T\phi _t(v_t,\varphi )\,dt, \quad F_1^{\varepsilon } (v):={\mathbb {E}}\int _0^T\phi _t\int _0^t(v_s,{\tilde{{\mathcal {L}}}}_s^{\varepsilon }\varphi )\,ds\,dt, \\{} & {} F_2^{\varepsilon }(v):={\mathbb {E}}\int _0^T\phi _t\int _0^t(v_s,{\mathcal {M}}_s^{\varepsilon k}\varphi )\,dV^k_s\,dt, \\{} & {} F_3^{\varepsilon }(v):={\mathbb {E}}\int _0^T\phi _t\int _0^t\int _{{\mathfrak {Z}}_0} (v_s,J_s^{\eta ^{\varepsilon }}\varphi )\,\nu _0(d{\mathfrak {z}})ds\,dt, \\{} & {} F_4^{\varepsilon }(v):={\mathbb {E}}\int _0^T\phi _t\int _0^t\int _{{\mathfrak {Z}}_1} (v_s,J_s^{\xi ^{\varepsilon }}\varphi )\,\nu _1(d{\mathfrak {z}})ds\,dt, \\{} & {} F_5^{\varepsilon }(v):={\mathbb {E}}\int _0^T\phi _t\int _0^t\int _{{\mathfrak {Z}}_1} (v_s,I_s^{\xi ^{\varepsilon }}\varphi )\,{{\tilde{N}}}_1(d{\mathfrak {z}},ds) \,dt \end{aligned}$$

for a fixed $\varphi \in C_0^{\infty }$. Define also $F_i$ as $F_i^{\varepsilon }$ for $i=1,2,\ldots ,5$, with ${\tilde{{\mathcal {L}}}}_s$, ${\mathcal {M}}_s^k$, $J_s^{\eta }$, $J^{\xi }_s$ and $I_s^{\xi }$ in place of ${\tilde{{\mathcal {L}}}}^{\varepsilon }_s$, ${\mathcal {M}}_s^{\varepsilon k}$, $J_s^{\eta ^{\varepsilon }}$, $J_s^{\xi ^{\varepsilon }}$ and $I_s^{\xi ^{\varepsilon }}$, respectively. It is an easy exercise to show that F and $F_i$ and $F_i^{\varepsilon }$, $i=1,2,3,4,5$ are continuous linear functionals on ${\mathbb {L}}_{p,q}$ for all $q>1$ such that

$$\begin{aligned} \lim _{\varepsilon \downarrow 0} \sup _{|v|_{{\mathbb {L}}_{p,q}} = 1}|F_i(v)-F_i^{\varepsilon }(v)|=0 \quad \text {for every }q>1. \end{aligned}$$

Since $u^{\varepsilon _n}$ converges weakly to ${{\bar{u}}}$ in ${\mathbb {L}}_{p,q}$, and $F^{\varepsilon _n}_i$ converges strongly to $F_i$ in ${\mathbb {L}}^{*}_{p,q}$, the dual of ${\mathbb {L}}_{p,q}$, we get that $F_i^{\varepsilon _n}(u^{\varepsilon _n})$ converges to $F_i({{\bar{u}}})$ for $i=1,2,3,4,5$. Therefore letting $\varepsilon \downarrow 0$ in (6.22) we obtain

(6.23)

Since this equation holds for all bounded ${\mathcal {F}}_t$-optional processes $\phi =(\phi _{t})_{t\in [0,T]}$ and functions $\varphi \in C_0^{\infty }$ we conclude that ${{\bar{u}}}$ is a ${\mathbb {V}}_p$-solution to (3.9). Letting $n\rightarrow \infty$ in equation (6.16) after taking $\varepsilon _n$ in place of $\varepsilon$, T in place of t, multiplying both sides of the equation with an arbitrary ${\mathcal {F}}_T$-measurable bounded random variable $\rho$ and taking expectation we get

$$\begin{aligned} {\mathbb {E}}\rho (g,\varphi ){} & {} ={\mathbb {E}}\rho (\psi ,\varphi ) + {\mathbb {E}}\rho \int _0^T({\bar{u}}_{s},{\tilde{{\mathcal {L}}}}_s\varphi )\,ds \\{} & {} \quad + {\mathbb {E}}\rho \int _0^T({{\bar{u}}}_{s},{\mathcal {M}}_s^{k}\varphi )\,dV^k_s + {\mathbb {E}}\rho \int _0^t\int _{{\mathfrak {Z}}_0}({\bar{u}}_{s},J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})ds \\{} & {} \quad +{\mathbb {E}}\rho \int _0^T\int _{{\mathfrak {Z}}_1}({\bar{u}}_{s},J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})ds +{\mathbb {E}}\rho \int _0^T\int _{{\mathfrak {Z}}_1}({{\bar{u}}}_{s},I_s^{\xi }\varphi )\,{\tilde{N}}_1(d{\mathfrak {z}},ds), \end{aligned}$$

which implies that almost surely

$$\begin{aligned} (g,\varphi )= & {} (\psi ,\varphi ) + \int _0^T({{\bar{u}}}_{s},{\tilde{{\mathcal {L}}}}_s\varphi )\,ds + \int _0^T({{\bar{u}}}_{s},{\mathcal {M}}_s^{k}\varphi )\,dV^k_s\\{} & {} +\int _0^T\int _{{\mathfrak {Z}}_0}({{\bar{u}}}_{s},J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})ds +\int _0^T\int _{{\mathfrak {Z}}_1}({{\bar{u}}}_{s},J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})ds\\{} & {} +\int _0^T\int _{{\mathfrak {Z}}_1}({{\bar{u}}}_{s},I_s^{\xi }\varphi )\,{\tilde{N}}_1(d{\mathfrak {z}},ds). \end{aligned}$$

Letting $q\rightarrow \infty$ in (6.21) we get

$$\begin{aligned} {\mathbb {E}}\mathrm{ess\,sup}_{t\in [0,T]}|{{\bar{u}}}_t|_{L_p}^p\leqslant N{\mathbb {E}}|\psi |_{L_p}^p<\infty . \end{aligned}$$

Consequently, by virtue of Lemma 5.8 we get the existence of a $P\otimes dt$-modification u of ${{\bar{u}}}$, which is an $L_p$-solution to (3.9), and hence

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|u_t|_{L_p}^p\leqslant N{\mathbb {E}}|\psi |_{L_p}^p. \end{aligned}$$

(6.24)

By (6.19) almost surely $u_t=0$ for dx-almost every $x\in \{x\in {\mathbb {R}}^d: |x|\geqslant {{\bar{R}}}\}$, for all $t\in [0,T]$, which due to (6.24) by Hölder’s inequality implies

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|u_t|_{L_1}\leqslant N{\bar{R}}^{d(p-1)/p}{\mathbb {E}}|\psi |_{L_p}<\infty . \end{aligned}$$

Hence by (5.49) in Lemma 5.5 the uniqueness of the $L_p$-solution follows, which completes the proof of the lemma. $\square$

Corollary 6.5

Let Assumptions 2.1, 2.2 and 2.4 hold with $K_1=0$. Assume, moreover, that the“support condition" (6.13) holds for some $R>0$. Then for every $p\geqslant 2$ there is a linear operator ${\mathbb {S}}$ defined on ${\mathbb {L}}_p$ such that ${\mathbb {S}}\psi$ admits a $P\otimes dt$-modification $u=(u_t)_{t\in [0,T]}$ which is an $L_p$-solution to Eq. (3.9) for every $\psi \in {\mathbb {L}}_p$, with initial condition $u_0=\psi$, and

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|u_t|_{L_p}^p\leqslant N{\mathbb {E}}|\psi |_{L_p}^p \end{aligned}$$

(6.25)

with a constant $N= N(d,p,T,K, K_{\xi }, K_{\eta }, L,\lambda ,|\bar{\xi }|_{L_2},|\bar{\eta }|_{L_2})$. Moreover, if $\psi \in {\mathbb {L}}_p$ such that almost surely $\psi (x)=0$ for $|x|\geqslant R$, then there is a constant ${\bar{R}}={\bar{R}}(R,K,K_0, K_{\xi }, K_{\eta })$ such that almost surely $u_t(x)=0$ for dx-a.e. $x\in \{x\in {\mathbb {R}}^d:|x|\geqslant {{\bar{R}}}\}$ for all $t\in [0,T]$.

Proof

If p is an even integer, then the corollary follows from Lemma 6.4. Assume p is not an even integer. Then let $p_0$ be the greatest even integer such that $p_0\leqslant p$ and let $p_1$ be the smallest even integer such that $p\leqslant p_1$. By Lemma 6.4 there are linear operators ${\mathbb {S}}$ and ${\mathbb {S}}_T$ defined on ${\mathbb {B}}_0$ such that ${\mathbb {S}}\psi :=(u_t)_{t\in [0,T]}$ is the unique $L_{p_i}$-solution of equation (3.9) with initial condition $u_0=\psi \in {\mathbb {B}}_0$ and ${\mathbb {S}}_T\psi =u_T$. for $i=0,1$. Moreover, by (6.25) we have

$$\begin{aligned} |{\mathbb {S}}_T\psi |_{{\mathbb {L}}_{p_i}}+|{\mathbb {S}}\psi |_{{\mathbb {L}}_{p_i,q}} \leqslant N|\psi |_{{\mathbb {L}}_{p_i}}\quad \text {for }i=0,1 \end{aligned}$$

for every $q\in [1,\infty )$ with a constant $N= N(d,p,T,K, K_{\xi }, K_{\eta }, L,\lambda ,|\bar{\xi }|_{L_2},|\bar{\eta }|_{L_2})$. Hence by a well-known generalization of the Riesz-Thorin interpolation theorem we have

$$\begin{aligned} |{\mathbb {S}}_T\psi |_{{\mathbb {L}}_{p}}\leqslant N|\psi |_{{\mathbb {L}}_{p}}, \quad |{\mathbb {S}}\psi |_{{\mathbb {L}}_{p,q}}\leqslant N|\psi |_{{\mathbb {L}}_{p}} \quad \text {for every }q\in [1,\infty ), \end{aligned}$$

(6.26)

for $\psi \in {\mathbb {B}}_0$ with a constant $N= N(d,p,T,K, K_{\xi }, K_{\eta }, L,\lambda ,|\bar{\xi }|_{L_2},|\bar{\eta }|_{L_2})$. Assume $\psi \in {\mathbb {L}}_p$. Then there is a sequence $(\psi ^n)_{n=1}^{\infty }\subset {\mathbb {B}}_0$ such that $\psi ^n\rightarrow \psi$ in ${\mathbb {L}}_p$ and $u^n={\mathbb {S}}\psi ^n$ has a $P\otimes dt$-modification, again denoted by $u^{n}=(u_t^{n})_{t\in [0,T]}$ which is an $L_p$-solution for every n with initial condition $u^n_0=\psi ^n$. In particular, for each $\varphi \in C_0^{\infty }$ almost surely

$$\begin{aligned} (u_t^{n},\varphi )&=(\psi ^{n},\varphi )+ \int _0^t(u_s^{n},{\tilde{{\mathcal {L}}}}_s\varphi )\,ds +\int _0^t(u_s^{n},{\mathcal {M}}_s^{k}\varphi )\,dV^k_s \nonumber \\&\quad +\int _0^t\int _{{\mathfrak {Z}}_0} (u_s^n,J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})\,ds \nonumber \\&\quad +\int _0^t\int _{{\mathfrak {Z}}_1} (u_s^{n},J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})\,ds +\int _0^t\int _{{\mathfrak {Z}}_1}(u_s^n,I_s^{\xi }\varphi )\,\tilde{N}_1(d{\mathfrak {z}},ds), \end{aligned}$$

(6.27)

holds for all $t\in [0,T]$. By virtue of (6.26) $u^n$ converges in ${\mathbb {L}}_{p,q}$ to some ${{\bar{u}}}\in {\mathbb {L}}_{p,q}$ for every $q>1$, and $u_T^n$ converges in ${\mathbb {L}}_p$ to some $g\in {\mathbb {L}}_p$. Hence, letting $n\rightarrow \infty$ in equation (6.27) (after multiplying both sides of it with any bounded ${\mathcal {F}}_t$-optional process $\phi =(\phi _t)_{t\in [0,T]}$ and integrating it over $\Omega \times [0,T]$ against $P\otimes dt$) we can see that ${{\bar{u}}}$ is a ${\mathbb {V}}_p$-solution such that (6.26) holds. Letting $n\rightarrow \infty$ in equation (6.27) with $t:=T$ (after multiplying both sides with an arbitrary ${\mathcal {F}}_T$-measurable bounded random variable $\rho$ and taking expectation) we get

$$\begin{aligned} {\mathbb {E}}\rho (g,\varphi )= & {} {\mathbb {E}}\rho (\psi ,\varphi )+ {\mathbb {E}}\rho \int _0^T({\bar{u}}_s,{\tilde{{\mathcal {L}}}}_s\varphi )\,ds +{\mathbb {E}}\rho \int _0^T({\bar{u}}_s,{\mathcal {M}}_s^{k}\varphi )\,dV^k_s \\{} & {} +{\mathbb {E}}\rho \int _0^T\int _{{\mathfrak {Z}}_0} ({\bar{u}}_s,J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})\,ds +{\mathbb {E}}\rho \int _0^T\int _{{\mathfrak {Z}}_1} ({\bar{u}}_s,J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})\,ds \\{} & {} +{\mathbb {E}}\rho \int _0^T\int _{{\mathfrak {Z}}_1}({\bar{u}}_s,I_s^{\xi }\varphi )\,{\tilde{N}}_1(d{\mathfrak {z}},ds), \end{aligned}$$

which implies

$$\begin{aligned} (g,\varphi )= & {} (\psi ,\varphi )+ \int _0^T({\bar{u}}_s,{\tilde{{\mathcal {L}}}}_s\varphi )\,ds +\int _0^T({\bar{u}}_s,{\mathcal {M}}_s^{k}\varphi )\,dV^k_s \\{} & {} +\int _0^T\int _{{\mathfrak {Z}}_0} ({\bar{u}}_s,J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})\,ds +\int _0^T\int _{{\mathfrak {Z}}_1} ({\bar{u}}_s,J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})\,ds\\{} & {} +\int _0^T\int _{{\mathfrak {Z}}_1}({\bar{u}}_s,I_s^{\xi }\varphi )\,{\tilde{N}}_1(d{\mathfrak {z}},ds) \quad (a.s.). \end{aligned}$$

Letting $q\rightarrow \infty$ in (6.26) we get

$$\begin{aligned} {\mathbb {E}}\mathrm{ess\,sup}_{t\in [0,T]}|{\bar{u}}_t|^p_{L_p}\leqslant N|\psi |_{L_p}^p. \end{aligned}$$

Hence by virtue of Lemma 5.8 the process ${\bar{u}}$ has a $P\otimes dt$ modification $u=(u_t)_{t\in [0,T]}$ which is an $L_p$-solution to Eq. (3.9) and (6.25) holds. Finally the last statement of the corollary about the compact support of u can be proved in the same way as it was shown for $u^{\varepsilon }$ in the proof of Lemma 6.4. $\square$

7 Proof of Theorem 2.1

To prove Theorem 2.1 we want to show that for $p\geqslant 2$ Eq. (3.9) has an $L_p$-solution which we can identify as the unnormalised conditional density of the conditional distribution of $X_t$ given the observation $\{Y_s:s\leqslant t\}$. To this end we need some lemmas. To formulate the first one, we recall that ${\mathbb {W}}^m_p$ denotes the space of $W^m_p$-valued ${\mathcal {F}}_0$-measurable random variables Z such that

$$\begin{aligned} |Z|^p_{{\mathbb {W}}^m_p}={\mathbb {E}}|Z|^p_{W^m_p}<\infty . \end{aligned}$$

Lemma 7.1

Let (X, Y) be an ${\mathcal {F}}_0$-measurable ${\mathbb {R}}^{d+d'}$-valued random variable such that the conditional density $\pi =P(X\in dx|Y)/dx$ exists. Assume $(\Omega ,{\mathcal {F}}_0, P)$ is “rich enough" to carry an ${\mathbb {R}}^d$-valued random variable $\zeta$ which is independent of (X, Y) and has a smooth probability density g supported in the unit ball centred at the origin. Then there exists a sequence of ${\mathcal {F}}_0$-measurable random variables $(X_n)_{n=1}^\infty$ such that the conditional density $\pi _n=P(X_n\in dx|Y)/dx$ exists, almost surely $\pi _n(x)=0$ for $|x|\geqslant n+1$ for each n,

$$\begin{aligned} \lim _{n\rightarrow \infty }X_n=X\quad \text {for every }\omega \in \Omega , \end{aligned}$$

and, if $\pi \in {\mathbb {W}}^m_p$ for some $p\geqslant 1$, $m\geqslant 0$, then $\pi _n\in {\mathbb {W}}^m_p$ for every $n\geqslant 1$, and

$$\begin{aligned} \lim _{n\rightarrow \infty }|\pi _n-\pi |_{{\mathbb {W}}^m_p}=0. \end{aligned}$$

Moreover, for every $n\geqslant 1$ we have

$$\begin{aligned} {\mathbb {E}}|X_n|^q\leqslant N(1+{\mathbb {E}}|X|^q) \quad \text {for every }q\in (0,\infty ) \end{aligned}$$

with a constant N depending only on q.

Proof

For $\varepsilon \in (0,1)$ define

$$\begin{aligned} X^{\varepsilon } _{k}:=X\textbf{1}_{|X|\leqslant k}+\varepsilon \zeta \quad \text {for integers }n\geqslant 1. \end{aligned}$$

Let $g_{\varepsilon }$ denote the density function of $\varepsilon \zeta$, and let $\mu _{k}$ be the regular conditional distribution of $Z_{k}:=X\textbf{1}_{|X|\leqslant k}$ given Y. Then

$$\begin{aligned} \mu _{k}^{(\varepsilon )}(x)=\int _{{\mathbb {R}}^d}g_\varepsilon (x-y)\,\mu _{k}(dy) \quad \text {and} \quad \pi ^{(\varepsilon )}(x)=\int _{{\mathbb {R}}^d}g_{\varepsilon }(x-y)\pi (y)dy, \quad x\in {\mathbb {R}}^d, \end{aligned}$$

are the conditional density functions of $X^{\varepsilon }_{k}$ and $X+\varepsilon \zeta$, given Y, respectively. Clearly, if $\pi \in {\mathbb {W}}^m_p$, then $\mu ^{(\varepsilon )}_{k}$ and $\pi ^{{(\varepsilon )}}$ belong to ${\mathbb {W}}^m_p$ for every k and $\varepsilon$. Moreover, by Fubini’s theorem, for each multi-index $\alpha =(\alpha _1,\dots ,\alpha _d)$, such that $0\leqslant |\alpha |\leqslant m$ we have

$$\begin{aligned}{} & {} |D^\alpha \mu ^{(\varepsilon )}_{k}-D^\alpha \pi ^{{(\varepsilon )}}|^p_{{\mathbb {L}}_p} ={\mathbb {E}}\int _{{\mathbb {R}}^d} \Big |\int _{{\mathbb {R}}^d}D^\alpha g_\varepsilon (x-y)\mu _{k}(dy) \nonumber \\{} & {} \quad - \int _{{\mathbb {R}}^d}D^\alpha g_\varepsilon (x-y)\pi (y)dy\Big |^pdx\nonumber \\{} & {} \quad =\int _{{\mathbb {R}}^d} {\mathbb {E}}\Big |\int _{{\mathbb {R}}^d}D^\alpha g_\varepsilon (x-y)\mu _{k}(dy) - \int _{{\mathbb {R}}^d}D^\alpha g_\varepsilon (x-y)\pi (y)dy\Big |^pdx \nonumber \\{} & {} \quad =\int _{{\mathbb {R}}^d} {\mathbb {E}}\big |{\mathbb {E}}(D^\alpha g_\varepsilon (x-Z_k)-D^\alpha g_\varepsilon (x-X)|Y)\big |^p\,dx \nonumber \\{} & {} \quad \leqslant \int _{{\mathbb {R}}^d}{\mathbb {E}}\big |D^\alpha g_\varepsilon (x-Z_{k})-D^\alpha g_\varepsilon (x-X)\big |^p\,dx \nonumber \\{} & {} \quad = {\mathbb {E}}\int _{{\mathbb {R}}^d}\big |D^\alpha g_\varepsilon (x-Z_{k})-D^\alpha g_\varepsilon (x-X)\big |^p\,dx, \end{aligned}$$

(7.1)

where the inequality is obtained by an application of Jensen’s inequality. Clearly, for every $0\leqslant |\alpha |\leqslant m$,

$$\begin{aligned} \int _{{\mathbb {R}}^d}\big |D^\alpha g_\varepsilon (x-Z_{k})-D^\alpha g_\varepsilon (x-X)\big |^p\,dx \leqslant 2^{p} |g_{\varepsilon }|^p_{W^m_p}<\infty \quad \text {for every }\omega \in \Omega \text { and }k\geqslant 1. \end{aligned}$$

Hence by Lebesgue’s theorem on dominated convergence, for each $0\leqslant |\alpha |\leqslant m$,

$$\begin{aligned}{} & {} \lim _{k\rightarrow \infty } {\mathbb {E}}\int _{{\mathbb {R}}^d}\big |D^\alpha g_\varepsilon (x-Z_k)-D^\alpha g_\varepsilon (x-X)\big |^p\,dx\\{} & {} \quad ={\mathbb {E}}\lim _{k\rightarrow \infty }\int _{{\mathbb {R}}^d} \big |D^\alpha g_\varepsilon (x-Z_{k})-D^\alpha g_\varepsilon (x-X)\big |^p\,dx=0. \end{aligned}$$

Consequently, by virtue of (7.1) we have $\lim _{k\rightarrow \infty } |\mu ^{(\varepsilon )}_{k}-\pi ^{(\varepsilon )}|_{{\mathbb {W}}^m_p}=0$ for every $\varepsilon \in (0,1)$. Since almost surely $|\pi ^{(\varepsilon )}-\pi |_{W^m_p}\rightarrow 0$ as $\varepsilon \downarrow 0$, and $|\pi ^{(\varepsilon )}-\pi |_{W^m_p}\leqslant 2|\pi |_{W^m_p}$ for every $\omega \in \Omega$, we have $\lim _{\varepsilon \downarrow 0}|\pi ^{{(\varepsilon )}}-\pi |_{{\mathbb {W}}^m_p}=0$ by Lebesgue’s theorem on dominated convergence. Hence there is a sequence of positive integers $k_n\uparrow \infty$ such that for $\pi _n:=\mu _{k_n}^{ (1/n)}$ we have $\lim _{n\rightarrow \infty }|\pi _n-\pi |_{{\mathbb {W}}^m_p}=0$. Clearly, for $X_n:=X^{\varepsilon _n}_{ k_n}$ with $\varepsilon _n=1/n$ we have $\lim _{n\rightarrow \infty }X_n=X$ for every $\omega \in \Omega$. Moreover, for every integer $n\geqslant 1$

$$\begin{aligned} {\mathbb {E}}|X_{n}|^q \leqslant N\big ({\mathbb {E}}|X\textbf{1}_{|X|\leqslant k_n}|^q + \varepsilon _n^q{\mathbb {E}}|\zeta |^q\big ) \leqslant N({\mathbb {E}}|X|^q+1) \quad \text {for }q\in (0,\infty ) \end{aligned}$$

with a constant $N=N(q)$, which completes the proof of the lemma. $\square$

To formulate our next lemma let $\chi$ be a smooth function on ${\mathbb {R}}$ such that $\chi (r)=1$ for $r\in [-1,1]$, $\chi (r)=0$ for $|r|\geqslant 2$, $\chi (r)\in [0,1]$ and $\chi '(r)=\tfrac{d}{dr}\chi (r)\in [-2,2]$ for all $r\in {\mathbb {R}}$.

Lemma 7.2

Let $b=(b^i)$ be an ${\mathbb {R}}^d$-valued function on ${\mathbb {R}}^m$ such that for a constant L

$$\begin{aligned} |b(v)-b(z)|\leqslant L|v-z| \quad \text {for all }v,z\in {\mathbb {R}}^m. \end{aligned}$$

(7.2)

Then for $b_n(z)=\chi (|z|/n)b(z)$, $z\in {\mathbb {R}}^m$, for integers $n\geqslant 1$ we have

$$\begin{aligned}{} & {} |b_n(z)|\leqslant 2nL+|b(0)|, \quad |b_n(v)-b_n(z)|\leqslant (5L+2|b(0)|)|v-z| \nonumber \\{} & {} \quad \text {for all }v,z\in {\mathbb {R}}^m.\nonumber \\ \end{aligned}$$

(7.3)

Proof

We leave the proof as an easy exercise for the reader. $\square$

We will truncate the functions $\xi$ and $\eta$ in equation (1.1) by the help of the following lemma, in which for each fixed $R>0$ and $\epsilon >0$ we use a function $\kappa ^R_\epsilon$ defined on ${\mathbb {R}}^d$ by

$$\begin{aligned} \kappa ^R_\epsilon (x)=\int _{{\mathbb {R}}^d}\phi ^R_{\epsilon }(x-y)k(y)\,dy, \quad \phi ^R_{\epsilon }(x)={\left\{ \begin{array}{ll} 1,&{} |x|\leqslant R+1,\\ 1+\epsilon \log \big (\tfrac{R+1}{|x|}\big ),&{} R+1<|x|< (R+1)e^{1/\epsilon },\\ 0,&{} |x|\geqslant (R+1)e^{1/\epsilon }, \end{array}\right. } \nonumber \\ \end{aligned}$$

(7.4)

where k is a nonnegative $C^{\infty }$ mapping on ${\mathbb {R}}^d$ with support in $\{x\in {\mathbb {R}}^d:|x|\leqslant 1\}$. Notice that $\kappa ^R_\epsilon \in C^{\infty }_0$ for each $R,\epsilon >0$, such that if $x,y\in {\mathbb {R}}^d$ and $|y|\leqslant |x|$, then

$$\begin{aligned} |\phi ^{R}_{\epsilon }(x)-\phi ^{R}_{\epsilon }(y)|\leqslant \frac{\epsilon |x-y|}{\max (R,|y|)}, \end{aligned}$$

and hence

$$\begin{aligned} |\kappa ^R_\epsilon (x)-\kappa ^R_\epsilon (y)| \leqslant \int _{{\mathbb {R}}^d}|\phi ^{R}_{\epsilon }(x-u)-\phi ^{R}_{\epsilon }(y-u)|k(u)\,du \leqslant \frac{\epsilon |x-y|}{\max (R,|y|-1)}. \nonumber \\ \end{aligned}$$

(7.5)

Lemma 7.3

Let $\xi :{\mathbb {R}}^d\mapsto {\mathbb {R}}^d$ be such that for a constant $L\geqslant 1$ and for every $\theta \in [0,1]$ the function $\tau _\theta (x)=x+\theta \xi (x)$ is L-biLipschitz, i.e.,

$$\begin{aligned} L^{-1}|x-y|\leqslant |\tau _\theta (x)-\tau _\theta (y)|\leqslant L|x-y| \end{aligned}$$

(7.6)

for all $x,y\in \mathbb {{R}}^d$. Then for any $M>L$ and any $R>0$ there is an $\epsilon =\epsilon (L,M,R,|\xi (0)|)>0$ such that with $\kappa ^R:=\kappa ^R_\epsilon$ the function $\xi ^R:=\kappa ^R\xi$ vanishes for $|x|\geqslant {{\bar{R}}}$ for a constant ${\bar{R}}={\bar{R}}(L,M,R,|\xi (0)|)>R$, $|\xi ^R|$ is bounded by a constant $N=N(L,M,R, |\xi (0)|)$, and for every $\theta \in [0,1]$ the mapping

$$\begin{aligned} \tau ^R_\theta (x) = x + \theta \xi ^R(x), \quad x\in {\mathbb {R}}^d \end{aligned}$$

is M-biLipschitz.

Proof

To show $\tau ^R_\theta$ is M-biLipschitz, we first note that if $x,y\in \mathbb {{R}}^d$ with $|x|\geqslant |y|$ then $\tau ^R_\theta (x)-\tau ^R_\theta (y)=A+B$ where $A=\tau _{\theta \kappa ^R(x)}(x)-\tau _{\theta \kappa ^R(x)}(y)$ and $B=\xi (y)(\kappa ^R(x)-\kappa ^R(y))$. The biLipschitz hypothesis (7.6), with $\theta$ replaced by $\theta \kappa ^R(x)$, implies $L^{-1}|x-y|\leqslant |A|\leqslant L|x-y|$. Due to (7.5) and that $\xi$ has linear growth, we can choose a sufficiently small $\epsilon =\epsilon (L,M,R,|\xi (0)|)$ to get $|B|<(L^{-1}-M^{-1})|x-y|$ and hence

$$\begin{aligned} M^{-1}|x-y|\leqslant |\tau ^R_\theta (x)-\tau ^R_\theta (y)|\leqslant M|x-y| \end{aligned}$$

as required. Finally the boundedness of $|\xi ^R|$ follows from the fact that it vanishes for $|x|>Re^{1/\epsilon }$ and that $\xi$ has linear growth. $\square$

Remark 7.1

Note that if $\tau$ is a continuously differentiable L-biLipschitz function on ${\mathbb {R}}^d$ then

$$\begin{aligned} L^{-d}\leqslant |\det (D\tau (x))|\leqslant L^d\quad \text {for }x\in {\mathbb {R}}^d. \end{aligned}$$

Proof

This remark must be well-known, since for $d=1$ it is obvious, and for $d>1$ it can be easily shown by using the singular value decomposition for the matrices $D\tau$, $D\tau ^{-1}$, or by applying Hadamard’s inequality to their determinants. $\square$

Proof of Theorem 2.1

The proof is structured into three steps. First we prove the theorem for the case where $p=2$. As second step we prove the results for all $p\geqslant 2$ for compactly supported coefficients and compactly supported initial conditional densities. The third step then involves an approximation procedure to obtain the desired results for coefficients and initial conditional densities with unbounded support.

Step I: Let Assumptions 2.1, 2.2 and 2.4 hold. Then by Theorem 3.1, the process $(P_t)_{t\in [0,T]}$ of the regular conditional distribution $P_t$ of $X_t$ given ${\mathcal {F}}^Y_t$, and $\mu =(\mu _t)_{t\in [0,T]}=(P_t(^o\!\gamma _t)^{-1})_{t\in [0,T]}$, the “unnormalised" (regular) conditional distribution process, are measure-valued weakly cadlag processes, and $\mu$ is a measure-valued solution to Eq. (3.3). (Recall that $(^o\!\gamma _t)_{t\in [0,T]}$ is the positive normalising process from Remark 3.2.) Assume that $u_0:=P(X_0\in dx|Y_0)/dx$ exists almost surely such that ${\mathbb {E}}|u_0|_{L_p}^p<\infty$ for $p=2$. In order to apply Lemma 5.7 if $K_1\ne 0$, we need to verify that

$$\begin{aligned} G(\mu ) = \sup _{t\in [0,T]} \int _{{\mathbb {R}}^d}|x|^2\,\mu _t(dx)<\infty \quad \text {almost surely.} \end{aligned}$$

(7.7)

For integers $k\geqslant 1$ let $\Omega _k:= \{|Y_0|\leqslant k\}\in {\mathcal {F}}_0^Y$. Then $\Omega _k\uparrow \Omega$ as $k\rightarrow \infty$. Taking $r>2$ from Assumption 2.3, by Doob’s inequality, and by Jensen’s inequality for optional projections we get

$$\begin{aligned}{} & {} {\mathbb {E}}\sup _{t\in [0,T]} \big ({\mathbb {E}}(|X_t|^2\textbf{1}_{\Omega _k}|{\mathcal {F}}_t^Y)\big )^{r/2} \leqslant {\mathbb {E}}\sup _{t\in [0,T]} \big ({\mathbb {E}}(\sup _{s\in [0,T]}|X_s|^2\textbf{1}_{\Omega _k}|{\mathcal {F}}_t^Y)\big )^{r/2} \\{} & {} \quad \leqslant N {\mathbb {E}}\big ({\mathbb {E}}(\sup _{s\in [0,T]}|X_s|^{2}{} \textbf{1}_{\Omega _k}|{\mathcal {F}}_T^Y)\big )^{r/2} \leqslant N {\mathbb {E}}\sup _{s\in [0,T]}|X_s|^{r}{} \textbf{1}_{\Omega _k}, \end{aligned}$$

for all k with a constant N depending only on r. Thus, by Fubini’s theorem and Hölder’s inequality, if $K_1\ne 0$, for all k we have

$$\begin{aligned}{} & {} G_k(\mu ):={\mathbb {E}}\sup _{t\in [0,T]}\int _{{\mathbb {R}}^d}|x|^2\mu _t(dx)\textbf{1}_{\Omega _k} \\{} & {} \quad = {\mathbb {E}}\sup _{t\in [0,T]}{\mathbb {E}}(|X_t|^{2}|{\mathcal {F}}_t^Y)({^o\!\gamma _t})^{-1}{} \textbf{1}_{\Omega _k} = {\mathbb {E}}\sup _{t\in [0,T]}{\mathbb {E}}(|X_t|^{2}{} \textbf{1}_{\Omega _k}|{\mathcal {F}}_t^Y)({^o\!\gamma })_t^{-1}\\{} & {} \quad \leqslant {\mathbb {E}}\sup _{t\in [0,T]}{\mathbb {E}}(|X_t|^{2}{} \textbf{1}_{\Omega _k}|{\mathcal {F}}_t^Y) \sup _{t\in [0,T]}({^o\!\gamma })_t^{-1} \\{} & {} \quad \leqslant \big ({\mathbb {E}}\sup _{t\in [0,T]}\big ({\mathbb {E}}(|X_t|^2\textbf{1}_{\Omega _k}|{\mathcal {F}}_t^Y)\big )^{r/2} \big )^{2/r} \big ({\mathbb {E}}\sup _{t\in [0,T]}({^o\!\gamma }_t)^{-r'}\big )^{1/r'}\\{} & {} \quad \leqslant N \big ( {\mathbb {E}}\sup _{t\in [0,T]}|X_t|^{r}{} \textbf{1}_{\Omega _k}\big )^{2/r}, \end{aligned}$$

where $2/r + 1/r'=1$, $N=N(r,d,C)$ is a constant, and we use that by Jensen’s inequality for optional projections and the boundedness of |B|

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}({^o\!\gamma }_t)^{-r'} \leqslant {\mathbb {E}}\sup _{t\in [0,T]}\gamma _t^{-r'}:=C<\infty \end{aligned}$$

(7.8)

with a constant C only depending on the bound in magnitude of |B| and r. Hence, using (2.1) with $q=r$ we have

$$\begin{aligned} G_k(\mu )\leqslant N\big (1+{\mathbb {E}}\sup _{t\in [0,T]}|X_t|^{r}{} \textbf{1}_{\Omega _k}\big ) \leqslant N'\big (k^{r}+{\mathbb {E}}|X_0|^{r} \big )<\infty , \end{aligned}$$

for constant $N=N(r,d,C)$ and $N' = N'(d,d',r,K,K_0,K_1,K_{\xi }, K_{\eta }, T,|{\bar{\xi }}|_{L_2},|{\bar{\eta }}|_{L_2})$. Since for all $k\geqslant 1$ we have that $G_k(\mu ) <\infty$ we can conclude that (7.7) holds. Hence, by Lemma 5.7, almost surely $d\mu _t/dx$ exists, and there is an $L_2$-valued weakly cadlag stochastic process $(u_t)_{t\in [0,T]}$ such that almost surely $u_t=d\mu _t/dx$ for all $t\in [0,T]$ and

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|u_t|_{L_2}^2\leqslant N{\mathbb {E}}|\pi _0|_{L_2}^2 \end{aligned}$$

(7.9)

for every T with a constant $N=N(d, d',K,K_\xi , K_\eta , L, T,|{\bar{\xi }}|_{L_2}, |{\bar{\eta }}|_{L_2},\lambda )$. Thus $\pi _t=dP_t/dx=u_t{^o\!\gamma _t}$, $t\in [0,T]$, is an $L_2$-valued weakly cadlag process, which proves Theorem 2.1 for $p=2$.

Step II. Let the assumptions of Theorem 2.1 hold with $K_1=0$ in Assumption 2.1. Assume that $\pi _0=P(X_0\in dx|Y_0)/dx\in {\mathbb {L}}_p$ for some $p>2$, such that almost surely $u_0(x)=0$ for $|x|\geqslant R$ for a constant R. Assume moreover, that the support condition (6.13) holds. Then by Corollary 6.5 there is an $L_p$-solution $(v_t)_{t\in [0,T]}$ to (3.9) with initial condition $v_0=\pi _0$ such that

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|v_t|^p_{L_p}\leqslant N{\mathbb {E}}|\psi |^p_{L_p} \end{aligned}$$

(7.10)

with a constant $N=N(d, d',K, L, K_{\xi }, K_{\eta }, T, p,\lambda ,|{\bar{\xi }}|_{L_2}, |{\bar{\eta }}|_{L_2})$, and almost surely

$$\begin{aligned} v_t(x)=0 \quad \text {for }dx\text {-a.e. }x\in \{x\in {\mathbb {R}}^d:|x|\geqslant {{\bar{R}}}\}\text { for all }t\in [0,T] \end{aligned}$$

with a constant ${{\bar{R}}}={{\bar{R}}}(R,K,K_0,K_\xi ,K_\eta )$. Hence $(v_t)_{t\in [0,T]}$ is also an $L_2$-solution to Eq. (3.9), and clearly,

$$\begin{aligned} \sup _{t\in [0,T]}|v_t|_{L_1} \leqslant {\bar{R}}^{d(p-1)/p}\sup _{t\in [0,T]}|v_t|_{L_p}<\infty . \end{aligned}$$

Since in particular ${\mathbb {E}}|\pi _0|^2_{L_2}<\infty$, by Step I there is an $L_2$-solution $(u_t)_{t\in [0,T]}$ to equation (3.9) such that almost surely $u_t=d\mu _t/dx$ for all $t\in [0,T]$, where $\mu _t=P_t(^o\!\,\gamma _t)^{-1}$ is the unnormalised (regular) conditional distribution of $X_t$ given ${\mathcal {F}}^Y_t$. Clearly,

$$\begin{aligned} \sup _{t\in [0,T]}|u_t|_{L_1}=\sup _{t\in [0,T]}(^o\!\gamma _t)^{-1}<\infty \,(\text {a.s.}). \end{aligned}$$

Hence by virtue of (5.49) in Lemma 5.5 we obtain $\sup _{t\in [0,T]}|u_t-v_t|_{L_2}=0$ (a.s.), which completes the proof of Theorem 2.1 under the additional assumptions of Step II.

Step III. Finally, we dispense with the assumption that the coefficients and the initial condition are compactly supported, and that $K_1=0$ in Assumption 2.1. Define the functions $b^n=(b^{ni}(t,z))$, $B^n = (B^{nj}(t,z))$, $\sigma ^n=(\sigma ^{nij}(t,z))$, $\eta ^n=(\eta ^{ni}(t,z,{\mathfrak {z}}_0))$ and $\xi ^n=(\xi ^{ni}(t,z,{\mathfrak {z}}_1))$ by

$$\begin{aligned} (b^n, B^n,\sigma ^n, \rho ^n) = (b, B,\sigma , \rho )\chi _n,\quad (\eta ^n,\xi ^n)=(\eta ,\xi ){\bar{\chi }}_n \end{aligned}$$

for every integer $n\geqslant 1$, where $\chi _n(z)=\chi (|z|/n)$ and ${\bar{\chi }}_n(x,y)=\kappa ^n(x)\chi (|y|/n)$, with $\chi$ defined before Lemma 7.2 and with $\kappa ^n$ stemming from Lemma 7.3 applied to $\xi$ and $\eta$ as functions of $x\in {\mathbb {R}}^d$. By Lemma 7.2, Assumptions 2.1 and 2.2 hold for $b^n$, $B^n$, $\sigma ^n$, $\rho ^n$, $\eta ^n$ and $\xi ^n$, in place of b, $\sigma$, $\rho$, $\eta$ and $\xi$, respectively, with $K_1=0$ and with appropriate constants $K_0'=K_0'(n,K,K_0,K_1,K_\eta ,K_\xi ,L))$ and $L'=L'(K,K_0, K_1, L,K_{\xi },K_{\eta })$ in place of $K_0$ and L. Moreover, by Lemma 7.3, Assumption 2.4 is satisfied with a constant $\lambda '=\lambda '(K_0,K_1,K_\xi ,K_\eta ,\lambda )$ in place of $\lambda$. Since $\pi _0=P(X_0\in dx|Y_0)/dx\in {\mathbb {L}}_p$ for $p>2$ by assumption (the case $p=2$ was proved in Step I) and clearly $\pi _0\in {\mathbb {L}}_1$, by Hölder’s inequality we have

$$\begin{aligned} |\pi _0|_{{\mathbb {L}}_2}\leqslant |\pi _0|^{1-\theta }_{{\mathbb {L}}_1}|\pi _0|^{\theta }_{{\mathbb {L}}_p}<\infty \quad \text {with }\theta =\tfrac{p}{2(p-1)}\in (0,1). \end{aligned}$$

Thus by Lemma 7.1 there exists a sequence $(X_0^n)_{n=1}^{\infty }$ of ${\mathcal {F}}_0$-measurable random variables such that the conditional density $\pi _0^n = P(X_0^n\in dx|{\mathcal {F}}_0^Y)/dx$ exists, $\pi _0^n(x)=0$ for $|x|\geqslant n+1$ for every n, $\lim _{n\rightarrow \infty }X_0^n=X_0$ for every $\omega \in \Omega$,

$$\begin{aligned} \lim _{n\rightarrow \infty }|\pi _0^n-\pi _0|_{{\mathbb {L}}_r}=0 \quad \text {for }r=2,p, \end{aligned}$$

(7.11)

and

$$\begin{aligned} {\mathbb {E}}|X^n_0|^{q}\leqslant N(1+{\mathbb {E}}|X_0|^{q}) \quad \text {for any }q>0 \end{aligned}$$

with a constant $N=N(q)$. Let $(X^n_t,Y^n_t)_{t\in [0,T]}$ denote the solution of equation (1.1) with initial value $(X_0^n,Y_0)$ and with $b^n$, $\sigma ^n$, $\rho ^n$, $\xi ^n$, $\eta ^n$ and $B^n$ in place of b, $\sigma$, $\rho$, $\xi$, $\eta$ and B. Define the random fields,

$$\begin{aligned} b^n_t(x)= & {} b^n(t,x,Y^n_{t-}), \quad \sigma ^n_t(x)=\sigma ^n(t,x,Y^n_{t-}), \quad \rho ^n_t(x)=\sigma ^n(t,x,Y^n_{t-}), \nonumber \\{} & {} \quad B^n_t(x)=B^n(t,x,Y^n_{t-})\nonumber \\ \eta ^n_t(x,{\mathfrak {z}}_0)= & {} \eta ^n(t,x,Y^n_{t-}, {\mathfrak {z}}_0), \quad \xi ^n_t(x,{\mathfrak {z}}_1)=\xi ^n(t,x,Y^n_{t-},{\mathfrak {z}}_1), \quad \beta ^n_t=B^n(t,X^n_t,Y_{t-}^n)\nonumber \\ \end{aligned}$$

(7.12)

for $\omega \in \Omega$, $t\geqslant 0$, $x\in {\mathbb {R}}^d$, ${\mathfrak {z}}_i\in {\mathfrak {Z}}_i$, $i=0,1$. Consider the equation

$$\begin{aligned} du_t^{n}=&{\tilde{{\mathcal {L}}}}_t^{n*}u_t^{n}\,dt +{\mathcal {M}}_t^{n k*}u_t^{n}\,dV^k_t +\int _{{\mathfrak {Z}}_0}J_t^{\eta ^{n}*}u_t^{n}\,\nu _0(d{\mathfrak {z}})dt \nonumber \\&+\int _{{\mathfrak {Z}}_1}J_t^{\xi ^{n}*}u_t^{n}\,\nu _1(d{\mathfrak {z}})dt +\int _{{\mathfrak {Z}}_1}I_t^{\xi ^{n}*}u_t^{n}\,{{\tilde{N}}}_1(d{\mathfrak {z}},dt), \quad \text {with }u_0^{n}=\pi _0^{n}, \end{aligned}$$

(7.13)

where for each fixed n and $k=1,2,\ldots ,d'$

$$\begin{aligned}{} & {} {\tilde{{\mathcal {L}}}}_t^{n}:=a_t^{n ij}D_{ij} +b_t^{ni}D_i+ \beta ^{nk}_t\rho _t^{nik}D_i +\beta _t^{nk}B_t^{nk}, \quad {\mathcal {M}}^{nk}_t:=\rho _t^{nik}D_i+B^{nk}_t, \\{} & {} a_t^{n ij}:=\tfrac{1}{2}\sum _k\sigma _t^{nik}\sigma _t^{njk} +\tfrac{1}{2}\sum _k \rho _t^{nik}\rho _t^{njk}, \quad \beta ^n_t:=B^n(t,X^n_{t-},Y^n_{t-}), \quad i,j=1,2,\ldots ,d, \end{aligned}$$

the operators $J_t^{\eta ^{n}}$ and $J_t^{\xi ^{n}}$ are defined as $J^{\xi }_t$ in (3.1) with $\eta ^{n}_t$ and $\xi ^{n}_t$ in place of $\eta _t$ and $\xi _t$, respectively, and the operator $I_t^{\xi ^{n}}$ is defined as $I_t^{\xi }$ in (3.1) with $\xi _t^{n}$ in place of $\xi _t$. For each n let $\gamma ^n$ denote the solution to $d\gamma ^n_t=-\gamma ^n_t\beta ^n_t\,dV_t$, $\gamma ^n_0=1$. By virtue of Step II (7.13) has an $L_p$-solution $u^n=(u^n_t)_{t\in [0,T]}$, which is also its unique $L_2$-solution, i.e., for each $\varphi \in C_0^{\infty }$ almost surely

$$\begin{aligned} (u_t^{n},\varphi )=&(\pi ^n_0,\varphi )+\int _0^t(u_s^{n},{\tilde{{\mathcal {L}}}}_s^{n}\varphi )\,ds +\int _0^t(u_s^{n},{\mathcal {M}}_s^{n k}\varphi )\,dV^k_s\nonumber \\&+\int _0^t\int _{{\mathfrak {Z}}_0}(u_s^{n},J_s^{\eta ^{n}}\varphi )\,\nu _0(d{\mathfrak {z}})ds \nonumber \\&+\int _0^t\int _{{\mathfrak {Z}}_1}(u_s^{n},J_s^{\xi ^{n}}\varphi )\,\nu _1(d{\mathfrak {z}})ds +\int _0^t\int _{{\mathfrak {Z}}_1}(u_s^{n},I_s^{\xi ^{n}}\varphi )\,{\tilde{N}}_1(d{\mathfrak {z}},ds) \end{aligned}$$

(7.14)

for all $t\in [0,T]$. Moreover, almost surely $u^n_t = d\mu ^n_t/dx$ for all $t\in [0,T]$, where $\mu ^n_t=P^n_t(^o\!\gamma ^n_t)^{-1}$ is the unnormalised conditional distribution, $P^n_t$ is the regular conditional distribution of $X_t^n$ given ${\mathcal {F}}^{Y^n}_t$, and $^o\!\gamma ^n$ denotes the ${\mathcal {F}}_t^{Y^n}$-optional projection of $\gamma ^n$ under P. Furthermore, for sufficiently large n we have

$$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|u^n_t|_{L_r}^r\leqslant N|\pi _0^n|_{{\mathbb {L}}_r}\quad \text {for }r=p,2 \end{aligned}$$

(7.15)

with a constant $N=N(d, d',p,K, K_{\xi }, K_{\eta }, L, T,|{\bar{\xi }}|_{L_2}, |{\bar{\eta }}|_{L_2},\lambda )$, which together with (7.11) implies

$$\begin{aligned} \sup _{n\geqslant 1}(|u^n_T|_{{\mathbb {L}}_r}+|u^n|_{{\mathbb {L}}_{r,q}})<\infty \quad \text {for }r=2,p \text {and every }q>1. \end{aligned}$$

Hence there exist a subsequence, denoted again by $(u^n)_{n=1}^\infty$, ${{\bar{u}}}\in \bigcap _{q=2}^{\infty }{\mathbb {L}}_{r,q}$ and $g\in {\mathbb {L}}_r$ for $r=2,p$ such that

$$\begin{aligned} u^n\rightarrow {{\bar{u}}} \quad \text {weakly in }{\mathbb {L}}_{r,q} \text {for }r=p,2 \text {and all integers }q>1, \end{aligned}$$

(7.16)

and

$$\begin{aligned} u^n_T\rightarrow g\quad \text {weakly in }{\mathbb {L}}_{r} \text {for }r=p,2. \end{aligned}$$

(7.17)

One knows, see e.g. [5], that $(X^n_t,Y^n_t)_{t\geqslant 0}$ converges to $(X_t,Y_t)_{t\geqslant 0}$ in probability, uniformly in t in finite intervals. Hence it is not difficult to show (see Lemma 3.8 in [7]) that there is a subsequence of $Y^n$, denoted for simplicity also by $Y^n$, and there is an ${\mathcal {F}}_t$-adapted cadlag process $(U_t)_{t\in [0,T]}$, such that almost surely $|Y^n_t|+|Y_t|\leqslant U_t$ for every $t\in [0,T]$ and integers $n\geqslant 1$. For every integer $m\geqslant 1$ define the stopping time

$$\begin{aligned} \tau _m=\inf \{t\in [0,T]: U_t\geqslant m\} \end{aligned}$$

To show that ${{\bar{u}}}$ is a ${\mathbb {V}}_r$-solution for $r=p,2$ to (3.9) with initial condition $u_0=\pi _0$, we pass to the limit $u^n\rightarrow {{\bar{u}}}$ in equation (7.14) in a similar way to that as we passed to the limit $u^{\varepsilon _n}\rightarrow {{\bar{u}}}$ in equation (6.16) in the proof of Lemma 6.4. We fix an integer $m\geqslant 1$ and multiply both sides of (7.14) with $(\phi _t\textbf{1}_{t\leqslant \tau _m})_{t\in [0,T]}$, where $(\phi _t)_{t\in [0,T]}$ is an arbitrary bounded ${\mathcal {F}}_t$-optional process $\phi =(\phi _t)_{t\in [0,T]}$. Then we integrate both sides of the equation we obtained over $\Omega \times [0,T]$ against $P\otimes dt$ to get

$$\begin{aligned} F(u^n)=F(\pi ^n_0) + \sum _{i=1}^5 F_i^n(u^n), \end{aligned}$$

(7.18)

where F and $F_i^{n}$ are linear functionals over ${\mathbb {L}}_{r,q}$, defined by

$$\begin{aligned}{} & {} F(v):={\mathbb {E}}\int _0^{T\wedge \tau _m}\phi _t(v_t,\varphi )\,dt, \quad F_1^{n} (v):={\mathbb {E}}\int _0^{T\wedge \tau _m}\phi _t\int _0^t(v_s,{\tilde{{\mathcal {L}}}}_s^{n}\varphi )\,ds\,dt,\\{} & {} F_2^{n}(v):={\mathbb {E}}\int _0^{T\wedge \tau _m}\phi _t\int _0^t(v_s,{\mathcal {M}}_s^{n k}\varphi )\,dV^k_s\,dt, \\{} & {} F_3^{n}(v):={\mathbb {E}}\int _0^{T\wedge \tau _m}\phi _t\int _0^t\int _{{\mathfrak {Z}}_0} (v_s,J_s^{\eta ^{n}}\varphi )\,\nu _0(d{\mathfrak {z}})ds\,dt,\\{} & {} F_4^{n}(v):={\mathbb {E}}\int _0^{T\wedge \tau _m}\phi _t\int _0^t\int _{{\mathfrak {Z}}_1} (v_s,J_s^{\xi ^{n}}\varphi )\,\nu _1(d{\mathfrak {z}})ds\,dt,\\{} & {} F_5^{n}(v):={\mathbb {E}}\int _0^{T\wedge \tau _m}\phi _t\int _0^t\int _{{\mathfrak {Z}}_1} (v_s,I_s^{\xi ^{n}}\varphi )\,{{\tilde{N}}}_1(d{\mathfrak {z}},ds) \,dt \end{aligned}$$

for a fixed $\varphi \in C_0^{\infty }$. Define also $F_i$ as $F_i^{n}$ for $i=1,2,\ldots ,5$, with ${\tilde{{\mathcal {L}}}}_s$, ${\mathcal {M}}_s^k$, $J_s^{\eta }$, $J^{\xi }_s$ and $I_s^{\xi }$ in place of ${\tilde{{\mathcal {L}}}}^{n}_s$, ${\mathcal {M}}_s^{n k}$, $J_s^{\eta ^{n}}$, $J_s^{\xi ^{n}}$ and $I_s^{\xi ^{n}}$, respectively. It is an easy exercise to show that hat F and $F_i^{n}$, $i=1,2,3,4,5$, are continuous linear functionals on ${\mathbb {L}}_{r,q}$ for $r=p,2$ and all $q>1$. We are going to show now that for $r=p,2$

$$\begin{aligned} \lim _{n\rightarrow \infty } \sup _{|v|_{{\mathbb {L}}_{r,q}} = 1}|F_i(v)-F_i^{n}(v)|=0 \quad \text {for every} q>1, \text {for}\, i=1,2,\ldots ,5.\nonumber \\ \end{aligned}$$

(7.19)

Let $r'=r/(r-1)$, $q'=q/(q-1)$. Then for $v\in {\mathbb {L}}_{r,q}$ by Hölder’s inequality we have

$$\begin{aligned} |F_1(v)-F^n_1(v)| \leqslant KT|v|_{{\mathbb {L}}_{r,q}} |({\tilde{{\mathcal {L}}}}-{\tilde{{\mathcal {L}}}}^n)\varphi |_{{\mathbb {L}}_{r',q'}} \end{aligned}$$

(7.20)

with $K=\sup _{\omega \in \Omega }\sup _{t\in [0,T]}|\phi _t|<\infty$. Clearly, $\lim _{n\rightarrow \infty }({\tilde{{\mathcal {L}}}}_s-{\tilde{{\mathcal {L}}}}^n_s)\varphi (x)=0$ almost surely for all $s\in [0,T]$ and $x\in {\mathbb {R}}^d$, and there is a constant N independent of n and m such that

$$\begin{aligned} |({\tilde{{\mathcal {L}}}}_s-{\tilde{{\mathcal {L}}}}^n_s)\varphi (x)| \leqslant N(1+|x|^2+2m^2)\textbf{1}_{|x|\leqslant R} \end{aligned}$$

(7.21)

for $\omega \in \Omega$, $s\in [0,T\wedge \tau _m]$ and $x\in {\mathbb {R}}^d$, where R is the diameter of the support of $\varphi$. Hence a repeated application of Lebesgue’s theorem on dominated convergence gives

$$\begin{aligned} \lim _{n\rightarrow \infty }|({\tilde{{\mathcal {L}}}}-{\tilde{{\mathcal {L}}}}^{n})\varphi |_{{\mathbb {L}}_{r',q'}}=0, \end{aligned}$$

and by (7.20) proves (7.19) for $i=1$. By the Davis inequality and Hölder’s inequality we have

$$\begin{aligned} |F_2(v)-F^n_2(v)| \leqslant 3KT {\mathbb {E}}\left( \int _0^{T\wedge \tau _m}\sum _k|(v_s,({\mathcal {M}}^k_s-{\mathcal {M}}^{nk}_s)\varphi )|^2\,ds \right) ^{1/2} \leqslant C_n^{(2)}|v|_{{\mathbb {L}}_{r,q}} \end{aligned}$$

with

$$\begin{aligned} C_n^{(2)}=3KT\Big ({\mathbb {E}}\Big ( \int _0^{T\wedge \tau _m} \Big ( \sum _k |({\mathcal {M}}^k_s-{\mathcal {M}}^{nk}_s)\varphi |^{2}_{L_{r'}}\Big )^{q/(q-2)}\,ds \Big )^{r'(q-2)/2q}\Big )^{1/r'}. \end{aligned}$$

Clearly, $\lim _{n\rightarrow \infty }({\mathcal {M}}^k_s-{\mathcal {M}}^{nk}_s)\varphi (x)=0$, and with a constant N independent of n and m we have

$$\begin{aligned} \sum _k|({\mathcal {M}}^k_s-{\mathcal {M}}^{nk}_s)\varphi (x)| \leqslant \sup _{s\in [0,T]}N(1+|x|+2m)\textbf{1}_{|x|\leqslant R} \end{aligned}$$

for all $\omega \in \Omega$, $s\in [0,T\wedge \tau _m]$ and $x\in {\mathbb {R}}^d$. Thus repeating the above argument we obtain (7.19) for $i=2$. By Hölder’s inequality we have

$$\begin{aligned} |F^3_n(v)-F^3(v)|\leqslant KT|v|_{{\mathbb {L}}_{r,q}}C^{(3)}_n \end{aligned}$$

with

$$\begin{aligned} C^{(3)}_n= \Big ( {\mathbb {E}}\Big (\int _0^{T\wedge \tau _m}\big |\int _{{\mathfrak {Z}}_0} |(J_s^{\eta }-J_s^{\eta ^n})\varphi |_{L_{r'}}\nu _0(d{\mathfrak {z}})\big |^{q'}\,ds \Big )^{r'/q'} \Big )^{1/r'}, \end{aligned}$$

where we have suppressed the variable ${\mathfrak {z}}\in {\mathfrak {Z}}_0$ in the integrand. Clearly,

$$\begin{aligned} \lim _{n\rightarrow \infty }(J^{\eta }-J^{\eta ^n})\varphi (x)=0 \quad \text {almost surely for all }s\in [0,T], x\in {\mathbb {R}}^d\text { and }{\mathfrak {z}}\in {\mathfrak {Z}}_0. \end{aligned}$$

By Taylor’s formula

$$\begin{aligned}{} & {} |J_s^{\eta ^n}\varphi (x)| \leqslant \sup _{\theta \in [0,1]}|D^2\varphi (x+\theta \eta ^n_s(x,{\mathfrak {z}}))||\eta _s(x,{\mathfrak {z}})|^2, \\{} & {} |J_s^{\eta }\varphi (x)| \leqslant \sup _{\theta \in [0,1]}|D^2\varphi (x+\theta \eta _s(x,{\mathfrak {z}}))||\eta _s(x,{\mathfrak {z}})|^2, \end{aligned}$$

and by Lemma 7.3 with $\lambda '$ from above we have

$$\begin{aligned}{} & {} \lambda '|x|\leqslant |x+\theta (\eta ^n_s(x,{\mathfrak {z}})-\eta ^n_s(0,{\mathfrak {z}}))| \leqslant |x+\theta \eta ^n_s(x,{\mathfrak {z}})|+|\eta ^n_s(0,{\mathfrak {z}})|,\\{} & {} \lambda '|x|\leqslant |x+\theta (\eta _s(x,{\mathfrak {z}})-\eta _s(0,{\mathfrak {z}}))| \leqslant |x+\theta \eta _s(x,{\mathfrak {z}})|+|\eta _s(0,{\mathfrak {z}})| \end{aligned}$$

for all $\theta \in [0,1]$, $\omega \in \Omega$, $s\in [0,T\wedge \tau _m]$. Hence, taking into account the the linear growth condition on $\eta$, see Assumption (2.1) (ii), for any given $R>0$ we have a constant ${\tilde{R}}={\tilde{R}}(R,K_0,K_1,K_\eta ,m)>R$ such that

$$\begin{aligned} |x+\theta \eta _s(x,{\mathfrak {z}})|\geqslant R, \quad |x+\theta \eta ^n_s(x,{\mathfrak {z}})|\geqslant R \quad \text {for }|x|\geqslant {\tilde{R}}, \end{aligned}$$

for all $\theta \in [0,1]$, $\omega \in \Omega$, $s\in [0,T\wedge \tau _m]$. Taking R such that $\varphi (x)=0$ for $|x|\geqslant R$ we have

$$\begin{aligned} |J_s^{\eta ^n}\varphi (x)-J_s^{\eta }\varphi (x)| \leqslant |J_s^{\eta ^n}\varphi (x)|+|J_s^{\eta }\varphi (x)| \leqslant 2\sup _{x\in {\mathbb {R}}^d}|D^2\varphi (x)|{\bar{\eta }}^2({\mathfrak {z}})\textbf{1}_{|x|\leqslant {\tilde{R}}} \end{aligned}$$

for $x\in {\mathbb {R}}$, $\omega \in \Omega$, $s\in [0,T\wedge \tau _m]$ and ${\mathfrak {z}}\in {\mathfrak {Z}}_0$. Hence by Lebesgue’s theorem on dominated convergence $\lim _{n\rightarrow \infty }C_n^{(3)}=0$ which gives (7.19) for $i=3$. We get (7.19) for $i=4$ in the same way. By the Davis inequality and Hölder’s inequality we have

$$\begin{aligned} |F_5(v)-F^n_5(v)| \leqslant 3KT {\mathbb {E}}\Big ( \int _0^{T\wedge \tau _m}\int _{{\mathfrak {Z}}_1}|(v_s,(I^{\xi ^n}_s-I^{\xi }_s)\varphi )|^2\nu _1(d{\mathfrak {z}})\,ds \Big )^{1/2} \leqslant C^{(5)}_n|v|_{{\mathbb {L}}_{r,q}} \end{aligned}$$

with

$$\begin{aligned} C^{(5)}_n = 3KT\Big ({\mathbb {E}}\Big ( \int _0^{T\wedge \tau _m} \Big ( \int _{{\mathfrak {Z}}_1} |(I^{\xi ^n}_s-I^{\xi }_s)\varphi |^{2}_{L_{r'}} \nu _1(d{\mathfrak {z}})\Big )^{q/(q-2)}\,ds \Big )^{r'(q-2)/2q}\Big )^{1/r'}. \end{aligned}$$

Clearly, $\lim _{n\rightarrow \infty }(I^{\xi ^n}_s-I^{\xi }_s)\varphi (x)=0$ almost surely for all $s\in [0,T]$, $x\in {\mathbb {R}}^d$ and ${\mathfrak {z}}\in {\mathfrak {Z}}_1$. By Taylor’s formula

$$\begin{aligned}{} & {} |I_s^{\xi ^n}\varphi (x)| \leqslant \sup _{\theta \in [0,1]}|D\varphi (x+\theta \xi ^n_s(x,{\mathfrak {z}}))||\xi _s(x,{\mathfrak {z}})|, \\{} & {} |I_s^{\xi }\varphi (x)| \leqslant \sup _{\theta \in [0,1]}|D\varphi (x+\theta \xi _s(x,{\mathfrak {z}}))||\xi _s(x,{\mathfrak {z}})|. \end{aligned}$$

Hence, using Assumptions 2.1, 2.2 and 2.4 in the same way as above, we get a constant ${\tilde{R}}={{\tilde{R}}}(R,K_0,K_1,K_\eta ,m)$ such that

$$\begin{aligned} |I_s^{\xi ^n}\varphi (x)-I_s^{\xi }\varphi (x)| \leqslant |I_s^{\xi ^n}\varphi (x)|+|I_s^{\xi }\varphi (x)| \leqslant 2\sup _{x\in {\mathbb {R}}^d}|D\varphi (x)|{\bar{\xi }}({\mathfrak {z}})\textbf{1}_{|x|\leqslant \tilde{R}} \end{aligned}$$

for $x\in {\mathbb {R}}$, $\omega \in \Omega$, $s\in [0,T\wedge \tau _m]$ and ${\mathfrak {z}}\in {\mathfrak {Z}}_0$. Consequently, by Lebesgue’s theorem on dominated convergence we obtain (7.19) for $i=5$, which completes the proof of (7.19). Since $u^{n}$ converges weakly to ${{\bar{u}}}$ in ${\mathbb {L}}_{r,q}$, and $F^{n}_i$ converges strongly to $F_i$ in ${\mathbb {L}}^{*}_{p,q}$, the dual of ${\mathbb {L}}_{p,q}$, we get that $F_i^{n}(u^{n})$ converges to $F_i({{\bar{u}}})$ for for $i=1,2,3,4,5$. Therefore letting $n\rightarrow \infty$ in (7.18) we obtain

Since this equation holds for all bounded ${\mathcal {F}}_t$-optional processes $\phi =(\phi _{t})_{t\in [0,T]}$, we get

$$\begin{aligned} \textbf{1}_{t\leqslant \tau _m}({{\bar{u}}}_t,\varphi ){} & {} =\textbf{1}_{t\leqslant \tau _m}\left( (\psi ,\varphi ) +\int _0^t({\bar{u}}_{s},{\tilde{{\mathcal {L}}}}_s\varphi )\,ds +\int _0^t({\bar{u}}_{s},{\mathcal {M}}_s^{k}\varphi )\,dV^k_s\right) \\{} & {} \quad +\textbf{1}_{t\leqslant \tau _m}\left( \int _0^t\int _{{\mathfrak {Z}}_0}({\bar{u}}_{s},J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})ds +\int _0^t\int _{{\mathfrak {Z}}_1}({\bar{u}}_{s},J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})ds\right. \\{} & {} \quad \left. +\int _0^t\int _{{\mathfrak {Z}}_1}({\bar{u}}_{s},I_s^{\xi }\varphi )\,{{\tilde{N}}}_1(d{\mathfrak {z}},ds)\right) \end{aligned}$$

for $P\otimes dt$-almost every $(t,\omega )\in [0,T]\times \Omega$ for every $\varphi \in C_0^{\infty }$ and integer $m\geqslant 1$, which implies that ${{\bar{u}}}$ is a ${\mathbb {V}}_r$-solution to (3.9) for $r=2,p$. In the same way as in the proof Lemma 6.4 we can show first that almost surely

$$\begin{aligned} \textbf{1}_{\tau _m>T}(g,\varphi ){} & {} =\textbf{1}_{\tau _m>T}\left( (\psi ,\varphi ) +\int _0^T({{\bar{u}}}_{s},{\tilde{{\mathcal {L}}}}_s\varphi )\,ds + \int _0^T({\bar{u}}_{s},{\mathcal {M}}_s^{k}\varphi )\,dV^k_s\right) \nonumber \\{} & {} \quad +\textbf{1}_{\tau _m>T}\left( \int _0^T\int _{{\mathfrak {Z}}_0}({\bar{u}}_{s},J_s^{\eta }\varphi )\,\nu _0(d{\mathfrak {z}})ds +\int _0^T\int _{{\mathfrak {Z}}_1}({\bar{u}}_{s},J_s^{\xi }\varphi )\,\nu _1(d{\mathfrak {z}})ds\right. \nonumber \\{} & {} \quad \left. +\int _0^T\int _{{\mathfrak {Z}}_1}({\bar{u}}_{s},I_s^{\xi }\varphi )\,{{\tilde{N}}}_1(d{\mathfrak {z}},ds) \right) \end{aligned}$$

(7.22)

for every $m\geqslant 1$. Hence taking into account $P(\cup _{m=1}^{\infty }\{\tau _m>T\})=1$, we get that equation (7.22) remains valid if we omit $\textbf{1}_{\tau _m>T}$ everywhere in it. From (7.15) we get that for sufficiently large n

$$\begin{aligned} |u^n|_{{\mathbb {L}}_{r,q}}\leqslant N|\pi ^n_0|_{{\mathbb {L}}_r} \quad \text {for }r=2,p\text {, for integers }q>1 \end{aligned}$$

with a constant $N=N(d, d',p,K, K_{\xi }, K_{\eta }, L, T,|{\bar{\xi }}|_{L_2}, |{\bar{\eta }}|_{L_2},\lambda )$. Letting here $n\rightarrow \infty$ and taking into account (7.11) and (7.17) we obtain

$$\begin{aligned} |{{\bar{u}}}|_{{\mathbb {L}}_{r,q}}\leqslant \liminf _{n\rightarrow \infty }|u^n|_{{\mathbb {L}}_{r,q}} \leqslant N\lim _{n\rightarrow \infty }|\pi ^n_0|_{L_r}\leqslant N|\pi _0|_{{\mathbb {L}}_r} \quad \text {for }r=2,p\text {, and for integers }q>1. \end{aligned}$$

Letting here $q\rightarrow \infty$ we get

$$\begin{aligned} {\mathbb {E}}\mathrm{ess\,sup}_{t\in [0,T]}|{{\bar{u}}}_t|_{{\mathbb {L}}_r}^r\leqslant N^r{\mathbb {E}}|\pi _0|^r_{{\mathbb {L}}_r},\quad \text {for }r=2,p. \end{aligned}$$

Hence, taking into account (7.22), by Lemma 5.8 we get a $P\otimes dt$-modification u of ${{\bar{u}}}$, which is an $L_r$-solution for $r=2,p$ to equation (3.9) with initial condition $u_0=\pi _0$. As the limit of $P\otimes dt\otimes dx$-almost everywhere nonnegative functions, u is also $P\otimes dt\otimes dx$ almost everywhere nonnegative. We now show that u satisfies

$$\begin{aligned} G(u):=\sup _{t\in [0,T]}\int _{{\mathbb {R}}^d}|x|^2u_t(dx)<\infty \,(a.s.). \end{aligned}$$

(7.23)

To show this recall that for each n and $\varphi \in C_b^2$, by Theorem 3.1, Remark 3.2 and by what we have proven above,

$$\begin{aligned} \mu ^n_t(\varphi ) = P_t^n(\varphi )\mu ^n_t(\textbf{1}) ={\mathbb {E}}(\varphi (X_t^n)|{\mathcal {F}}_t^{Y^n})({^o\!\gamma }^n_t)^{-1}, \end{aligned}$$

where $\mu ^n_t(dx) = u^n_t(x)dx$, $P^n_t(dx) = \pi ^n_t(x)dx$ and ${^o\!\gamma }^n$ denotes the ${\mathcal {F}}_t^{Y^n}$-optional projection of $(\gamma _t^n)_{t\in [0,T]}$. Further, for integers $m\geqslant 1$ let again $\Omega _m:= \{|Y_0|\leqslant m\} \in {\mathcal {F}}_0^Y$. Thus by Doob’s inequality and Jensen’s inequality for optional projections, for $r>1$ we have, in the same way as in Step I,

$$\begin{aligned} G_{m}(u^n){} & {} :={\mathbb {E}}\sup _{t\in [0,T]}\int _{{\mathbb {R}}^d}|x|^{2}u^n_t(x)\,dx\textbf{1}_{\Omega _m} = {\mathbb {E}}\sup _{t\in [0,T]} {\mathbb {E}}(|X_t^n|^{2}|{\mathcal {F}}_t^{Y^n})({^o\!\gamma }^n_t)^{-1}{} \textbf{1}_{\Omega _m} \\{} & {} \leqslant N\big ({\mathbb {E}}\sup _{t\in [0,T]}|X_t^n|^{r}{} \textbf{1}_{\Omega _m} \big )^{2/r} \quad \text {for }t\in [0,T] \end{aligned}$$

with a constant $N = N(r,C)$, where C is the constant from (7.8), which depends only on K, r and T. Taking r from Assumption 2.3, by Young’s inequality, (2.1) for all m and n we have

$$\begin{aligned} G_{m}(u^n)\leqslant N\big (m^{r}+{\mathbb {E}}|X^n_0|^{r})\big ) \leqslant N\big (m^{r}+\sup _n{\mathbb {E}}|X^n_0|^{r}\big )=:N'(m)<\infty . \nonumber \\ \end{aligned}$$

(7.24)

By Mazur’s theorem there exists a sequence of convex linear combinations $v^k = \sum _{i=1}^{k}c_{i,k}u^i$ converging to u (strongly) in ${\mathbb {L}}_{p,q}$ as $k\rightarrow \infty$. Thus there exists a subsequence, also denoted by $(v^k)_{k=1}^{\infty }$ which converges to u for $P\otimes dt\otimes dx$-almost every $(\omega ,t,x)$. Then, by Fatou’s lemma and (7.24),

$$\begin{aligned} G_m(u)= & {} {\mathbb {E}}\sup _{t\in [0,T]}\int _{{\mathbb {R}}^d}|x|^{2} \liminf _{k\rightarrow \infty } v_t^{k}(x)\,dx \textbf{1}_{\Omega _m} \leqslant \liminf _{k\rightarrow \infty }G_m(v^{k}) \\= & {} \liminf _{k\rightarrow \infty }\sum _{i=1}^{k} c_{k,i}G_{m}(u^i) \leqslant N'(m) \quad \text {for each integer }m\geqslant 1, \end{aligned}$$

which proves (7.23). Next, due to Lemma 3.2, using $|B^n|\leqslant |B|\leqslant K$, we have

$$\begin{aligned} \sup _{n\in {\mathbb {N}}}{\mathbb {E}}\sup _{t\in [0,T]}|u_t^n|_{L_1}\leqslant N, \end{aligned}$$

(7.25)

for a constant $N=N(d,K,T)$. The estimate above implies that $u^n \in {\mathbb {L}}_{1,q}$ for all $q\geqslant 1$. Returning to the sequence $(v^k)_{k\in {\mathbb {N}}}\subset {\mathbb {L}}_{1,q}\cap {\mathbb {L}}_{p,q}$ converging point-wise to u for $P\otimes dt\otimes dx$-almost every $(\omega ,t,x)$, we can compute by use of Fatou’s lemma

$$\begin{aligned} {\mathbb {E}}\mathrm{ess\,sup}_{t\in [0,T]}|u_t|_{L_1}= & {} {\mathbb {E}}\mathrm{ess\,sup}_{t\in [0,T]}|\liminf _{k\rightarrow \infty }v^k_t|_{L_1} \leqslant \liminf _{k\rightarrow \infty }{\mathbb {E}}\sup _{t\in [0,T]}|v^k_t|_{L_1}\nonumber \\{} & {} \leqslant \liminf _{k\rightarrow \infty }\sum _{i=1}^kc_{i,k}{\mathbb {E}}\sup _{t\in [0,T]}|u^i_t|_{L_1}\leqslant N, \end{aligned}$$

(7.26)

with the constant N from (7.25). As also (7.23) holds and since u is in particular an $L_2$-solution to (3.9) we can apply Lemma 5.5 to see that indeed for all $t\in [0,T]$, $u_t = d\mu _t/dx$ almost surely and thus $\pi _t = u_t{^o\!\gamma }_t$. This finishes the proof. $\square$

Inspecting the proof above, together with Lemma 7.1, it is useful to make the following observations.

Corollary 7.4

Let the conditions of Theorem 2.1 hold. Assume the initial conditional density $\pi _0 = P(X_0\in dx|{\mathcal {F}}^Y_0)$ additionally satisfies ${\mathbb {E}}|\pi _0|_{W^m_p}^p<\infty$ for some integer $m\geqslant 0$. Then there exist sequences

$$\begin{aligned} (X_0^n)_{n=1}^\infty , ((X_t^n,Y_t^n)_{t\in [0,T]})_{n=1}^\infty ,\quad \text {as well as}\quad (\pi _0^n)_{n=1}^\infty \quad \text {and}\quad ((\pi _t^n)_{t\in [0,T]})_{n=1}^\infty \end{aligned}$$

such that the following are satisfied:

• (i) For each $n\geqslant 1$ the coefficients $b^n,B^n,\sigma ^n,\rho ^n,\xi ^n$ and $\eta ^n$, defined in (7.12), satisfy Assumptions 2.1 and 2.2 with $K_1=0$ and constants $K_0'=K'_0(n,L, K, K_0, K_1,K_{\xi },K_{\eta })$ and $L'=L'(K,K_0, K_1, L,K_{\xi },K_{\eta })$ in place of $K_0$ and L, as well as Assumption 2.4 with $\lambda '=\lambda '(\lambda ,K_0,K_1,K_\xi ,K_\eta )$ in place of $\lambda$. Moreover, for each $n\geqslant 1$ they satisfy the support condition (6.13) of Lemma 6.4 with some $R>0$ depending only on n.

• (ii) For each $n\geqslant 1$ the random variable $X_0^n$ is ${\mathcal {F}}_0$-measurable and such that

  $$\begin{aligned} \lim _{n\rightarrow \infty } X_0^n = X_0\,\,, \omega \in \Omega ,\quad \text {and}\quad {\mathbb {E}}|X_0^n|^r \leqslant N(1+{\mathbb {E}}|X_0|^r) \end{aligned}$$

  with a constant N independent of n.

• (iii) $Z_t^n=(X_t^n,Y_t^n)$ is the solution to (1.1) with the coefficients $b^n,B^n,\sigma ^n,\rho ^n,\xi ^n$ and $\eta ^n$ in place of $b,B,\sigma ,\rho ,\xi$ and $\eta$, respectively, and with initial condition $Z_0^n = (X_0^n,Y_0)$.

• (iv) For each $n\geqslant 1$ we have $\pi _0^n = P(X_0^n\in dx|{\mathcal {F}}^Y_0)/dx$, $\pi _0^n(x)=0$ for $|x|\geqslant n+1$ and

  $$\begin{aligned} \lim _{n\rightarrow \infty }|\pi ^n_0-\pi _0|_{{\mathbb {W}}^m_p}=0. \end{aligned}$$

• (v) For each $n\geqslant 1$ there exists an $L_r$-solution $u^n$ to (3.9), $r=2,p$, with initial condition $\pi ^n_0$, such that $u^n$ is the unnormalised conditional density of $X^n$ given $Y^n$, almost surely

  $$\begin{aligned} u_t^n(x)=0 \quad \text {for }dx\text {-a.e. }x\in \{x\in {\mathbb {R}}^d:|x|\geqslant \bar{R}\}\text { for all }t\in [0,T] \end{aligned}$$

  with a constant ${{\bar{R}}}={{\bar{R}}}(n,K,K_0,K_\xi ,K_\eta )$ and

  $$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}|u^n_t|_{L_p}^p\leqslant N{\mathbb {E}}|\pi _0^n|^p_{L_p} \end{aligned}$$

  (7.27)

  with a constant $N=N(d, d',K, L, K_{\xi }, K_{\eta }, T, p,\lambda ,|{\bar{\xi }}|_{L_2}, |{\bar{\eta }}|_{L_2})$. Moreover,

  $$\begin{aligned} u^n\rightarrow u \quad \text {weakly in }{\mathbb {L}}_{r,q}\text { for }r=p,2\text { and all integers }q>1, \end{aligned}$$

  where u is the unnormalised conditional density of X given Y, satisfying (7.27) with the same constant N and u in place of $u^n$.

• (vi) Consequently, for each $n\geqslant 1$ and $t\in [0,T]$ we have

  $$\begin{aligned} \pi ^n_t = P(X_t^n\in dx|{\mathcal {F}}^{Y^n}_t)/dx = u^n_t(x){^o\!\gamma _t^n},\quad \text {almost surely}, \end{aligned}$$

  as well as

  $$\begin{aligned} \pi _t = P(X_t\in dx|{\mathcal {F}}^{Y}_t)/dx = u_t(x){^o\!\gamma _t},\quad \text {almost surely}, \end{aligned}$$

  where ${^o\!\gamma ^n}$ and ${^o\!\gamma }$ are cadlag positive normalising processes.