On classical solutions of linear stochastic integro-differential equations

Abstract

We prove the existence of classical solutions to parabolic linear stochastic integro-differential equations with adapted coefficients using Feynman–Kac transformations, conditioning, and the interlacing of space-inverses of stochastic flows associated with the equations. The equations are forward and the derivation of existence does not use the “general theory” of SPDEs. Uniqueness is proved in the class of classical solutions with polynomial growth.

Appendix

1.1 Martingale and point measure moment estimates

Set $(Z,{\mathcal {Z}},\pi )=(Z^{1},{\mathcal {Z}}^{1},\pi ^{1})$, $p(dt,dz)=p^{1}(dt,dz)$, and $q(dt,dz)=q^{1}(dt,dz)$. The following moment estimates are used to derive the estimates of $\Gamma _t$ and $\Psi _t$ in Lemma 3.1. The notation $a\underset{p,T}{ \sim }b$ is used to indicate that the quantity a is bounded above and below by a constant depending only on p and T times b.

Lemma 4.1

Let $h:\Omega \times [ 0,T]\times Z\rightarrow \mathbf {R}^{d_1}$ be ${\mathcal {P}}_{T}\otimes {\mathcal {Z}}$ -measurable

• (1) For any stopping time $\tau \le T$ and $p\ge 2$,

  $$\begin{aligned} \mathbf {E}\left[ \sup _{t\le \tau }\left| \int _{]0,\tau ] }\int _{Z}h_s(z)q(ds,dz)\right| ^{p}\right]&\underset{p,T}{\sim }\mathbf {E}\left[ \int _{]0,\tau ] }\int _{Z}\left| h_s(z)\right| ^{p}\pi (dz)ds\ \right] \\&\quad +\mathbf {E}\left[ \left( \int _{]0,\tau ] }\int _{Z}\left| h_s(z)\right| ^{2}\pi (dz)ds\right) ^{p/2}\right] . \end{aligned}$$

• (2) For any stopping time $\tau \le T$ and $\bar{p}\ge 1$,

  $$\begin{aligned} \mathbf {E}\left[ \sup _{t\le \tau }\left( \int _{]0,\tau ] }\int _{Z}|h_s(z)|p(ds,dz)\right) ^{\bar{p}}\right]&\underset{p,T}{\sim }\mathbf {E}\left[ \int _{]0,\tau ] }\int _{Z}\left| h_s(z)\right| ^{\bar{p}}\pi (dz)ds\right] \\&\quad +\mathbf {E}\left[ \left( \int _{]0,\tau ] }\int _{Z}|h_s(z)|\pi (dz)ds\right) ^{\bar{p}}\right] . \end{aligned}$$

Proof

We will only prove part (2), since part (1) is well-known (see, e.g., [20] or [30]). Assume that $h_t(\omega ,z)>0$ for all $\omega ,t$ and z. Let

$$\begin{aligned} A_{t}=\int _{]0,t]}\int _{Z}h_s(z)p(ds,dz) \quad \text {and} \quad L_{t}=\int _{]0,t]}\int _Zh_s(z)\pi (dz)ds,\;\;t\le T. \end{aligned}$$

It suffices to prove (2) for $p>1$, since the case $p= 1$ is obvious. Fix an arbitrary stopping time $\tau \le T$ and $p>1$. For all t, we have

$$\begin{aligned} A_{t}^{p}=\sum _{s\le t}\left[ \left( A_{s-}+\Delta A_{s}\right) ^{p}-A_{s-}^{p}\right] . \end{aligned}$$

Thus, by the inequality

$$\begin{aligned} b^p\le (a+b)^p-a^p\le p(a+b)^{p-1}b\le p2^{p-1}[a^{p-1}b+b^p], \;\;a,b\ge 0, \end{aligned}$$

we get

$$\begin{aligned} A_t^p \ge \int _{]0,t]}\int _{Z}h_s(z)^{p}p(ds,dz) \end{aligned}$$

and

$$\begin{aligned} A_t^p \le p 2^{p-2}\left[ \int _{]0,t]}\int _{h}h_s(z)^{p}p(ds,dz)+ \int _{0}^{t}\int _ZA_{s-}^{p-1} h_s(z)p(ds,dz)\right] . \end{aligned}$$

Since $A_t$ is increasing, we obtain

$$\begin{aligned}&\mathbf {E}\int _{]0,\tau ]}\int _{Z}h_s(z)^{p}p(ds,dz)\le \mathbf {E}A_{\tau }^p \\&\quad \le p 2^{p-2}\mathbf {E}\left[ \int _{]0,\tau ]}\int _{Z}h_s(z)^{p}p(ds,dz)+A_{ \tau }^{p-1}L_{\tau }\right] . \end{aligned}$$

It is easy to see that

$$\begin{aligned} \mathbf {E}[L_{\tau }^{p}]=p\mathbf {E}\int _{]0,\tau ] }L_{s}^{p-1}dL_{s}=p \mathbf {E}\int _{]0,\tau ] }L_{s}^{p-1}dA_{s}\le p\mathbf {E}[L_{\tau }^{p-1}A_{\tau }]. \end{aligned}$$

Applying Young’s inequality, for any $\varepsilon >0$, we get

$$\begin{aligned} A_{\tau }^{p-1}L_{\tau }\le \varepsilon A_{\tau }^{p}+\frac{(p-1)^{p-1}}{\varepsilon ^{p-1}p^p}L_{\tau }^{p}\quad \text {and}\quad L_{\tau }^{p-1}A_{\tau }\le \varepsilon L_{\tau }^{p}+\frac{(p-1)^{p-1}}{\varepsilon ^{p-1}p^p}A_{\tau }^{p}. \end{aligned}$$

Combining the estimates for any $\varepsilon _1\in (0,\frac{1}{p})$, we have

$$\begin{aligned} \left( \frac{\varepsilon _1^{p-1}p^p(1-p\varepsilon _1) }{p(p-1)^{p-1}}\mathbf {E }L_{\tau }^{p}\right) \vee \mathbf {E}\int _{]0,\tau ]} \int _{Z}h_s(z)^{p}p(ds,dz) \le \mathbf {E}[A_{\tau }^{p}]. \end{aligned}$$

and for any $\varepsilon _2 \in (0,\frac{1}{p2^{p-2}})$

$$\begin{aligned} \mathbf {E}[ A_{\tau }^p]\le \frac{p 2^{p-2}}{(1-p2^{p-2}\varepsilon _2)} \mathbf {E}\left[ \int _{]0,\tau ]}\int _{Z}h_s(z)^{p}p(ds,dz)+\frac{(p-1)^{p-1}}{ \varepsilon _2^{p-1}p^p}L^p_{\tau }\right] , \end{aligned}$$

which completes the proof.$\square$

1.2 Optional projection

The following lemma concerning the optional projection plays an integral role in Sect. 3.4 and the proof of Theorem 2.2. For more information on the Skorokhod $\mathcal {J}_{1}$ -topology, we refer the reader to Chapter 6, Sect. 1 of [23]. Also, we refer the reader to Theorem 5.3 [13] for the construction of regular conditional probability measures on Borel spaces.

Lemma 4.2

([25, cf. Theorem 1]) Let $\mathcal {X}$ be a Polish space and $D\left( [0,T];\mathcal {X}\right)$ be the space of $\mathcal {X}$-valued càdlàg trajectories with the Skorokhod $\mathcal { J}_{1}$-topology. If $\mathfrak {A}$ is a random variable taking values in $D\left( [0,T];\mathcal {X}\right)$, then there exists a family of $\mathcal {B }([0,T])\times \mathcal {F}$-measurable non-negative measures $E^{t}(dU),$ $(\omega ,t)\in \Omega \times [0,T],$ on $D\left( [0,T];\mathcal {X}\right)$ and a random-variable $\zeta$ satisfying $\mathbf {P}\left( \zeta <T\right) =0$ such that $E^{t}(D\left( [0,T];\mathcal {X}\right) )=1$ for $t<\zeta$ and $E^{t}(D([0,T];\mathcal {X}) )=0$ for $t\ge \zeta .$ In addition, $E^{t}$ is c àdlàg in the topology of weak convergence, $E^{t}=E^{t+}$ for all $t\in [0,T]$, and for each continuous and bounded functional F on $D\left( [0,T];\mathcal {X}\right) ,$ the process $E^{t}\left( F\right)$ is the càdlàg version of $\mathbf {E}[ F\left( \mathfrak {A}\right) | \mathcal {F}_{t}]$. If $G:\Omega \times [0,T]\times [0,T]\times D\left( [0,T]; \mathcal {X}\right) \rightarrow \mathbf {R}^{d_2}$ is bounded and $\mathcal {O\times B}\left( [0,T]\right) \times {\mathcal {B}}\left( D\left( [0,T]; \mathcal {X}\right) \right)$-measurable, then

$$\begin{aligned} \int _{D( [0,T];\mathcal {X}) } G_t(\omega ,t,U)E^{t}(dU)=E^{t}( G_t) \end{aligned}$$

is the optional projection of $G_{t}(\mathfrak {A})=G_t(\omega ,t,\mathfrak {A} )$. Furthermore, if $G=G_t(\omega ,t,U)$ is bounded and ${\mathcal {P}}\times {\mathcal {B}}( [0,T]) \times {\mathcal {B}}( D( [0,T]; \mathcal {X}))$ -measurable, then $E^{t-}(G_{t})$ is the predictable projection of $G_{t}( \mathfrak {A}) =G_t(\omega ,t,\mathfrak {A}).$

Proof

We follow the proof of Theorem 1 in [25]. Since $D([0,T];\mathcal {X})$ is a Polish space, for each $t\in [0,T]$, there is family of probability measures $\tilde{E}_{\omega }^{t}(dw)$, $\omega \in \Omega$, on $D([0,T];\mathcal {X})$ such that for each $A\in {\mathcal {B}}(D([0,T];\mathcal { X})),$ $\tilde{E}^{t}(A)$ is $\mathcal {F}_{t}$-measurable and $\mathbf {P}$ -a.s. 

$$\begin{aligned} \mathbf {P}\left( \mathfrak {A}\in A|\mathcal {F}_{t}\right) =\tilde{E} ^{t}\left( A\right) . \end{aligned}$$

For each $\omega \in \Omega$, let $I\left( \omega \right)$ be the set of all $t\in (0,T]$ such that for any bounded continuous function F on $D([0,T];\mathcal {X})$, the function

$$\begin{aligned} r\mapsto \tilde{E}_{\omega }^{r}(F)=\int _{D\left( [0,T];\mathcal {X}\right) }F(w)\tilde{E}^{r}(dw) \end{aligned}$$

has a right-hand limit on $[0,s)\cap \mathbf {Q}$ and a left-hand limit on $(0,s]\cap \mathbf {Q}$ for every rational $s\in [0,t]\cap \mathbf {Q}$. Let $\zeta \left( \omega \right) =\sup \left( t:t\in I(\omega )\right) \wedge T.$ It is easy to see that $\mathbf {P}\left( \zeta <T\right) =0.$ We set $\tilde{E}_{\omega }^{t}=0$ if $\zeta (\omega )<t\le T$. The function $\tilde{E}_{\omega }^{t}$ has left-hand and right-hand limits for all $t\in \mathbf {Q}\cap \left[ 0,T\right]$. We define $E_{\omega }^{t}=\tilde{E} _{\omega }^{t+}$ for each $t\in [0,T)$ (the limit is taken along the rationals), and $E_{\omega }^{T}$ is the left-hand limit at T along the rationals. The statement follows by repeating the proof of Theorem 1 in [25] in an obvious way.$\square$

1.3 Estimates of Hölder continuous functions

For a Banach space V with norm $|\cdot |_{V}$ and continuous function $f:\mathbf {R}^{d_1}\rightarrow V$, we define

$$\begin{aligned} |f|_{0;V}=\sup _{x\in \mathbf {R}^{d_1}}|f(x)|_V \end{aligned}$$

and

$$\begin{aligned}{}[f]_{\beta ;V}=\sup _{x,y\in \mathbf {R}^{d_1},x\ne y}\frac{| f(x)-f(y)|_{V}}{|x-y|^{\beta }},\;\;\beta \in (0,1]. \end{aligned}$$

For each real number $\beta \in \mathbf {R}$, we write $\beta =[\beta ]^-+\{\beta \}^+$, where $[\beta ]^-$ is an integer and $\{\beta \}^+\in (0,1]$. For a Banach space V with norm $|\cdot |_{V}$ and real number $\beta >0$, we denote by $\mathcal {C}^{\beta }(\mathbf {R}^{d_1};V)$ the Banach space of all $[\beta ]^-$-times continuously differentiable bounded functions $f:\mathbf {R}^{d_1}\rightarrow V$ having finite norm

$$\begin{aligned} |f|_{\beta ;V}:=\sum _{| \gamma |\le [\beta ]^- }|\partial ^{\gamma }f|_{0;V}+\sum _{|\gamma |=[\beta ]^-}[\partial ^{\gamma }f]_{\{\beta \}^+ ;V}. \end{aligned}$$

When V is clear from the context, we drop the subscript V from the norm $| \cdot |_{\beta ;V}$ and semi-norm $[\cdot ]_{\{\beta \}^+;V}$ and write $|\cdot |_{\beta }$ and $[\cdot ]_{\{\beta \}^+}$, respectively.

In the coming lemmas, we establish some properties of weighted Hölder spaces that are used Sect. 3.5 and the proof of Theorem 2.4.

Lemma 4.3

Let $\beta \in (0,1]$ and $\theta _1,\theta _2\in \mathbf {R}$ with $\theta _{1}-\theta _{2}\le \beta .$

• (1) There is a constant $c_{1}=c_{1}\left( \theta _{2},\beta \right)$ such that for all $\phi :\mathbf {R}^{d_1}\rightarrow \mathbf {R}$ with $|r_{1}^{-\theta _{1}}\phi |_{0}+[r_{1}^{-\theta _{2}}\phi ]_{\beta }=:N_{1}<\infty ,$

  $$\begin{aligned} |\phi (x)-\phi (y)|\le c_{1}N_{1}(r_{1}(x)^{\theta _{2}}\vee r_{1}(y)^{\theta _{2}})|x-y|^{\beta }, \end{aligned}$$

  for all $x,y\in \mathbf {R}^{d_1}$, where $r_{1 }(x):=\sqrt{ 1+| x|^{2}},\;x\in \mathbf {R}^{d_1}$.

• (2) Conversely, if $\phi :\mathbf {R}^{d_1}\rightarrow \mathbf {R}$ satisfies $|r_{1}^{-\theta _{1}}\phi |_{0}<\infty$ and there is a constant $N_{2}$ such that for all $x,y\in \mathbf {R}^{d_1}$,

  $$\begin{aligned} |\phi (x)-\phi (y)|\le N_{2}( r_{1}(x)^{\theta _{2}}\vee r_{1}(y)^{\theta _{2}}) |x-y|^{\beta }, \end{aligned}$$

  then

  $$\begin{aligned}{}[r_{1}^{-\theta _{2}}\phi ]_{\beta }\le c_{1}|r_{1}^{-\theta _{1}}\phi |_{0}+N_{2}. \end{aligned}$$

Proof

(1) For all x, y such that $r_{1}\left( x\right) ^{\theta _{2}}\ge r_{1}\left( y\right) ^{\theta _{2}}$, we have

$$\begin{aligned} |\phi (x)-\phi (y)|&\le r_{1}(x)^{\theta _{2}}[r_{1}^{-\theta _{2}}\phi ]_{\beta }|x-y|^{\beta }+r_{1}(y) ^{\theta _{1}-\theta _{2}}|r_{1}^{-\theta _{1}}\phi |_{0} |r_{1}^{\theta _{2}}(x)-r_{1}(y) ^{\theta _{2}}| \\&\le ([r_{1}^{-\theta _{2}}\phi ]_{\beta }+c_{1}|r_{1}^{-\theta _1}\phi |_{0})r_{1}(x)^{\theta _{2}}|x-y|^{\beta }, \end{aligned}$$

where $c_{1} :=1+\sup _{t\in (0,1) }\frac{1-t^{\theta _{2}}}{( 1-t)^{\beta }}$ if $\theta _{2}\ge 0$ and $c_{1} :=1+\sup _{t\in (1,\infty )}\frac{ (t^{\theta _{2}}-1)t^{\beta }}{( t-1) ^{\beta }}$ if $\theta _{2}<0,$ which proves the first claim. (2) For all x and y with $r_{1}(x)^{\theta _{2}}>r_{1}(y)^{\theta _{2}}$, we have

$$\begin{aligned}&|r_{1}(x)^{-\theta _{2}}\phi (x)-r_{1}(y)^{-\theta _{2}}\phi (y)|\\&\quad \le r_{1}(x)^{-\theta _{2}}|\phi (x)-\phi (y)|+r_{1}(y)^{\theta _{1}-\theta _{2}}|r_{1}^{-\theta _{1}}(y)\phi (y)| |r_{1}(y)^{\theta _{2}}r_{1}(x)^{-\theta _{2}}-1|\\&\quad \le (c_{1}| r^{-\theta _{1}}\phi | _{0}+N_{2})|x-y|^{\beta }, \end{aligned}$$

which proves the second claim.$\square$

Lemma 4.4

Let $\beta ,\mu \in (0,1]$ and $\theta _1,\theta _2,\theta _3,\theta _4\in \mathbf {R}$ with $\theta _{1}-\theta _{2}\le \beta$, $\theta _{3}-\theta _{4}\le \mu$, and $\theta _{3}\ge 0.$ If $\phi :\mathbf {R}^{d_1}\rightarrow \mathbf {R}$ and $H:\mathbf {R} ^{d_1}\rightarrow \mathbf {R}^{d_1}$ are such that

$$\begin{aligned} |r_{1}^{-\theta _{1}}\phi |_{0}+[r_{1}^{-\theta _{2}}\phi ]_{\beta }=:N_1<\infty \quad \text {and} \quad |r_{1}^{-\theta _{3}}H|_{0}+[r_{1}^{-\theta _{4}}H]_{\mu }=:N_2<\infty , \end{aligned}$$

then

$$\begin{aligned} |r_1^{-\theta _1\theta _3} \cdot (\varphi \circ H ) |_{0} \le |r_{1}^{-\theta _{1}}\phi |_{0} (1+|r_{1}^{-\theta _{3}}H|_{0})^{\theta _1}\le N_{1}\left( 1+N_{2}\right) ^{\theta _{1}} \end{aligned}$$

and there is a constant $N=N(\beta ,\mu ,\theta _{1},\theta _{2})$ such that

$$\begin{aligned} {[} r_1^{-\theta _{1}\theta _{3}} \cdot (\varphi \circ H ){]}_{\beta \mu }\le NN_{1}( 1+N_{2})^{\theta _{2}+\beta }, \end{aligned}$$

where $r_{1 }(x):=\sqrt{ 1+| x|^{2}},\;x\in \mathbf {R}^{d_1}$.

Proof

For all x, we have

$$\begin{aligned} r_{1}(H(x))\le (1+|r_{1}^{-\theta _{3}}H|_{0})r_{1}(x)^{\theta _{3}}\le ( 1+N_{2})r_{1}(x)^{\theta _{3}}, \end{aligned}$$

and hence

$$\begin{aligned} |r_1^{-\theta _1\theta _3} \cdot (\varphi \circ H )|_{0}\le |r_{1}^{-\theta _{1}}\phi |_{0}|r_{1}^{\theta _{1}}\circ H\cdot r_{1}^{-\theta _{1}\theta _{3}}|_{0}\le N_{1}(1+N_{2})^{\theta _{1}}. \end{aligned}$$

Using Lemma 4.3, for all x and y, we get

$$\begin{aligned} |\phi (H(x))-\phi (H(y))|&\le N N_{1}( r_{1}(H(x))\vee r_{1}(H(y)))^{\theta _{2}}|H(x)-H(y)|^{\beta } \\&\le NN_{1}(1+N_{2})^{\theta _{2}}(r_{1}(x)\vee r_{1}(y))^{\theta _{2}\theta _{3}}N_{2}^{\beta }( r_{1}(x)\\&\quad \vee r_{1}(y)) ^{\beta \theta _{4}}|x-y|^{\beta \mu } \\&\le NN_{1}(1+N_{2})^{\theta _{2}+\beta }(r_{1}(x)\vee r_{1}(y))^{\theta _{2}\theta _{3}+\beta \theta _{4}}| x-y| ^{\beta \mu }, \end{aligned}$$

for some constant $N=N(\beta ,\mu ,\theta _{1},\theta _{2})$. Noting that

$$\begin{aligned} \theta _1\theta _3-\theta _2\theta _3-\beta \theta _4=(\theta _1-\theta _2)\theta _3- \beta \theta _4\le \beta (\theta _3-\theta _4)\le \beta \mu , \end{aligned}$$

we apply Lemma 4.3 to complete the proof.$\square$

Remark 4.5

Let $\beta \in (0,1]$ and $\theta _1,\theta _2\in \mathbf {R}.$ Then there is a constant $N=N(\beta ,\theta _1,\theta _2)$ such that for all $\phi : \mathbf {R}^{d_1}\rightarrow \mathbf {R}$ with $|r_{1}^{-\theta _{1}}\phi |_{0}+[r_{1}^{-\theta _{2}}\phi ]_{\beta }=:N_{1}<\infty ,$ we have $|r^{-\theta }\phi |_{\beta }\le NN_{1}$, where $\theta =\max \left\{ \theta _{1},\theta _{2}\right\} .$ Thus, imposing the assumptions of Lemma 4.4 and assuming further that $\theta _{1}=\theta _{2}$ and $\theta _{4}\ge 0$, we have

$$\begin{aligned} |r^{-\theta _1\theta _3-\beta \theta _4}\cdot (\phi \circ H)|_{\beta \mu }\le NN_{1}( 1+N_{2}) ^{\theta _{1}+\beta }. \end{aligned}$$

Proof

If $\theta _{2}\ge \theta _{1}$, then the claim is obvious. If $\theta _{1}>\theta _{2}$, then for all x and y we find

$$\begin{aligned} |r_{1}(x)^{-\theta _{1}}\phi (x)-r_{1}(y)^{-\theta _{1}}\phi (y)|&\le r_{1}(x) ^{\theta _{2}-\theta _{1}}| r_{1}(x)^{-\theta _{2}}\phi (x)-r_{1}(y)^{-\theta _{2}}\phi (y)| \\&\quad +\left| \frac{r_1(y)^{\theta _{1}-\theta _{2}}}{r_1(x) ^{\theta _{1}-\theta _{2}}}\!-\!1\right| |r_{1}^{-\theta _{1}}\phi |_0 \le N_1(1\!+\!c_1)|x\!-\!y|^{\beta }, \end{aligned}$$

where $c_{1}:=\sup _{t\in (0,1)}\frac{1-t^{\theta _{1}-\theta _{2}}}{\left( 1-t\right) ^{\beta }}.$ $\square$

Lemma 4.6

Let $\theta \ge 0$ and $\beta >1$. Then there are constants $N_{1}=N_{1}(d_1,\theta ,\beta )$ and $N_{2}(d_1,\theta ,\beta )$ such that for all $\phi :\mathbf {R}^{d_1}\rightarrow \mathbf {R}$ with $r_{1}^{-\theta _{1}}\phi \in \mathcal {C}^{\beta }(\mathbf {R}^{d_1},\mathbf {R})$ ,

$$\begin{aligned} N_{1}|r_{1}^{-\theta }\phi |_{\beta }\le \sum _{| \gamma | \le \left[ \beta \right] ^{-}}|r_{1}^{-\theta }\partial ^{\gamma }\phi |_{0}+\sum _{|\gamma |=[\beta ]^{-}}|r_{1}^{-\theta }\partial ^{\gamma }\phi |_{\{\beta \}^{+}}\le N_{2}|r_{1}^{-\theta }\phi |_{\beta }, \end{aligned}$$

(4.1)

where $r_{1 }(x):=\sqrt{ 1+| x|^{2}},\;x\in \mathbf {R}^{d_1}$.

Proof

For any multi-index $\gamma$ with $|\gamma |\le [\beta ]^{-}$ and x , we have

$$\begin{aligned} \partial ^{\gamma }(r_{1}^{-\theta }\phi )(x)=\sum _{\underset{|\gamma _{1}|\ge 1}{\gamma _{1}+\gamma _{2}+=\gamma }}r_{1}(x)^{\theta }\partial ^{\gamma _{1}}(r_{1}^{-\theta })(x)r_{1}(x)^{-\theta }\partial ^{\gamma _{2}}\phi (x)+r_{1}(x)^{-\theta }\partial ^{\gamma }\phi (x). \end{aligned}$$

It is easy to show by induction that for all multi-indices $\gamma$, $|r_{1}^{\theta }\partial ^{\gamma }(r_{1}^{-\theta })|_{1}<\infty .$ Moreover, for all multi-indices $\gamma$ with $|\gamma |<[\beta ]^{-}$,

$$\begin{aligned} |r_{1}^{-\theta }\partial ^{\gamma }\phi |_{1}\le |\nabla ( r_{1}^{-\theta }\partial ^{\gamma }\phi )|_0\le |r_{1}^{-\theta }\nabla ( r_{1}^{-\theta })|_{0}|r_{1}^{-\theta }\partial ^{\gamma }\nabla \phi |_{0}. \end{aligned}$$

Thus, for any multi-index $\gamma$ with $|\gamma |\le [\beta ]^{-}$ ,

$$\begin{aligned} |\partial ^{\gamma }(r_{1}^{-\theta }\phi )|_{0}\le \sum _{\underset{|\gamma _{1}|\ge 1}{\gamma _{1}+\gamma _{2}+=\gamma }}|r_{1}^{\theta }\partial ^{\gamma _{1}}(r_{1}^{-\theta })|_{0}|r_{1}^{-\theta }\partial ^{\gamma _{2}}\phi |_{0}+|r_{1}^{-\theta }\partial ^{\gamma }\phi |_{0} \end{aligned}$$

and for any multi-index $\gamma$ with $|\gamma |=[\beta ]^{-}$,

$$\begin{aligned} |\partial ^{\gamma }(r_{1}^{-\theta }\phi )|_{\{\beta \}^{+}}\le \sum _{ \underset{|\gamma _{1}|\ge 1}{\gamma _{1}+\gamma _{2}+=\gamma } }|r_{1}^{\theta }\partial ^{\gamma _{1}}(r_{1}^{-\theta })|_{1}|r_{1}^{-\theta }\nabla ( r_{1}^{-\theta })|_{0}|r_{1}^{-\theta }\partial ^{\gamma _{2}}\nabla \phi |_{0}+|r_{1}^{-\theta }\partial ^{\gamma }\phi |_{0}. \end{aligned}$$

This proves the leftmost inequality in (4.1). For all $i\in \{1,\ldots ,d\}$ and x,

$$\begin{aligned} r_{1}^{-\theta }\partial _{i}\phi (x)=\partial _{i}(r_{1}^{-\theta }\phi )(x)-r_{1}(x)^{-\theta }\phi (x)r_{1}(x)^{\theta }\partial _{i}(r_{1}^{-\theta })(x). \end{aligned}$$

It follows by induction that for all multi-indices $\gamma$ with $|\gamma |\le [\beta ]^{-}$ and x, $r_{1}^{-\theta }\partial ^{\gamma }\phi (x)$ is a sum of $\partial ^{\gamma }(r_{1}^{-\theta }\phi )(x),$ a finite sum of terms, each of which is a product of one term of the form $\partial ^{\tilde{\gamma }}(r_{1}^{-\theta }\phi )(x),$ $|\tilde{\gamma } |<|\gamma |$, and a finite number of terms of the form $\partial ^{\gamma _{1}}(r_{1}^{\theta })\partial ^{\gamma _{2}}(r_{1}^{-\theta }),$ $|\gamma _{1}|,|\gamma _{2}|\le |\gamma |$. Since for all multi-indices $\gamma _{1}$ and $\gamma _{2}$, we have $|\partial ^{\gamma _{1}}(r_{1}^{\theta })\partial ^{\gamma _{2}}(r_{1}^{-\theta })|_{1}<\infty ,$ the rightmost inequality in (4.1) follows.$\square$

Corollary 4.7

For any $\theta \ge 0$ and $\beta >1$ , there are constants $N_{1}=N_{1}(d_1,\theta ,\beta )$ and $N_{2}(d_1,\theta ,\beta )$ such that for all $\phi :\mathbf {R}^{d_1}\rightarrow \mathbf {R}$ with $r_{1}^{-\theta _{1}}\phi \in \mathcal {C}^{\beta }(\mathbf {R}^{d_1},\mathbf {R})$,

$$\begin{aligned} N_{1}|r_{1}^{-\theta }\phi |_{\beta }\le |r_{1}^{-\theta }\phi |_{0}+\sum _{|\gamma |=[\beta ]^{-}}|r_{1}^{-\theta }\partial ^{\gamma }\phi |_{\{\beta \}^{+}}\le N_{2}|r_{1}^{-\theta }\phi |_{\beta }, \end{aligned}$$

where $r_{1 }(x):=\sqrt{ 1+| x|^{2}},\;x\in \mathbf {R}^{d_1}$.

Proof

It is well known that for an arbitrary unit ball $B\subset \mathbf {R}^{d_1}$ and any $1\le k<[\beta ]^{-}$, there is a constant N such that for any $\varepsilon >0,$

$$\begin{aligned} \sup _{x\in B,|\gamma |=k}|\partial ^{\gamma }\phi (x)|\le N\left( \varepsilon \sup _{x\in B,|\gamma | =[\beta ] ^{-}}|\partial ^{\gamma }\phi (x)|+\varepsilon ^{-k}\sup _{x\in B}|\phi (x)|\right) . \end{aligned}$$

Let $U_{0}=\{ x\in \mathbf {R}^{d_1}:|x|\le 1\}$ and $U_{j}=\{ x\in \mathbf {R }^{d_1}:2^{j-1}\le | x| \le 2^{j}\} ,j\ge 1.$ For all j, we have

$$\begin{aligned} \sup _{x\in U_{j},|\gamma | =k}| \partial ^{\gamma }\phi (x)|&=\sup _{B\subseteq U_{j}}\sup _{x\in B,| \gamma | =k}| \partial ^{\gamma }\phi (x)| \\&\le N\left( \varepsilon \sup _{B\subseteq U_{j}}\sup _{x\in B,| \gamma | =[ \beta ] ^{-}}| \partial ^{\gamma }\phi (x)| +\varepsilon ^{-k}\sup _{B\subseteq U_{j}}\sup _{x\in B}| \phi (x)| \right) \\&\le N\left( \varepsilon \sup _{x\in U_{j},| \gamma | =[\beta ]^{-}}| \partial ^{\gamma }\phi (x)| +\varepsilon ^{-k}\sup _{x\in U_{j}}| \phi (x)| \right) . \end{aligned}$$

Since for every j,

$$\begin{aligned}&2^{-\theta /2}2^{-j\theta }\sup _{x\in U_{j},| \gamma | =k}| \partial ^{\gamma }\phi (x)|\\&\quad \le \sup _{x\in U_{j},| \gamma | =k}| r^{-\theta }\partial ^{\gamma }\phi (x)| \le 2^{^{\theta }}2^{-(j-1)\theta }\sup _{x\in U_{j},| \gamma | =k}| \partial ^{\gamma }\phi (x)| , \end{aligned}$$

we see that

$$\begin{aligned} 2^{-\theta /2}\sup _{j}2^{-j\theta }\sup _{x\in U_{j},| \gamma | =k}| \partial ^{\gamma }\phi (x)|&\le \sup _{j}\sup _{x\in U_{j},| \gamma | =k}| r^{-\theta }\partial ^{\gamma }\phi (x)| =| r^{-\theta }\partial ^{\gamma }\phi | _{0} \\&\le 2^{\theta }\sup _{j}2^{-j\theta }\sup _{x\in U_{j},| \gamma | =k}| \partial ^{\gamma }\phi (x)|, \end{aligned}$$

and the statement follows.$\square$

Remark 4.8

If $\phi :\mathbf {R}^{d_1}\rightarrow \mathbf {R}$ is such that $| r^{-\theta _{1}}\phi | _{0}+| r^{-\theta _{2}}\nabla \phi | _{0}<\infty$ for $\theta _1,\theta _2\in \mathbf {R}$ with $\theta _{1}-\theta _{2}\le 1$, then

$$\begin{aligned}{}[ r^{-\theta _{2}}\phi ]_{1}\le N(| r^{-\theta _{1}}\phi | _{0}+| r^{-\theta _{2}}\nabla \phi | _{0}), \end{aligned}$$

where $r_{1 }(x):=\sqrt{ 1+| x|^{2}},\;x\in \mathbf {R}^{d_1}$.

Proof

Indeed, for all x and y, we have

$$\begin{aligned} | \phi (x)-\phi (y) |&\le | r^{-\theta _{2}}\nabla \phi | _{0}\int _{0}^{1}r^{\theta _{2}}( x+s( y-x) ) ds| y-x| \\&\le | r^{-\theta _{2}}\nabla \phi | _{0}( r( y) ^{\theta _{2}}\vee r( x) ^{\theta _{2}}) | y-x|, \end{aligned}$$

and hence the claim follows from Lemma 4.3.$\square$

Lemma 4.9

Let $n\in \mathbf {N}$, $\beta ,\mu \in (0,1]$, $\theta _1,\theta _{3},\theta _{4}\ge 0$ be such that $\theta _{3}-\theta _{4}\le 1$. There is a constant $N=N(d_1,\theta _{1},\theta _{3},\theta _{4},n,\beta )$ such that for all $\phi :\mathbf {R} ^{d_1}\rightarrow \mathbf {R}$ with $r_{1}^{-\theta _{1}}\phi \in \mathcal {C} ^{n+\beta }(\mathbf {R}^{d_1},\mathbf {R})$ and $H:\mathbf {R}^{d_1}\rightarrow \mathbf {R}^{d_1}$ with

$$\begin{aligned} |r_{1}^{-\theta _{3}}H|_{0}+|r_{1}^{-\theta _{4}}\nabla H|_{n-1+\mu }=:N_2<\infty , \end{aligned}$$

we have

$$\begin{aligned} r_1^{-\theta _1\theta _3}\cdot (\phi \circ H )|_{0} \le | r_{1}^{-\theta _{1}}\phi | _{0} (1+|r_{1}^{-\theta _{3}}H|_{0})^{\theta _{1}} \end{aligned}$$

and

$$\begin{aligned} |r_{1}^{-\theta _{1}\theta _{3}-\theta _{4}(n+\mu \wedge \beta )}\nabla (\phi \circ H)|_{n-1+\mu \wedge \beta } \le N|r_{1}^{-\theta _{1}}\phi |_{n+\beta }( 1+N_{2}) ^{\theta _{1}+\mu \wedge \beta +n}, \end{aligned}$$

where $r_{1 }(x):=\sqrt{ 1+| x|^{2}},\;x\in \mathbf {R}^{d_1}$.

Proof

It follows immediately from Lemma 4.4 and Remark 4.8 that

$$\begin{aligned} r_1^{-\theta _1\theta _3}\cdot (\phi \circ H )|_{0}\le |r_{1}^{-\theta _{1}}\phi |_{0}(1+|r_{1}^{-\theta _{3}}H|_{0})^{\theta _{1}}. \end{aligned}$$

Using induction, we get that for all x and $|\gamma |=n$,

$$\begin{aligned} \partial ^{\gamma }(\phi (H(x)))=\mathcal {I}_{1}^{\gamma }(x)+\mathcal {I} _{2}^{\gamma }(x)+I_{3}^{\gamma }(x), \end{aligned}$$

where

$$\begin{aligned} \mathcal {I}_{1}^{\gamma }(x)=\sum _{i=1}^{d_{1}}\partial _{i}\phi (H(x))\partial ^{\gamma }H^{i}(x), \end{aligned}$$

$\mathcal {I}_{2}^{\gamma }(x)$ is a finite sum of terms of the form

$$\begin{aligned} \partial _{i_{1}}\cdots \partial _{i_{|\gamma |}}\phi (H(x))\partial ^{ \tilde{\gamma }_{1}}H^{i_{1}}\cdots \partial ^{\tilde{\gamma }_{|\gamma |}}H^{i_{|\gamma |}} \end{aligned}$$

with $i_{1},\ldots ,i_{|\gamma |}\in \{1,2,\ldots ,d\}$, $|\tilde{\gamma } _{1}|=\cdots =|\tilde{\gamma }_{|\gamma |}|=1$, and $\sum _{k=1}^{|\gamma |} \tilde{\gamma }_{k}=\gamma$, if $n\ge 2$ and zero otherwise, and where $\mathcal {I}_{3}^{\gamma }(x)$ is a finite sum of terms of the form

$$\begin{aligned} \partial _{i_{1}}\cdots \partial _{i_{k}}\phi (H(x))\partial ^{\tilde{\gamma } _{1}}H^{i_{1}}(x)\cdots \partial ^{\tilde{\gamma }_{k}}H^{i_{k}}(x) \end{aligned}$$

with $2\le k<n$, $i_{1},i_{2},\ldots ,i_{k}\in \{1,\ldots ,d\}$, and $\sum _{j=1}^{k}\tilde{\gamma }_{j}=\gamma$, $1\le |\tilde{\gamma } _{j}|<|\gamma |,$ if $n\ge 3$, and zero otherwise. Thus, owing to Lemmas 4.4 and 4.6, for any multi-index $\gamma$ with $|\gamma |=n$, we have

$$\begin{aligned}&|r_{1}^{-\theta _{1}\theta _{3}-\theta _{4}}\mathcal {I}_{1}^{\gamma }|_{0}\le N|r_{1}^{-\theta _{1}}\nabla \phi |_{0}(1+|r_{1}^{-\theta _{3}}H|_{0})^{\theta _{1}}|r_{1}^{-\theta _{4}}\partial ^{\gamma }H|_{0},\\&\quad |r_{1}^{-\theta _{1}\theta _{3}-n\theta _{4}}\mathcal {I}_{2}^{\gamma }|_{0}\le N|r_{1}^{-\theta _{1}}\partial ^{\gamma }\phi |_{0}(1+|r_{1}^{-\theta _{3}}H|_{0})^{\theta _{1}}|r_{1}^{-\theta _{4}}\nabla H|_{0}^{n}, \end{aligned}$$

and

$$\begin{aligned} |r_{1}^{-\theta _{1}\theta _{3}-(n-1)\theta _{4}}\mathcal {I}_{3}^{\gamma }|_{0}\le N|r_{1}^{-\theta _{1}}\phi |_{n-1}(1+|r_{1}^{-\theta _{3}}H|_{0}+|r_{1}^{-\theta _{4}}\nabla H|_{n-2})^{\theta _{1}+n-1}, \end{aligned}$$

and hence

$$\begin{aligned} |r^{-\theta _{1}\theta _{3}-n\theta _{4}}\partial ^{\gamma }(\phi \circ H)|_{0}\le N|r_{1}^{-\theta _{1}}\phi |_{n}(1+|r_{1}^{-\theta _{3}}H|_{0}+|r_{1}^{-\theta _{4}}\nabla H|_{n-1})^{\theta _{1}+n}. \end{aligned}$$

Since

$$\begin{aligned} |r_{1}^{-\theta _{1}}\partial _i\phi |_{0}+ [r_1^{-\theta _1}\partial _i\phi ]_{\beta \wedge \mu }:=N_1<\infty , \quad |r_{1}^{-\theta _{3}}H|_{0}+ [r_1^{-\theta _4}H]_1\le N_2 \end{aligned}$$

by Lemma 4.4, we have

$$\begin{aligned}{}[r_1^{-\theta _1\theta _3-\mu \wedge \beta \theta _4}\partial _{i}\phi (H)]_{\beta \wedge \mu }\le NN_1( 1+N_{2})^{\theta _{1}+\mu \wedge \beta }. \end{aligned}$$

Thus, since $[r_1^{-\theta _4}\partial ^{\gamma }H^{i}]_{\mu }\le N_2$, we obtain

$$\begin{aligned}{}[r_{1}^{-\theta _{1}\theta _{3}-(1+\mu \wedge \beta )\theta _{4}}\partial _{i}\phi (H)\partial ^{\gamma }H^{i}]_{\mu \wedge \beta }\le NN_1(1+N_{2})^{\theta _{1}+\mu \wedge \beta +1}. \end{aligned}$$

Appealing to Lemma 4.6, for all multi-indices $\gamma$ with $|\gamma |=n$, we get

$$\begin{aligned} |r_{1}^{-\theta _{1}\theta _{3}-(1+\mu \wedge \beta )\theta _{4}}\mathcal {I} _{1}^{\gamma }|_{\mu \wedge \beta }\le N|r_{1}^{-\theta _{1}}\phi |_{1+\mu \wedge \beta }\left( 1+N_{2}\right) ^{\theta _{1}+\mu \wedge \beta +1}. \end{aligned}$$

Similarly, applying Lemmas 4.4 and 4.6, we have

$$\begin{aligned}&|r_{1}^{-\theta _{1}\theta _{3}-(n+\mu \wedge \beta )\theta _{4}}\mathcal {I} _{2}^{\gamma }|_{\mu \wedge \beta }+|r_{1}^{-\theta _{1}\theta _{3}-(n-1+\mu \wedge \beta )\theta _{4}}\mathcal {I}_{3}^{\gamma }|_{\mu \wedge \beta }\\&\quad \le N|r_{1}^{-\theta _{1}}\phi |_{n+\mu \wedge \beta }\left( 1+N_{2}\right) ^{\theta _{1}+n+\mu \wedge \beta }. \end{aligned}$$

Another application of Lemma 4.6 completes the proof.$\square$

We will now provide some useful estimates of composite functions of diffeomorphisms.

Lemma 4.10

Let $H:\mathbf {R}^{d_1}\rightarrow \mathbf {R} ^{d_1}$ be continuously differentiable and assume that for all $x\in \mathbf {R}^{d_1}$,

$$\begin{aligned} | H(x)| \le L_{0}+L_{1}|x|\quad \text {and}\quad |\nabla H(x)|\le L_{2}. \end{aligned}$$

Assume that for all $x\in \mathbf {R}^{d_1}$ $\kappa (x)=(I_{d_1}+\nabla H(x))^{-1}$ exists and $|\kappa (x)|\le N_{\kappa }$. In what follows $r_{1 }(x):=\sqrt{ 1+| x|^{2}},\;x\in \mathbf {R}^{d_1}$.

• (1) Then the mapping $\tilde{H}(x):=x+H(x)$ is a diffeomorphism with $\tilde{H}^{-1}(x)=x-H( \tilde{H}^{-1}(x))=:x+F(x)$ and for all $x\in \mathbf {R}^{d_1}$,

  $$\begin{aligned}&| F(x)| \le L_0+L_1L_0N_{\kappa }+L_1N_{\kappa }| x|, \\&|\nabla F(x)| \le N_{\kappa }L_{2}, \quad \quad |\left( I_{d_1}+\nabla F(x)\right) ^{-1}| \le 1+L_{2}. \end{aligned}$$

  For all $p\in \mathbf {R}$, there is a constant $N=N(L_0,L_1,N_{\kappa },p)$ such that for all $x\in \mathbf {R}^{d_1}$,

  $$\begin{aligned} \frac{r_1^p(\tilde{H}(x))}{r_1^p(x)}+\frac{r_1^p(\tilde{H}^{-1}(x))}{r_1^p(x)}\le N, \quad r_1^{-1}(x)|H^{i}(x)+F^{i}(x)|\le N[H]_1|r_1^{-1}H|_0. \end{aligned}$$

  Moreover, there is a constant $N=N(L_0,L_1,N_{\kappa },p)$ such that

  $$\begin{aligned}&\left| \frac{r_1^p(\tilde{H})}{r_1^p}-1+\mathbf {1}_{(1,2]}(\alpha )pH^{i} r_1^{-2}x^i\right| _\alpha +\left| \frac{r_1^p(\tilde{H}^{-1})}{r_1^p}-1-\mathbf {1}_{(1,2]}(\alpha )pF^{i}r_1^{-2}x^i\right| _{\alpha }\\&\le N(|r_1^{-1}H|_0^{[\alpha ]^-+1}+[H]^{[\alpha ]^-+1}_1). \end{aligned}$$

• (2) If for some $\beta >1$, $|\nabla H|_{\beta -1}\le L_3$, then there is a constant $N=N(d_1,\beta ,N_{\kappa },L_{3})$ such that

  $$\begin{aligned} |\nabla F| _{\beta -1}\le N|\nabla H| _{\beta -1}. \end{aligned}$$

  (4.2)

• (3) If for some $\beta \ge 1$, $|\nabla H|_{\beta -1}\le L_3$, then for all $\theta \ge 0$, there is a constant $N=N(d_1,$ $\beta ,N_{\kappa },$ $L_1,L_3,\theta )$ such that

  $$\begin{aligned} \left| \frac{r^{\theta }_1\circ \tilde{H}^{-1}}{r_1^{\theta }} -1\right| _{\beta }\le N( |r_1^{-1}H|_0+|\nabla H|_{\beta -1}). \end{aligned}$$

• (4) If $|H|_{0 }\le L_{4}$, and for some $\beta >0$, $|\nabla H|_{\beta \vee 1-1 }\le L_{5}$ and $\phi :\mathbf {R} ^{d_1}\rightarrow \mathbf {R}$ is such that for some $\mu \in (0,1]$ and $\theta \ge 0$, $r_1^{-\theta }\phi \in C^{\beta +\mu }( \mathbf {R}^{d_1};\mathbf {R})$, then there is a constant $N=N(d_1,\beta ,\mu ,N_{ \kappa },L_{4},L_5,\theta )$ such that

  $$\begin{aligned}&|r_1^{-\theta }(\phi \circ \tilde{H}^{-1}-\phi )| _{\beta } \le N|r_1^{-\theta }\phi |_{\beta }(|H|_0+|\nabla H|_{\beta \vee 1-1})\\&\quad + NL_4^{\mu }\mathbf {1}_{(0,1]}(\{\beta \}^++\mu )\sum _{|\gamma |=[\beta ]^-} [\partial ^{\gamma }(r_1^{-\theta }\phi )]_{\{\beta \}^++\mu } \\&\quad +N\mathbf {1}_{(1,2]}(\{\beta \}^++\mu )\sum _{|\gamma |=[\beta ]^-}\left( L_4^{\mu }[\nabla \partial ^{\gamma }(r_1^{-\theta }\phi )]_{\{\beta \}^++\mu -1}+|\nabla \partial ^{\gamma }(r_1^{-\theta }\phi )|_{0}|\nabla H|_0 \right) . \end{aligned}$$

Proof

(1) Since $(I_{d_1}+\nabla H(x))^{-1}$ exists for all x, it follows from Theorem 0.2 in [3] that the mapping $\tilde{H}$ is a global diffeomorphism. For all x, we easily verify $\tilde{H}^{-1}(x)=x-H(\tilde{H }^{-1} (x))$ by substituting $\tilde{H}(x)$ into the expression. Simple computations show that for all x, we have

$$\begin{aligned}&|\nabla \tilde{H}(x)|\le 1+L_{2},\;|\nabla \tilde{H}^{-1}(x)|=|\kappa (\tilde{H}^{-1}(x))|\le N_{\kappa }, \\&\quad |\nabla F(x)|=|\nabla H(\tilde{H}^{-1}(x))\nabla \tilde{H}^{-1}(x)|\le N_{\kappa }L_{2}, \\&\quad |(I_{d_1}+\nabla F(x))^{-1}|=|\nabla \tilde{H}^{-1}(x)^{-1}|=|\kappa (\tilde{ H}^{-1}(x))^{-1}|\\&\quad =|I_{d_1} +\nabla H(\tilde{H}^{-1}(x))|\le 1+L_{2}. \end{aligned}$$

For all x and y, we easily obtain

$$\begin{aligned} |\tilde{H}(x)-\tilde{H}(y)|\le (1+L_{2})|x-y|,\quad |\tilde{H}^{-1}(x)- \tilde{H}^{-1}(y)|\le N_{\kappa }|x-y|, \end{aligned}$$

and hence

$$\begin{aligned} N_{\kappa }^{-1}|x-y|\le |\tilde{H}(x)-\tilde{H}(y)|,\quad (1+L_{2})^{-1}|x-y|\le |\tilde{H}^{-1}(x)-\tilde{H}^{-1}(y)|. \end{aligned}$$

(4.3)

Making use of (4.3), for all x, we get

$$\begin{aligned} N_{\kappa }^{-1}|x|\le L_{0}+|\tilde{H}(x)|,\quad |\tilde{H}^{-1}(x)|\le N_{\kappa }L_{0}+N_{\kappa }|x|,\quad |x| \le L_{0}+L_{1}|\tilde{H} ^{-1}(x)|, \end{aligned}$$

and thus

$$\begin{aligned} |F(x)|\le L_{0}+L_{1}N_{\kappa }L_{0}+L_{1}N_{\kappa }|x|. \end{aligned}$$

The rest of the estimates then follow easily from the above estimates and Taylor’s theorem.

(2) Using the chain rule, for all x, we obtain

$$\begin{aligned} \nabla F(x)=-\nabla H(\tilde{H}^{-1}(x))\nabla \tilde{H}^{-1}(x)=-\nabla H( \tilde{H}^{-1}(x))\kappa (\tilde{H}^{-1}(x)), \end{aligned}$$

(4.4)

and hence $|\nabla F|_{0}\le N_{\kappa }|\nabla H|_{0}.$ For all x and y , we have

$$\begin{aligned} \kappa (\tilde{H}^{-1}(y))-\kappa (\tilde{H}^{-1}(x))=\kappa (\tilde{H} ^{-1}(y))[\nabla H(\tilde{H}^{-1}(x))-\nabla H(\tilde{H}^{-1}(y))]\kappa ( \tilde{H}^{-1}(x)), \end{aligned}$$

and thus since $[\tilde{H}^{-1}]_{1}\le (1+ N_{\kappa }L_{3})$ by part (1), we have for all $\delta \in (0,1\wedge (\beta -1)],$

$$\begin{aligned}{}[\kappa (\tilde{H}^{-1})]_{\delta }\le N_{\kappa }^{2}(1+ N_{\kappa }L_{3})^{\delta }[\nabla H]_{\delta }. \end{aligned}$$

It follows that there is a constant $N=N(N_{\kappa },L_{3})$ such that for all $\delta \in (0,1\wedge (\beta -1)]$,

$$\begin{aligned} |\nabla F|_{\delta }\le N|H|_{\delta }. \end{aligned}$$

It is well-known that the inverse map $\mathfrak {I}$ on the set of invertible $d_1\times d_1$ matrices is infinitely differentiable and for each n, there exists a constant $N=N(n,d_1)$ such that for all invertible matrices A , the n-th derivative of $\mathfrak {I}$ evaluated at A, denoted $\mathfrak {I}^{(n)}(A)$, satisfies

$$\begin{aligned} | \mathfrak {I}^{(n)}(A)| \le N|A^{-n-1}|\le N| A^{-1}| ^{n+1}. \end{aligned}$$

Using induction we find that for all multi-indices $\gamma$ with $|\gamma |\le [\beta ]^{-}$ and for all x, $\partial ^{\gamma }F(x)$ is a finite sum of terms, each of which is a finite product of

$$\begin{aligned}&\partial ^{\bar{\gamma }}H(\tilde{H}^{-1}(x)),\quad \kappa (\tilde{H} ^{-1}(x))^{\bar{n}},\quad \mathfrak {I}^{(\bar{n}-1)}(I+\nabla H(\tilde{H} ^{-1}(x))),\;\;|\bar{\gamma }|\\&\quad \le |\gamma |,\;\;\bar{n}\in \{1,\ldots ,|\gamma |\}. \end{aligned}$$

Therefore, differentiating (4.4) and estimating directly we easily obtain (4.2).

(3) For each x, we have

$$\begin{aligned} \frac{r_{1}(\tilde{H}^{-1}(x))^{\theta }}{r_{1}(x)^{\theta }} -1&=r_{1}(x)^{-\theta }\int _{0}^{1}r_{1}(G_s(x))^{\theta -2}G_s(x)^{* }F(x)ds\nonumber \\&=\int _{0}^{1}\frac{r_{1}^{\theta -1}( G_s(x)) }{r_{1}(x)^{\theta -1}} \kappa (G_s(x))^*dsr_{1}(x)^{-1}F(x), \end{aligned}$$

where $G_s(x): =x+sF(x)$, $s\in [0,1]$, and $J(x):=r_{1}(x)^{-1}x.$ According to part (1) and (2), we have $| r_{1}^{-1}F| _{0}\le N| r_{1}^{-1}H| _{0}$ and $|\nabla F|_{\beta -1}\le N|\nabla H|_{\beta -1}$, and hence

$$\begin{aligned} | r_{1}^{-1}G_s| _{0} \le N(1+| r_{1}^{-1}H| _{0}),\quad |\nabla G_s(x)|_{\beta -1}\le N(1+|\nabla H|_{\beta -1}). \end{aligned}$$

and

$$\begin{aligned} | J\circ G_s| _{\beta }\le N(1+| r_{1}^{-1}H| _{0}+|\nabla H|_{\beta -1}), \end{aligned}$$

for some constant N independent of s. Moreover, using Lemma 4.9 we find

$$\begin{aligned} | r_1^{1-\theta } \cdot (r_1^{\theta -1}\circ G_s )| _{\beta }\le N\left( 1+| r_{1}^{-1}H| _{0}+|\nabla H|_{\beta -1}\right) ^{\theta +\beta }. \end{aligned}$$

The statement then follows.

(4) First, we will consider the case $\theta =0$. By part (1), we have that for all $\bar{\mu }\in (0,(\beta +\mu )\wedge 1]$,

$$\begin{aligned} |\phi \circ \tilde{H}^{-1}-\phi |_0\le [\phi ]_{\bar{\mu }}|H\circ \tilde{H} ^{-1}|_0^{\mu }\le [\phi ]_{\bar{\mu }}|H|_0^{\bar{\mu }} . \end{aligned}$$

Let us consider the case $\beta \le 1$. For each x, let $\mathcal {J} (x)=\phi (\tilde{H}^{-1}(x))-\phi (x)$. For all x and y, it is clear that

$$\begin{aligned} |\mathcal {J}(x)-\mathcal {J}(y)|\le A(x,y)+B(x,y)+C(x,y), \end{aligned}$$

where

$$\begin{aligned} A(x,y):=|\mathcal {J}(x)|\mathbf {1}_{[L_4,\infty )}(|x-y|),\quad B(x,y):=| \mathcal {J}(y)|\mathbf {1}_{[L_4,\infty )}(|x-y|), \end{aligned}$$

and

$$\begin{aligned} C(x,y):=|\mathcal {J}(x)-\mathcal {J}(y)|\mathbf {1}_{[0,L_4)}(|x-y|). \end{aligned}$$

Moreover, owing to part (1), if $\beta +\mu \le 1$, then for all x and y, we have

$$\begin{aligned}&A(x,y)\le [\phi ]_{\beta +\mu }L_4^{\beta +\mu }\mathbf {1}_{[L_4, \infty )}(|x-y|)\le [\phi ]_{\beta +\mu } L_4^{\mu }|x-y|^{\{\beta \}^+}, \\&\quad B(x,y)\le [\phi ]_{\beta +\mu }L_4^{\mu }|x-y|^{\beta }, \end{aligned}$$

and

$$\begin{aligned}&C(x,y)\le [\phi ]_{\beta +\mu } [\tilde{H}^{-1}]^{\beta +\mu }_1|x-y|^{\beta +\mu }\mathbf {1}_{[0,L_4)}(|x-y|)\\&\quad + [\phi ]_{\beta +\mu }|x-y|^{\beta +\mu }\mathbf {1}_{[0,L_4)}(|x-y|) \\&\quad \le N[\phi ]_{\beta +\mu }L_4^{ \mu }|x-y|^{\beta } \end{aligned}$$

for some constant $N=N(\mu ,N_{\kappa },L_4)$. Using the identity

$$\begin{aligned}&\mathcal {J}(x)-\mathcal {J}(y) \\&\quad =-\int _0^1 \left( \nabla \phi \left( x-s H(\tilde{H}^{-1}(x))\right) - \nabla \phi \left( y-s H(\tilde{H}^{-1}(y))\right) \right) H(\tilde{H} ^{-1}(x))ds \\&\quad - \int _0^1\nabla \phi \left( y-s H(\tilde{H}^{-1}(y))\right) (H(\tilde{H} ^{-1}(y))-H(\tilde{H}^{-1}(x)))ds, \end{aligned}$$

and part (1), if $\beta +\mu >1$, then there is a constant $N=N(\mu ,N_{\kappa },L_4)$ such that for all x and y,

$$\begin{aligned} |\mathcal {J}(x)-\mathcal {J}(y)|\mathbf {1}_{[L_4,\infty )}(|x-y|)&\le N([\nabla \phi ]_{\beta +\mu -1} |x-y|^{\beta +\mu -1}L_4 \\&\quad +|\nabla \phi |_0|x-y[H]_1)\mathbf {1}_{[L_4,\infty )}(|x-y|) \\&\le N[\nabla \phi ]_{\beta +\mu -1}L_4^{\mu }|x-y|^{\beta }\!+\!N|\nabla \phi |_0|\nabla H|_0|x\!-\!y|. \end{aligned}$$

Moreover, since

$$\begin{aligned}&\mathcal {J}(x)-\mathcal {J}(y) \\&\quad =\int _0^1 \nabla \phi \left( \tilde{H}^{-1}(x+s(y-x))\right) \left( \nabla \tilde{H}^{-1}(x+s(y-x))-I_{d_1}\right) (x-y)ds \\&\quad +\int _0^1 \left( \nabla \phi \left( \tilde{H}^{-1}\left( x+s(y-x)\right) \right) -\nabla \phi \left( x+s(y-x)\right) \right) (x-y)ds, \end{aligned}$$

by part (1) and (4.2), if $\beta +\mu >1$, we attain that there is a constant $N=N(\mu ,N_{\kappa },L_4)$ such that for all x and y,

$$\begin{aligned}&|\mathcal {J}(x)-\mathcal {J}(y)|\mathbf {1}_{[0,L_4)}(|x-y|)\le (|\nabla \phi |_0|\nabla H|_0\\&\quad +[\nabla \phi ]_{\beta +\mu -1}L_4^{\beta +\mu -1})|x-y| \mathbf {1}_{[0,L_4)}(|x-y|) \\&\quad \le |\nabla \phi |_0|\nabla H|_0|x-y|+[\nabla \phi ]_{\beta +\mu -1}L_4^{\mu }|x-y|^{\beta }. \end{aligned}$$

Combining the above estimates, we get that for all $\beta \le 1$ and $\mu \in (0,1]$, there is a constant $N=N(\mu ,N_{\kappa },L_4)$ such that

$$\begin{aligned}&[\phi \circ \tilde{H}^{-1}-\phi ]_{\beta }\le N\mathbf {1}_{[0,1]}(\beta +\mu )[ \phi ]_{\beta +\mu }L_4^{\mu }\nonumber \\&\quad +N\mathbf {1}_{(1,2]}(\beta +\mu )\left( [\nabla \phi ]_{\beta +\mu -1}+|\nabla \phi |_{0}|\nabla H|_0\right) . \end{aligned}$$

(4.5)

This proves the desired estimate for $\beta \le 1$ and $\theta =0$. We now consider the case $\beta >1$. For $\beta >1$, it is straightforward to prove by induction that for all multi-indices $\gamma$ with $1\le |\gamma |\le [\beta ]^-$ and for all x,

$$\begin{aligned} \partial ^{\gamma } (\phi (\tilde{H}^{-1}))(x)=\mathcal {J}^{\gamma }_1(x)+ \mathcal {J}^{\gamma }_2(x)+\mathcal {J}^{\gamma }_3(x)+\mathcal {J} ^{\gamma }_4(x), \end{aligned}$$

where

$$\begin{aligned} \mathcal {J}^{\gamma }_1(x) := & {} \partial ^{\gamma }\phi (\tilde{H}^{-1}(x)),\\ \mathcal {J}^{\gamma }_2(x)= & {} \partial ^{\gamma }\phi (\tilde{H}^{-1})(\partial _{1} \tilde{H}^{-1;1})^{\gamma _1}\cdots (\partial _{d}\tilde{H}^{-1;d})^{ \gamma _d}-1, \end{aligned}$$

$\mathcal {J}^{\gamma }_3(x)$ is a finite sum of terms of the form

$$\begin{aligned} \partial _{j_{1}}\cdots \partial _{j_{k}}\phi (\tilde{H}^{-1}(x))\partial ^{ \tilde{\gamma }_{1}}\tilde{H}^{-1;j_{1}}(x)\cdots \partial ^{\tilde{\gamma } _{k}}\tilde{H}^{-1;j_{k}}(x) \end{aligned}$$

with $1\le k<[\beta ]^-$, $j_1,\ldots ,j_k\in \{1,\ldots ,d\}$, and $\sum _{j=1}^{k}\tilde{\gamma }_{j}=\gamma$, and $\mathcal {J}_4(x)$ is a finite sum of terms of the form

$$\begin{aligned} \partial _{j_{1}}\ldots \partial _{j_{\left[ \beta \right] ^{-}}}\phi (\tilde{ H}^{-1}(x))\partial _{i_{1}}\tilde{H}^{-1;j_{1}}(x)\cdots \partial _{i_{ \left[ \beta \right] ^{-}}}\tilde{H}^{-1;j_{\left[ \beta \right] ^{-}}}(x) \end{aligned}$$

with $i_1,j_1,\ldots ,i_{[\beta ]^-},j_{[\beta ]^-}\in \{1,\ldots ,d\}$ and at least one pair $i_{k}\ne j_{k}$. Since for all x,

$$\begin{aligned} \nabla \tilde{H}^{-1}(x)=I+\nabla F(x) \end{aligned}$$

and (4.2) holds, there is a constant $N=N(d_1,\beta )$ such that

$$\begin{aligned} \sum _{1\le |\gamma |\le \beta }\sum _{i=2}^{4}|\mathcal {J}^{\gamma }_{i}|_{0}+ \sum _{|\gamma |=\beta }\sum _{i=2}^{4}|\mathcal {J}^{\gamma }_{i}|_{\{\beta \}^+} \le N|\nabla \phi |_{\beta -1 }|\nabla F|_{\beta -1 } \!\le \! N|\nabla \phi |_{\beta -1} |\nabla H|_{\beta -1 }. \end{aligned}$$

If $\beta >2$, then for all multi-indices $\gamma$ with $1\le |\gamma |<[\beta ]^-$, we get

$$\begin{aligned} |\mathcal {J}^{\gamma }_1-\partial ^{\gamma }\phi |_0=|\partial ^{\gamma }\phi \circ \tilde{H}^{-1}-\partial ^{\gamma }\phi |_0\le [\partial ^{\gamma }\phi ]_{1}|H|_0. \end{aligned}$$

It is easy to see that there is a constant $N=N(L_4,N_{\kappa })$ such that for all $\gamma$ with $|\gamma |=[\beta ]^-$ and all $\bar{\mu }\in (0,(\{\beta \}^++\mu )\wedge 1]$,

$$\begin{aligned} |\mathcal {J}^{\gamma }_1-\partial ^{\gamma }\phi |_{0}=|\partial ^{\gamma }\phi \circ \tilde{H}^{-1}-\partial ^{\gamma }\phi |_{0}\le [\partial ^{\gamma }\phi ]_{\bar{\mu }}|H|_0^{\bar{\mu }}. \end{aligned}$$

Moreover, appealing to the estimate (4.5) we obtain

$$\begin{aligned}&[\mathcal {J}^{\gamma }_1-\partial ^{\gamma }\phi ]_{\{\beta \}^+} \\&\quad \le NL_4^{\mu }\mathbf {1}_{[0,1]}(\{\beta \}^++\mu )[\partial ^{\gamma }\phi ]_{\{\beta \}^++\mu }\\&\quad +N\mathbf {1}_{(1,2]}(\{\beta \}^++\mu )\left( [\nabla \partial ^{\gamma }\phi ]_{\{\beta \}^++\mu -1}+|\nabla \partial ^{\gamma } \phi |_{0}|\nabla H|_0\right) . \end{aligned}$$

Let us now consider the case $\theta >0$. The following decomposition obviously holds for all x:

$$\begin{aligned}&r_1(x)^{-\theta }\phi (\tilde{H}^{-1}(x))-r_1(x)^{-\theta }\phi (x)\\&\quad =\hat{\phi }( \tilde{H}^{-1})-\hat{\phi }(x)+\left( \frac{r_1(\tilde{H}^{-1}(x))^{\theta }}{ r_1(x)^{\theta }}-1\right) \hat{\phi }(\tilde{H}^{-1}(x)), \end{aligned}$$

where $\hat{\phi }=r_1^{-\theta }\phi \in C^{\beta }( \mathbf {R}^{d_1};\mathbf {R} ^{d_1}).$ Thus, to complete the proof we require

$$\begin{aligned} |\hat{\phi }\circ \tilde{H}^{-1}|_{\beta }\le N|\hat{\phi }|_{\beta } \quad \text {and}\quad \left| \frac{r^{\theta }_1\circ \tilde{H}^{-1}}{r_1^{\theta }} -1\right| _{\beta }\le N(|H|_0+|\nabla H|_{\beta \vee 1-1}). \end{aligned}$$

The latter inequality was proved in part (3) and the first inequality follows from part (2) and Lemma 4.9.$\square$

Remark 4.11

Let $H:\mathbf {R}^{d_1}\rightarrow \mathbf {R} ^{d_1}$ be continuously differentiable and assume that for all x,

$$\begin{aligned} |\nabla H(x)|\le \eta <1. \end{aligned}$$

Then for all $x\in \mathbf {R}^{d_1}$,

$$\begin{aligned} |(I_{d_1}+\nabla H(x))^{-1}|\le | I_{d_1}+\sum _{k=1}^{\infty }(-1)^{k}\nabla H(x)^{k}| \le \frac{1}{1-\eta }. \end{aligned}$$

1.4 Stochastic Fubini theorem

Let $m=(m_t^{\varrho })_{t\le T}$, $\varrho \ge 1$, be a sequence of $\mathbf {F }$-adapted locally square integrable continuous martingales issuing from zero such that $\mathbf {P}$-a.s. for all $t\in [0,T]$, $\langle m^{\varrho _1},m^{\varrho _2}\rangle _t=0$ for $\varrho _1\ne \varrho _2$ and $\langle m^{\varrho }\rangle _t=N_t$ for $\varrho \ge 1$, where $N_t$ is a ${\mathcal {P}}_T$-measurable continuous increasing processes issuing from zero. Let $\eta (dt,dz)$ be a $\mathbf {F}$-adapted integer-valued random measure on $([0,T]\times U,{\mathcal {B}}([0,T])\otimes \mathcal {U})$, where $(U,\mathcal {U })$ is a Blackwell space. We assume that $\eta (dt,dz)$ is optional, ${\mathcal {P}}_T\otimes \mathcal {U}$-sigma-finite, and quasi-left continuous. Thus, there exists a unique (up to a $\mathbf {P}$-null set) dual predictable projection (or compensator) $\eta ^p(dt,dz)$ of $\eta (dt,dz)$ such that $\mu (\omega ,\{t\}\times U)=0$ for all $\omega$ and t. We refer the reader to Chapter II, Sect. 1, in [12] for any unexplained concepts relating to random measures.

Let $(X,\Sigma ,\mu )$ be a sigma-finite measure space; that is, there is an increasing sequence of $\Sigma$-measurable sets $X_n$, $n\in \mathbf {N}$, such that $X=\cup _{n=1}^{\infty } X_n$ and $\mu (X_n)<\infty$ for each n. Let $f:\Omega \times [0,T] \times X\rightarrow \mathbf {R}^{d_2}$ be $\mathcal { R}_{T}\otimes \Sigma$-measurable, $g:\Omega \times [0,T] \times X\rightarrow \ell _{2}(\mathbf {R}^{d_2})$ be $\mathcal {R}_{T}\otimes \Sigma /{\mathcal {B}} (\ell _{2}(\mathbf {R}^{d_2}))$-measurable, and $h:\Omega \times [0,T] \times X\times U\rightarrow \mathbf {R}^{d_2}$ be ${\mathcal {P}}_{T}\otimes \Sigma \otimes \mathcal {U}$-measurable. Moreover, assume that for all $t\in [0,T]$ and $x\in X$, $\mathbf {P}$-a.s.

$$\begin{aligned} \int _{]0,T]}|g_t(x)|^{2}dN_t+\int _{]0,T]}\int _{U}|h_t(x,z)|^{2} \eta ^p(dt,dz)<\infty . \end{aligned}$$

Let $F=F_t(x):\Omega \times [0,T]\times X\rightarrow \mathbf {R}^{d_2}$ be $\mathcal {O}_T\otimes {\mathcal {B}}(X)$-measurable and assume that for $d \mathbf {P} \mu$-almost all $(t,x)\in [0,T]\times X$,

$$\begin{aligned} F_t(x) =\int _{]0,t]}g^{\varrho }_s(x)dm^{\varrho }_s+\int _{]0,t]}\int _{U}h_s(x,z) \tilde{\eta }(dt,dz), \end{aligned}$$

where $\tilde{\eta }(dt,dz)=\eta (dt,dz)-\eta ^p(dt,dz)$.

The following version of the stochastic Fubini theorem is a straightforward extension of [17, Lemma 2.6] and [26, Corollary 1]. See also in [43, Proposition 3.1], [40, Theorem 2.2], and [37, Theorem 1.4.8]. Indeed, to prove it for a bounded measure we can use a monotone class argument as in [35, Theorem 64]. To handle the general setting with possibly infinite $\mu$, we use assumptions (ii) and (iii) below and take limits on the sets $X_n$ using the Lenglart domination lemma [23, Theorem 1.4.5] and the following well-known inequalities:

$$\begin{aligned}&\mathbf {E} \sup _{t\le T}\left| \int _{]0,t]} g_s^{\varrho }dm_s^{\varrho }\right| \le N\mathbf {E} \left( \int _{]0,T]} |g_t(x)|^2dm_t^{\varrho }\right) ^{1/2}\\&\quad \mathbf {E} \sup _{t\le T}\left| \int _{]0,t]} \int _{U}h_t(x,z)\tilde{\eta } (dt,dz)\right| \le N\mathbf {E} \left( \int _{]0,T]} \int _{U}|h_t(x,z)|^2\eta ^p(dt,dz)\right) ^{1/2}, \end{aligned}$$

where $\tau \le T$ is an arbitrary stopping time and $N=N(T)$ is a constant independent of g and h.

Proposition 4.12

(c.f. Corollary 1 in [26] and Lemma 2.6 in [17]) Assume that

• (1) $\mathbf {P}$-a.s. for all $n\ge 1$,

  $$\begin{aligned}&\int _{X_n}\left( \int _{]0,T]}|g_t(x)|^2dN_t\right) ^{1/2}\mu (dx)\\&\quad +\int _{X_n}\left( \int _{]0,T]}\int _{U}|h_t(x,z)|^2\eta ^p (dt,dz)\right) ^{1/2}\mu (dx)<\infty ; \end{aligned}$$

• (2) $\mathbf {P}$-a.s.

  $$\begin{aligned} \int _{]0,T]} \left( \int _X |g_t(x)|\mu (dx)\right) ^2dt+\int _{]0,T]} \int _U \left( \int _X |h_t(x,z)|\mu (dx)\right) ^2\eta ^p(dt,dz); \end{aligned}$$

• (3) $\mathbf {P}$-a.s. for al $t\in [0,T]$,

  $$\begin{aligned} \int _X |F_t(x)|\mu (dx)<\infty . \end{aligned}$$

Then $\mathbf {P}$-a.s. for all $t\in [0,T]$,

$$\begin{aligned} \int _{X}F_t(x)\mu (dx) =\int _{]0,t]}\int _{X}g^{\varrho }_s(x)\mu (dx)dm^{\varrho }_{s}+\int _{]0,t]} \int _{U}\int _{X}h_s(x,z)\mu (dx)\tilde{\eta }(dr,dz). \end{aligned}$$

We obtain the following corollary by applying Minkowski’s integral inequaility.

Corollary 4.13

Assume that $\mathbf {P}$-a.s.

$$\begin{aligned}&\int _{X}\left( \int _{]0,T]}|g_t(x)|^2dN_t\right) ^{1/2}\mu (dx)\nonumber \\&\quad +\int _{X}\left( \int _{]0,T]}\int _{U_1}|h_t(x,z)|^2\eta ^p(dt,dz)\right) ^{1/2} \mu (dx)<\infty . \end{aligned}$$

(4.6)

Then $\mathbf {P}$-a.s. for all $t\in [0,T]$,

$$\begin{aligned} \int _{X}F_t(x)\mu (dx) =\int _{]0,t]}\int _{X}g^{\varrho }_s(x)\mu (dx)dm^{\varrho }_{s}+\int _{]0,t]} \int _{U}\int _{X}h_s(x,z)\mu (dx)\tilde{\eta }(dr,dz). \end{aligned}$$

Remark 4.14

If $\mu$ is a finite-measure and $\mathbf {P}$ -a.s.

$$\begin{aligned} \int _{X}\int _{]0,T]}|g_t(x)|^2dN_t\mu (dx)+\int _{X}\int _{]0,T]} \int _{U_1}|h_t(x,z)|^2\eta ^p(dt,dz)\mu (dx)<\infty , \end{aligned}$$

then (4.6) holds by Hölder’s inequality.

1.5 Itô-Wentzell formula

Definition 4.15

We say that an $\mathbf {R}^{d_1}$-valued $\mathbf {F}$-adapted quasi-left continuous semimartingale $L_{t}=(L_t^k)_{1\le k\le d_1}$, $t\ge 0$, is of $\alpha$-order for $\alpha \in (0,2]$, if $\mathbf {P}$-a.s. for all $t\ge 0$,

$$\begin{aligned} \sum _{s\le t}|\Delta L_{s}|^{\alpha }<\infty \end{aligned}$$

and

$$\begin{aligned} L_{t}&=L_{0}+\int _{]0,t]}\int _{\mathbf {R}_0^{d_1}} z p^{L}(ds,dz),\text { if }\alpha \in (0,1), \\ L_{t}&=L_{0}+A_{t}+\int _{]0,t]}\int _{| z| \le 1}z q^{L}(ds,dz)+\int _{]0,t]}\int _{| z| >1}z p^{L}(ds,dz), \text { if }\alpha \in [1,2), \\ L_{t}&=L_{0}+A_{t}+L_{t}^{c}+\int _{]0,t]}\int _{| z| \le 1}z q^{L}(ds,dz)+\int _{]0,t]}\int _{| z| >1}z p^{L}(ds,dz), \text { if }\alpha \!=\!2, \end{aligned}$$

where $p^{L}(dt,dz)$ is the jump measure of L with dual predictable projection $\pi ^L (dt,dz)$, $q^{L}$ (dt, dz) $=p^{L}(dt,dz) -\pi ^L (dt,dz)$ is a martingale measure, $A_{t}=(A_t^i)_{1\le i\le d_1}$ is a continuous process of finite variation with $A_0=0$, and $L_{t}^{c}=(L_t^{c;i})_{1\le i\le d_1}$ is a continuous local martingale issuing from zero.

Set $(w^{\varrho })_{\varrho \ge 1}=(w^{1;\varrho })_{\varrho \ge 1}$, $(Z, {\mathcal {Z}},\pi )=({\mathcal {Z}}^{1},{\mathcal {Z}}^{1},\pi ^{1}),$ $p(dt,dz )=p^{1}(dt,dz )$, and $q(dt,dz )=q^{1}(dt,dz )$. Also, set $D=D^{1}$, $E=E^{1},$ and assume $Z=D\cup E$.

Let $f:\Omega \times [0,T] \times \mathbf {R}^{d_1}\rightarrow \mathbf {R}^{d_2}$ be $\mathcal {R}_{T}\otimes {\mathcal {B}}(\mathbf {R}^{d_1})$-measurable, $g:\Omega \times [0,T] \times \mathbf {R}^{d_1}\rightarrow \ell _{2}(\mathbf {R} ^{d_2})$ be $\mathcal {R}_{T}\otimes {\mathcal {B}}(\mathbf {R}^{d_1})/{\mathcal {B}} (\ell _{2}(\mathbf {R}^{d_2}))$-measurable, and $h:\Omega \times [0,T] \times \mathbf {R}^{d_1}\times Z\rightarrow \mathbf {R}^{d_2}$ be ${\mathcal {P}} _{T}\otimes {\mathcal {B}}(\mathbf {R}^{d_1})\otimes {\mathcal {Z}}$-measurable. Moreover, assume that, $\mathbf {P}$-a.s. for all $x\in \mathbf {R}^{d_1}$,

$$\begin{aligned}&\int _{]0,T]}|f_t(x)|dt+\int _{]0,T]}|g_t(x)|^{2}dt<\infty \\&\quad +\int _{]0,T]}\int _{D}|h_t(x,z)|^{2}\pi (dz)dt+\int _{]0,T]}\int _{E}|h_t(x,z)| \pi (dz)dt<\infty . \end{aligned}$$

Let $F=F_t(x):\Omega \times [0,T]\times \mathbf {R}^{d_1}\rightarrow \mathbf {R} ^{d_2}$ be $\mathcal {O}_T\otimes {\mathcal {B}}(\mathbf {R}^{d_1})$-measurable and assume that for all x, $\mathbf {P}$-a.s. for all t,

$$\begin{aligned} F_t(x)&=F_0(x)+\int _{]0,t]}f_s(x)ds+\int _{]0,t]}g^{\varrho }_s(x)dw^{\varrho }_{s} \\&\quad +\int _{]0,t]}\int _{Z}h_s(x,z)[\mathbf {1}_{D}(z)q(ds,dz) +\mathbf {1} _{E}(z)p(ds,dz)]. \end{aligned}$$

For each $n\in \{1,2\}$, let $\bar{C} ^{n}_{loc}(\mathbf {R}^{d_1};\mathbf {R} ^{d_2})$ be space of n-times continuously differentiable functions $f: \mathbf {R}^{d_1}\rightarrow \mathbf {R}^{d_2}$. We now state our version of the Itô-Wentzell formula. For each $\omega ,t$ and x, we denote $\Delta F(x)=F_t(x)-F_{t-}(x)$.

Proposition 4.16

([26, cf. Proposition 1]) Let $(L_t)_{t\ge 0}$ be an $\mathbf {R}^{d_1}$ -valued quasi-left continuous semimartingale of order $\alpha \in (0,2]$. Assume that:

• (1)

  • (a) $\mathbf {P}$-a.s. $F\in D([0,T];\mathcal {C}_{loc}^{\alpha }(\mathbf {R}^d;\mathbf {R}^m)$ if $\alpha$ is fractional and $F\in D([0,T];\bar{C}_{loc}^{\alpha }(\mathbf {R}^d;\mathbf {R}^m)$ if $\alpha =1,2$ ;

  • (b) for $d\mathbf {P}dt$-almost-all $(\omega ,t)\in \Omega \times [0,T]$, $f_t(x)$ and $g_t(x)=(g^{i\varrho }_t(x))_{\varrho \ge 1}\in \ell _2(\mathbf {R}^{d_2})$ are continuous in x and

    $$\begin{aligned}&d\mathbf {P}dt-\lim _{y\rightarrow x}\left[ \int _{D}|h_t(y,z)-h_t(x,z)| ^{2}\pi (dz)\right. \\&\left. \quad +\int _{E}|h_t(y,z)-h_t(x,z)|\pi (dz)\right] = 0; \end{aligned}$$

  • (c) for all $\varrho \ge 1$ and $i\in \{1,\ldots ,d_1\}$ and for $d\mathbf {P}d |\langle L^{c;i},w^{\varrho }\rangle |_t$-almost-all $(\omega ,t)\in \Omega \times [0,T]$, $g^{i\varrho }_t\in C^1_{loc}(\mathbf {R}^d;\mathbf {R})$, if $\alpha = 2$, ;

• (2) for all compact subsets K of $\mathbf {R}^{d_1}$, $\mathbf {P}$-a.s.

  $$\begin{aligned}&\int _{]0,T]}\sup _{x\in K} \left( |f_t(x)|+|g_t(x)|^2+\int _{D}|h_t(x,z)|^2\pi (dz) + \int _{E}|h_t(x,z)|\pi (dz)\right) dt{<}\infty ,\\&\quad \sum _{\varrho \ge 1}\int _{]0,T]}\sup _{x\in K}| \nabla g^{i\varrho }_t(x)|d|\langle L^{c;i},w^{\varrho }\rangle |_t <\infty \\&\quad \sum _{t\le T}|\Delta F_t|_{\alpha \wedge 1;K}|\Delta L_t|^{\alpha \wedge 1}<\infty . \end{aligned}$$

Then $\mathbf {P}$-a.s for all $t\in [0,T]$,

$$\begin{aligned} F_t(L_t)&=F_0(L_{0})+\int _{]0,t]}f_s(L_{s})ds+\int _{]0,t]}g_s^{\varrho } (L_{s})dw^{\varrho }_{s} \nonumber \\&\quad +\int _{]0,t]}\int _{Z}h_s(L_{s-},z)[\mathbf {1}_{D}(z)q(dr,dz) +\mathbf {1}_{E}(z)p(dr,dz)] \nonumber \\&\quad +\int _{]0,t]}\partial _i F_{s-}(L_{s-})[\mathbf {1}_{[1,2]}( \alpha )dA^i_s+\mathbf {1}_{\{2\}}(\alpha )dL ^{c;i}_s] \nonumber \\&\quad +\sum _{s\le t}\left( F_{s-}(L_{s})-F_{s-}(L_{s-})-\mathbf {1} _{[1,2]}(\alpha )\nabla F_{s-}(L_{s-})\Delta L_s\right) \nonumber \\&\quad +\mathbf {1}_{\{2\}}(\alpha )\frac{1}{2}\int _{]0,t]} \partial _{ij}F_s(L_{s})d\langle L^{c;i},L^{c;j}\rangle _{s} \nonumber \\&\quad +\mathbf {1}_{\{2\}}(\alpha )\int _{]0,t]}\partial _ig^{ \varrho }_{s}(L_{s})d\langle w^{\varrho },L^{c;i}\rangle _{s}+\sum _{s\le t}\left( \Delta F_s(L_{s})-\Delta F_s(L_{s-})\right) . \end{aligned}$$

(4.7)

Proof

Since both sides have identical jumps and we can always interlace a finite set of jumps, we may assume that $|\Delta L_{t}|\le 1$ for all $t\in [0,T]$; that is, it is enough to prove the statement for $\tilde{L} _{t}=L_{t}-\sum _{s\le t}\mathbf {1}_{[1,\infty )}(|\Delta L_s|)\Delta L_{s}$, $t\in [0,T]$. It suffices to assume that for some K and all $\omega$, $|L_0|\le K$. For each $R>K$, let

$$\begin{aligned} \tau _R=\inf \left( t\in [0,T]:|A|_t+|\langle L^c\rangle | _t+\sum _{s\le t}|\Delta L_s|^{\alpha }+|L_{t}|> R\right) \wedge T \end{aligned}$$

and note that $\mathbf {P}$-a.s. $\tau _R\uparrow T$ as R tends to infinity. If instead of L, f, g, h,  and F, we take $L_{\cdot \wedge \tau _R}$, $f\mathbf {1}_{(0,\tau _R]}$, $g^{\varrho }\mathbf {1}_{(0,\tau _R]}$, $h\mathbf {1}_{(0,\tau _R]}$, $F\mathbf {1}_{(0,\tau _R]}$, then the assumptions of the proposition hold for this new set of processes. Moreover, if we can prove (4.7) for this new set of processes, then by taking the limit as R tends to infinity, we obtain (4.7). Therefore, we may assume that for some $R>0$ , $\mathbf {P}$-a.s. for all $t\in [0,T]$,

$$\begin{aligned} |A|_t+|\langle L^c\rangle |_t+\sum _{s\le t}|\Delta L_s|^{\alpha }+|L_{t}|\le R. \end{aligned}$$

(4.8)

Let $\phi \in C_{c}^{\infty }( \mathbf {R}^{d_1},\mathbf {R})$ have support in the unit ball in $\mathbf {R}^{d_1}$ and satisfy $\int _{\mathbf {R} ^{d_1}}\phi (x)dx=1,\phi (x)=\phi (-x),$ and $\phi (x)\ge 0$, for all $x\in \mathbf {R}^{d_1}$. For each $\varepsilon \in (0,1)$, let $\phi _{\varepsilon }(x)=\varepsilon ^{-d}\phi \left( x/\varepsilon \right) ,x\in \mathbf {R}^{d_1}$. By Itô’s formula, for all $x\in \mathbf {R}^{d_1}$, $\mathbf {P}$-a.s. for all $t\in [0,T]$,

$$\begin{aligned} F_t(x) \phi _{\varepsilon }(x-L_{t})&= F_0(x)\phi _{\varepsilon }(x-L_0)-\int _{]0,t]}F_{s-}(x) \partial _{i}\phi _{\varepsilon }(x-L_{s-})dL_{s}^{i} \\&\quad +\int _{]0,t]}\phi _{\varepsilon }\left( x-L_{s}\right) f_s(x)ds+\int _{]0,t]}\phi _{\varepsilon }\left( x-L_{s}\right) g^{\varrho }_s(x)dw^{\varrho }_{s} \\&\quad + \mathbf {1}_{\{2\}}(\alpha )\frac{1}{2}\int _{]0,t]}F_s(x)\partial _{ij}\phi _{\varepsilon }(x-L_{s})d\langle L^{c;i},L^{c;j}\rangle _{s} \\&\quad + \mathbf {1}_{\{2\}}(\alpha )\int _{]0,t]}g^{\varrho }_{s}(x)\partial _{i}\phi _{\varepsilon }(x-L_{s})d\langle w^{\varrho },L^{c;i}\rangle _{s} \\&\quad +\int _{]0,t]}\int _{Z}\phi _{\varepsilon }\left( x-L_{s-}\right) h_s(x,z)[\mathbf {1}_{D}(z)q(dr,dz)+\mathbf {1}_{E}(z)p(dr,dz)] \\&\quad +\sum _{s\le t}\Delta F_s(x) \left( \phi _{\varepsilon }(x-L_{s})-\phi _{\varepsilon }\left( x-L_{s-}\right) \right) \\&\quad +\sum _{s\le t}F_{s-}(x)\left( \phi _{\varepsilon }(x-L_{s})-\phi _{\varepsilon }(x-L_{s-})+\,\partial _{i}\phi _{\varepsilon }\left( x- L_{s-}\right) \Delta L_{s}\right) . \end{aligned}$$

Appealing to assumption (2) and (4.8) (i.e. for the integrals against F), we integrate both sides of the above in x, apply Corollary 4.13 (see, also, Remark 4.14) and the deterministic Fubini theorem, and then integrate by parts to get that $\mathbf {P}$-a.s. for all $t\in [0,T]$,

$$\begin{aligned} F^{(\varepsilon ) }_{t }(L_{t})&=F^{(\varepsilon )}_0(L_0)+\int _{]0,t]}\nabla F^{(\varepsilon ) }_{s-}(L_{s-})[\mathbf {1}_{[1,2]}(\alpha )dA^i_s+\mathbf {1} _{\{2\}}(\alpha )dL ^{c;i}_s]\nonumber \\&\quad +\int _{]0,t]}f^{(\varepsilon ) }_s(L_{s})ds+\int _{]0,t]}g^{(\varepsilon ) }_s(L_{s})dw^{\varrho }_{s} \nonumber \\&\quad +\int _{]0,t]}\int _{Z}h^{(\varepsilon ) }_s(L_{s-},z)[\mathbf {1}_{D}(z)q(dr,dz)+\mathbf {1}_{E}(z)p(dr,dz)] \nonumber \\&\quad +\mathbf {1}_{\{2\}}(\alpha )\frac{1}{2}\int _{]0,t]}\partial _{ij}F^{(\varepsilon ) }_s(L_{s})d\langle L^{c;i},L^{c;j}\rangle _{s} + \mathbf {1}_{\{2\}}(\alpha )\int _{]0,t]}\partial _{i}g ^{(\varepsilon );\varrho }_s (L_{s})d\langle w^{\varrho },L^{c;i}\rangle _s \nonumber \\&\quad +\sum _{s\le t}\left( \Delta F^{(\varepsilon ) }_s(L_{s})-\Delta F^{(\varepsilon ) }_s(L_{s-})\right) \nonumber \\&\quad +\sum _{s\le t}\left( F^{(\varepsilon ) }_{s-}(L_{s})-F^{(\varepsilon ) }_{s-}(L_{s-})-\mathbf {1}_{[1,2]}(\alpha )\nabla F^{(\varepsilon ) }_{s-}(L_{s-})\Delta L_s\right) \end{aligned}$$

(4.9)

where for all $\omega ,t,x,$ and z,

$$\begin{aligned} F^{(\varepsilon )}_t(x):= & {} \phi _{\varepsilon }*F_t(x),\;\;f^{(\varepsilon ) }_t=\phi _{\varepsilon }*f_t(x),\\ g ^{(\varepsilon );\varrho }_t(x)= & {} \phi _{\varepsilon }*g^{\varrho }_t (x),\;\;h^{(\varepsilon ) }_t(x,z)=\phi _{\varepsilon }*h_t(x,z), \end{aligned}$$

and $*$ denotes the convolution operator. Let $B_{R+1}=\{x\in \mathbf {R}^{d_1}: |x|\le R+1\}$. Owing to assumption (1)(a) and standard properties of mollifiers, for any multi-index $\gamma$ with $|\gamma |\le \alpha$, $\mathbf {P}$-a.s. for all t,

$$\begin{aligned} |\partial ^{\gamma }F^{(\varepsilon )}_t(L_t)|\le \sup _{t\le T}\sup _{x\in B_{R+1}}| \partial ^{\gamma }F_t(x)|<\infty \end{aligned}$$

and for all x,

$$\begin{aligned} d\mathbf {P} dt-\lim _{\varepsilon \downarrow 0}| \partial ^{\gamma }F^{(\varepsilon )}_t(x)-\partial ^{\gamma }F^{(\varepsilon )}_t(x)|=0. \end{aligned}$$

Similarly, by assumption 1(b), $d\mathbf {P} dt$-almost-all $(\omega ,t)\in \Omega \times [0,T]$,

$$\begin{aligned}&|f^{(\varepsilon )}_t(L_t)|\le \sup _{x\in B_{R+1}}|f_t(x)|<\infty ,\quad |g^{(\varepsilon )}_t(L_t)|\le \sup _{x\in B_{R+1}}|g_t(x)|<\infty , \\&\quad \int _{D}|h^{\varepsilon }_t(L_t,z)|^2\pi (dz)\le \sup _{x\in B_{R+1}}\int _{D}|h_t(x,z)|^2\pi (dz), \\&\quad \int _{E}|h^{\varepsilon }_t(L_t,z)|\pi (dz)\le \sup _{x\in B_{R+1}}\int _{E}|h_t(x,z)|\pi (dz) \end{aligned}$$

and for all x,

$$\begin{aligned} d\mathbf {P} dt-\lim _{\varepsilon \downarrow 0} |f^{(\varepsilon )}_t(x) -f_t(x)|=0, \quad d\mathbf {P} dt-\lim _{\varepsilon \rightarrow 0}| g^{(\varepsilon )}_t(x) -g_t(x)|=0 \end{aligned}$$

and

$$\begin{aligned} d\mathbf {P} dt-\lim _{\varepsilon \downarrow 0}\int _{Z}[\mathbf {1} _{D}(z)|h^{(\varepsilon )}_t(x,z)-h_t(x,z)|^2+\mathbf {1}_{E}(z)|h^{( \varepsilon )}_t(x,z)-h_t(x,z)|]\pi (dz)=0, \end{aligned}$$

where in the last-line we have also used Minkowski’s integral inequality and a standard mollifying convergence argument. Using assumption 1(d), for all $\varrho \ge 1$ and $i\in \{1,\ldots ,d_1\}$ and for $d\mathbf {P} d |\langle L^{c;i},w^{\varrho }\rangle |_t$-almost-all $(\omega ,t)\in \Omega \times [0,T]$

$$\begin{aligned} | \nabla g^{(\varepsilon );i\varrho }_t(L_t)|\le \sup _{x\in B_{R+1}}| \nabla g^{i\varrho }_t(x)| \end{aligned}$$

and for all x,

$$\begin{aligned} d\mathbf {P} d |\langle L^{c;i},w^{\varrho }\rangle |_t-\lim _{\varepsilon \rightarrow 0}| \nabla g^{(\varepsilon );i\varrho }_t(x) -\nabla g^{i\varrho }_t(x)|=0, \;\;\text {if } \alpha = 2. \end{aligned}$$

Owing to assumption 1(a) and (4.8), $\mathbf {P}$-a.s. 

$$\begin{aligned} \sum _{s\le t}|F^{(\varepsilon ) }_{s-}(L_{s})-F^{(\varepsilon ) }_{s-}(L_{s-})- \mathbf {1}_{[1,2]}(\alpha )\nabla F^{(\varepsilon ) }_{s-}(L_{s-})\Delta L_s|\\ \le \sup _{t\le T}|F_{t}|_{\alpha ;B_{R+1}} \sum _{s\le t}|\Delta L_s|^{\alpha }\le R\sup _{t\le T}|F_{t}|_{\alpha ;B_{R+1}}. \end{aligned}$$

Since $\mathbf {P}$-a.s. $F\in D([0,T];\mathcal {C}^{\alpha }(\mathbf {R}^d; \mathbf {R}^m)$, it follows that for all x, $\mathbf {P}$-a.s. for all t,

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}|\Delta F_t^{\varepsilon }(x)-\Delta F_t(x)|=0. \end{aligned}$$

By assumption (2), $\mathbf {P}$-a.s for all t, we have

$$\begin{aligned} \sum _{s\le t}\left( \Delta F^{(\varepsilon ) }_s(L_{s-}+\Delta L_{s})-\Delta F^{(\varepsilon ) }_s(L_{s-})\right) \le \sum _{s\le t}|\Delta F_{t}|_{\alpha \wedge 1;B_{R+1}}|\Delta L_s|^{\alpha \wedge 1}. \end{aligned}$$

Combining the above and using assumptions (1)(a) and (2) and the bounds given in (4.8) and the deterministic and stochastic dominated convergence theorem, we obtain convergence of all the terms in (4.9), which complete the proof.$\square$