Sequential Systems of Reflected Backward Stochastic Differential Equations with Application to Impulse Control

Abstract

We consider a system of finite horizon, sequentially interconnected, obliquely reflected backward stochastic differential equations (RBSDEs) with stochastic Lipschitz coefficients. We show existence of solutions to our system of RBSDEs by applying a Picard iteration approach. Uniqueness then follows by relating the limit to an auxiliary impulse control problem. Moreover, we show that the solution to our system of RBSDEs is connected to weak solutions of a stochastic differential game where one player implements an impulse control while the opponent plays a continuous control that enters the drift term. As all our arguments are probabilistic and hence hold in a non-markovian framework, we are able to consider the setting where the underlying uncertainty in the game stems from an impulsively and continuously controlled path-dependent stochastic differential equation driven by Brownian motion.

1 Introduction

Backward stochastic differential equations (BSDEs) has been a topic of rapid development during the last decades. Non-linear BSDEs were independently introduced in [24] and [7] and has since found numerous applications.

El Karoui et. al. introduced the notion of reflected backward stochastic differential equations (RBSDEs) and demonstrated a link between RBSDEs and optimal stopping in [12]. This was later exploited in a series of articles [15, 18, 20] as a means of finding solutions to optimal switching problems. In [18] existence and uniqueness of solutions to an interconnected systems of reflected BSDEs were shown. Furthermore, it was shown that the solutions are related to optimal switching problems under Knightian uncertainty. Important contributions from the perspective of the present work are also [1, 4] that consider BSDEs where the Lipschitz coefficient on the z-variable of the driver is a stochastic process and the more recent work presented in [10] where a RBSDE with stochastic Lipschitz coefficient is solved.

Although the literature on discretely indexed systems of RBSDEs related to switching problems has grown considerably in the last decade, there is this far no work that deals with systems of RBSDEs related to general impulse control problems. In the present work we aim to add to the literature on RBSDEs by considering a sequentially arranged system of RBSDEs, namely (the notation will be explained later)

$$\begin{aligned} {\left\{ \begin{array}{ll} Y^{v}_t=\xi ^v+\int _t^T f^v(s,Y^{v}_s,Z^{v}_s)ds-\int _t^T Z^{v}_sdW_s+ K^{v}_T-K^{v}_t,\quad \forall t\in [0,T], \\ Y^{v}_t\ge \sup _{b\in U}\{Y^{v\circ (t,b)}_t-c^v(t,b)\}, \quad \forall t\in [0,T],\\ \int _0^T(Y^{v}_t-\sup _{b\in U}\{Y^{v\circ (t,b)}_t-c^v(t,b)\})dK^{v}_t=0. \end{array}\right. } \end{aligned}$$

(1.1)

As opposed to the setting in previous works our family of RBSDEs is continuously parameterized (the parameter v is an impulse control). Moreover, to make our results more applicable, we allow the driver $f^v$ to satisfy a Lipschitz condition on the z-variable that is formulated in terms of a stochastic process. We rely on a Picard iteration approach and the main obstacle we face is showing continuity of the map $(t,b)\mapsto Y^{v\circ (t,b)}_t$ that appears in the barrier to (1.1). In particular, we cannot use a “no free loop” property (see e.g. Step 5 in the proof of Theorem 3.2 in [18]). Instead we rely on a uniform convergence argument that requires some intricate analysis.

A strong motivation for the introduction of non-linear BSDEs was their close connection to various types of stochastic control problems (see e.g. [13, 25, 30]) and stochastic differential games [16, 17, 19]. Analogously, our main motivation for studying (1.1) is its relation to stochastic differential games (SDGs) of impulse versus continuous control.

In impulse control the control-law takes the form $u=(\tau _1,\ldots ,\tau _N;\beta _1,\ldots ,\beta _N)$, where $\tau _1\le \tau _2\le \cdots \le \tau _N$ is a sequence of times when the operator intervenes on the system and $\beta _j$ is the impulse that the operator affects the system with at time $\tau _j$. We restrict our attention to the case of a Brownian filtration ${\mathbb {F}}:=\{{\mathcal {F}}_t\}_{t\ge 0}$ and assume that the $\tau _j$ are ${\mathbb {F}}$-stopping times and that $\beta _j$ is ${\mathcal {F}}_{\tau _j}$-measurable and take values in a compact subset U of ${\mathbb {R}}^d$.

We extend the results in [22] to the two-player, zero-sum game setting by considering a weak formulation of the problem of maximizing the reward functional

$$\begin{aligned} J(u,\alpha ):={\mathbb {E}}\Big [\int _0^T\phi (t,X^{u,\alpha }_t,\alpha _t)dt +\psi (X^{u,\alpha }_T)-\sum _{j=1}^N\ell (\tau _j,X^{[u]_{j-1}, \alpha }_{\tau _j},\beta _j)\Big ] \end{aligned}$$

(1.2)

over impulse controls, u, when simultaneously a minimization is performed over continuous controls $\alpha :=(\alpha _s)_{0\le s\le T}$, taking values in a compact subset A of ${\mathbb {R}}^d$. Here, $[u]_j:=(\tau _1,\ldots ,\tau _{N\wedge j};\beta _1,\ldots ,\beta _{N\wedge j})$ and $X^{u,\alpha }$ solves the impulsively controlled path-dependent SDE

$$\begin{aligned} X^{u,\alpha }_t&=x_0+\int _0^ta(s,(X^{u,\alpha }_r)_{ r\le s},\alpha _s)ds +\int _0^t\sigma (s,(X^{u,\alpha }_r)_{ r\le s})dW_s, \quad \forall t\in [0,\tau _{1}) \end{aligned}$$

(1.3)

$$\begin{aligned} X^{u,\alpha }_{t}&=\Gamma (\tau _j,X^{[u]_{j-1},\alpha }_{\tau _j},\beta _j) +\int _{\tau _j}^ta(s,(X^{u,\alpha }_r)_{ r\le s},\alpha _s)ds\nonumber \\&\quad +\int _{\tau _j}^t\sigma (s,(X^{u,\alpha }_r)_{ r\le s})dW_s, \quad \forall t\in [\tau _{j},\tau _{j+1}) \end{aligned}$$

(1.4)

for $j=1,\ldots ,N$, with $\tau _{N+1}:=\infty$. By considering systems of reflected BSDEs with drivers that satisfy a stochastic Lipschitz condition we are able to relax the common assumption that $|\sigma ^{-1}(t,x)a(t,x,\alpha )|$ is bounded and instead assume a linear growth, i.e. that $|\sigma ^{-1}(t,x)a(t,x,\alpha )|\le k_L(1+\sup _{s\in [0,t]}|x_s|)$, for some constant $k_L>0$.

The remainder of the article is organised as follows. In the next section we set the notation and specify what we mean by a solution to (1.1). Moreover, in this section we formulate a stability and a moment estimate for solutions to RBSDEs with a stochastic Lipschitz coefficient. The proofs of these estimates are postponed to Appendix A. In Section 3 we turn to sequential systems of RBSDEs and show that (1.1) admits a unique solution. Then, in Section 4, we show that the above formulated game has a saddle-point and give a representation of the corresponding optimal controls by relating solutions to (1.1) to weak formulations of the SDG at hand. This is followed by some concluding remarks in Section 5.

2 Preliminaries

We let $(\Omega ,{\mathcal {F}},{\mathbb {F}},{\mathbb {P}})$ be a complete filtered probability space, where ${\mathbb {F}}:=({\mathcal {F}}_t)_{0\le t\le T}$ is the augmented natural filtration of a d-dimensional Brownian motion W and ${\mathcal {F}}:={\mathcal {F}}_T$, where $T\in (0,\infty )$ is the horizon.

Throughout, we will use the following notation:

• We let ${\mathbb {E}}$ denote expectation with respect to ${\mathbb {P}}$ and for any other probability measure ${\mathbb {Q}}$ on $(\Omega ,{\mathcal {F}})$, we denote by ${\mathbb {E}}^{{\mathbb {Q}}}$ expectation with respect to ${\mathbb {Q}}$.

• ${\mathcal {P}}_{{\mathbb {F}}}$ is the $\sigma$-algebra of ${\mathbb {F}}$-progressively measurable subsets of $[0,T]\times \Omega$.

• For $p\ge 1$, we let ${\mathcal {S}}^{p}$ be the set of all ${\mathbb {R}}$-valued, ${\mathcal {P}}_{{\mathbb {F}}}$-measurable, continuous processes $(Z_t: t\in [0,T])$ such that $\Vert Z\Vert _{{\mathcal {S}}^p}:={\mathbb {E}}\big [\sup _{t\in [0,T]} |Z_t|^p\big ]^{1/p}<\infty$.

• For $p\ge 1$, we let ${\mathcal {S}}^{p}_l$ be the set of all ${\mathbb {R}}$-valued, ${\mathcal {P}}_{{\mathbb {F}}}$-measurable, càglàd processes $(Z_t: t\in [0,T])$ such that $\Vert Z\Vert _{{\mathcal {S}}^p}<\infty$.

• We let ${\mathcal {H}}^{p}$ denote the set of all ${\mathbb {R}}^d$-valued ${\mathcal {P}}_{{\mathbb {F}}}$-measurable processes $(Z_t: t\in [0,T])$ such that $\Vert Z\Vert _{{\mathcal {H}}^p}:={\mathbb {E}}\big [\big (\int _0^T |Z_t|^2 dt\big )^{p/2}\big ]^{1/p}<\infty$.

• For any probability measure ${\mathbb {Q}}$, we let ${\mathcal {S}}^p_{\mathbb {Q}}$ and ${\mathcal {H}}^p_{\mathbb {Q}}$ be defined as ${\mathcal {S}}^p$ and ${\mathcal {H}}^p$, respectively, with the exception that the norm is defined with expectation taken with respect to ${\mathbb {Q}}$, i.e. $\Vert Z\Vert _{{\mathcal {S}}^p_{\mathbb {Q}}}:={\mathbb {E}}^{\mathbb {Q}}\big [\sup _{t\in [0,T]} |Z_t|^p\big ]$ and $\Vert Z\Vert _{{\mathcal {H}}^p_{\mathbb {Q}}}:={\mathbb {E}}^{\mathbb {Q}}\Big [\big (\int _0^T |Z_t|^2 dt\big )^{p/2}\Big ]<\infty$.

• We let ${\mathcal {T}}$ be the set of all ${\mathbb {F}}$-stopping times and for each $\eta \in {\mathcal {T}}$ we let ${\mathcal {T}}_\eta$ be the corresponding subsets of stopping times $\tau$ such that $\tau \ge \eta$, ${\mathbb {P}}$-a.s.

• For each $\tau \in {\mathcal {T}}$, we let ${\mathcal {I}}(\tau )$ be the set of all ${\mathcal {F}}_\tau$-measurable random variables taking values in U, so that ${\mathcal {I}}(\tau )$ is the set of all admissible interventions at time $\tau$.

• We let ${\mathcal {U}}$ be the set of all $u=(\tau _1,\ldots ,\tau _N;\beta _1,\ldots ,\beta _N)$, where $(\tau _j)_{j=1}^\infty$ is a non-decreasing sequence of ${\mathbb {F}}$-stopping times taking values in [0, T], $\beta _j\in {\mathcal {I}}(\tau _j)$ and N is an $\mathcal {F}_T$-measurable, integer valued random variable such that $\{N \ge j \}$ on $\{\tau _j < T\}$. Throughout, we also set $\tau _0 := 0$.

• We let ${\mathcal {U}}^f$ denote the subset of $u\in {\mathcal {U}}$ for which N is ${\mathbb {P}}$-a.s. finite (i.e. ${\mathcal {U}}^f:=\{u\in {\mathcal {U}}:\, {\mathbb {P}}\big [\{\omega \in \Omega :$ $N>k, \,\forall k>0\}\big ]=0\}$) and for all $k\ge 0$ we let ${\mathcal {U}}^k:=\{u\in {\mathcal {U}}:\,N\le k,\,{\mathbb {P}}\mathrm{- a.s.}\}$.

• For $\eta \in {\mathcal {T}}$ we let ${\mathcal {U}}_{\eta }$ (and ${\mathcal {U}}_{\eta }^f$ resp. ${\mathcal {U}}_{\eta }^k$) be the subset of ${\mathcal {U}}$ (and ${\mathcal {U}}^f$ resp. ${\mathcal {U}}^k$) with $\tau _1\ge \eta$, ${\mathbb {P}}$-a.s.

• We let ${\mathcal {A}}$ be the set of all ${\mathcal {P}}_{{\mathbb {F}}}$-measurable processes $(\alpha _t)_{0\le t\le T}$ taking values in A and for each $t\in [0,T]$ we let ${\mathcal {A}}_t$ be the set of all ${\mathcal {P}}_{{\mathbb {F}}}$-measurable processes $(\alpha _s)_{t\le s\le T}$ taking values in A.

• We denote by ${\mathcal {D}}$ the set of all double sequences $(t_1,\ldots ;b_1,\ldots )$ where $(t_j)_{j\ge 1}$ is a non-decreasing sequence in [0, T] and $b_j\in U$ for $j\ge 1$.

• We let ${\mathcal {D}}^f$ be the subset of ${\mathcal {D}}$ with all finite sequences and for $k\ge 0$ we let ${\mathcal {D}}^k$ be the subset of sequences with precisely k interventions, i.e. sequences of the type $(t_1,\ldots ,t_k;b_1,\ldots ,b_k)$.

• Throughout, we let $\mathbf {v}=(\mathbf {t},\mathbf {b})$, with $\mathbf {t}:=(t_1,\ldots ,t_n)$ and $\mathbf {b}:=(b_1,\ldots ,b_n)$, where n is possibly infinite, denote a generic element of ${\mathcal {D}}$.

• For $\mathbf {v}=(\mathbf {t},\mathbf {b})\in {\mathcal {D}}^f$ and $\mathbf {v}'=(\mathbf {t}',\mathbf {b}')\in {\mathcal {D}}$ we introduce the concatenation, denoted by $\circ$, defined as $\mathbf {v}\circ \mathbf {v}':=(t_1,\ldots ,t_n,t'_1\vee t_n,\ldots ,t'_{n'}\vee t_n;b_1,\ldots ,b_n,b'_1,\ldots ,b'_{n'})$. Furthermore, for $\mathbf {v}\in {\mathcal {D}}$, we define the truncation to $k\ge 0$ interventions as $[\mathbf {v}]_{k}:=(t_1,\ldots ,t_{k\wedge n};b_1,\ldots ,b_{k\wedge n})$.

• We extend $\circ$ to a map from ${\mathcal {U}}^f \times {\mathcal {U}}$ to ${\mathcal {U}}$ by letting ${\mathbf {u}}\circ \mathbf {u}':=(\tau _1,\ldots ,\tau _N,\tau '_1\vee \vee \pi (u),\ldots ,\tau '_{N'}\vee \pi (u);\beta _1,\ldots ,\beta _N,\beta '_1,\ldots ,\beta '_{N'})$ where $\pi (u)$ is the smallest $\mathcal {F}$-stopping time such that $\tau _N \le \pi (u)$, ${\mathbb {P}}$-a.s.

• For each $u\in {\mathcal {U}}^f$ we let $u(t)=[u]_{N(t)}$ with $N(t):=\max \{j\ge 0:\tau _j\le t\}$.

• We introduce the norm $\Vert \mathbf {v}\Vert _{{\mathcal {D}}^f}:=\sum _{j=1}^n(|t_j|+|b_j|)$ on ${\mathcal {D}}^f$ and let $\Vert \mathbf {v}-\mathbf {v}'\Vert _{{\mathcal {D}}^f}:=\infty$ whenever $n\ne n'$.

• We let $*$ denote stochastic integration and set $(X*W)_{t,s}=\int _t^s X_rdW_r$.

• We let ${\mathcal {E}}$ denote the Doléans-Dade exponential and use the notation

  $$\begin{aligned} {\mathcal {E}}(X*W)_{t,s}=e^{\int _t^s X_rdW_r-\frac{1}{2}\int _t^s |X_r|^2dr}. \end{aligned}$$

  Also, we write ${\mathcal {E}}(X*W)_{t}:={\mathcal {E}}(X*W)_{0,t}$.

• For any ${\mathcal {P}}_{{\mathbb {F}}}$-measurable process $\zeta$ such that ${\mathbb {E}}[{\mathcal {E}}(\zeta *W)_T]=1$, we define ${\mathbb {Q}}^{\zeta }$ to be the probability measure equivalent to ${\mathbb {P}}$, such that $d{\mathbb {Q}}^\zeta ={\mathcal {E}}(\zeta *W)_Td{\mathbb {P}}$.

• For any non-negative, ${\mathcal {P}}_{{\mathbb {F}}}$-measurable càdlàg process L we let ${\mathfrak {P}}^L$ denote the set of all probability measures ${\mathbb {Q}}$ on $(\Omega ,{\mathcal {F}})$ such that $d{\mathbb {Q}}={\mathcal {E}}(\zeta *W)_Td{\mathbb {P}}$, for some ${\mathcal {P}}_{{\mathbb {F}}}$-measurable process $\zeta$, with $|\zeta _t|\le L_t$ for all $t\in [0,T]$ (outside of a ${\mathbb {P}}$-null set).

In addition, we will throughout assume that, unless otherwise specified, all inequalities hold in the ${\mathbb {P}}$-a.s. sense.

Furthermore, we define the following set:

Definition 2.1

We let ${\mathcal {O}}_c$ be the set of all ${\mathcal {P}}_{\mathbb {F}}\otimes {\mathcal {B}}(U)$-measurable maps¹$h:\Omega \times [0,T]\times U\rightarrow {\mathbb {R}}$ such that for each $\tau \in {\mathcal {T}}^f$ and $\beta \in {\mathcal {I}}(\tau )$ we have $h(\tau ,\beta )\in L^p(\Omega ,{\mathcal {F}},{\mathbb {P}})$ for all $p\ge 0$ and (outside of a ${\mathbb {P}}$-null set) the map $(t,b)\mapsto h(t,b)$ is jointly continuous.

Definition 2.2

We refer to a family of processes $((X^v_t)_{0\le t\le T}:v\in {\mathcal {U}}^f)$ as being consistent if for each $u\in {\mathcal {U}}^f$, the map $h:[0,T]\times \Omega \times U\rightarrow {\mathbb {R}}$ given by $h(t,b)=X^{u\circ (t,b)}_t$ is ${\mathcal {P}}_{\mathbb {F}}\otimes {\mathcal {B}}(U)$-measurable and for each $\tau \in {\mathcal {T}}$ and each $\beta \in {\mathcal {I}}(\tau )$ we have $X^{u\circ (\tau ,\beta )}_\tau =h(\tau ,\beta )$, ${\mathbb {P}}$-a.s.

One of the main objectives of the present work is to show that (1.1) admits a unique solution. We, therefore, need to define what we mean by a solution to (1.1).

Definition 2.3

A solution to (1.1) is a family $(Y^v,Z^v,K^v)_{v\in {\mathcal {U}}^f}$, where

• (i) the family $(Y^v)_{v\in {\mathcal {U}}^f}$ is consistent and for each $v\in {\mathcal {U}}^f$, we have $Y^v\in {\mathcal {S}}^2$ with a norm that is uniformly bounded in v (i.e. $\sup _{u\in {\mathcal {U}}^f}\Vert Y^u\Vert _{{\mathcal {S}}^2})<\infty$) and $(t,b)\mapsto Y^{v\circ (t,b)}_t\in {\mathcal {O}}_c$,

• (ii) $Z^v\in {\mathcal {H}}^2$ for each $v\in {\mathcal {U}}^f$; and

• (iii) $K^v\in {\mathcal {S}}^2$ is non-decreasing with $K^v_0=0$ for each $v\in {\mathcal {U}}^f$.

2.1 RBSDEs with Stochastic Lipschitz Coefficient

Our approach will rely heavily on the available theory of reflected backward SDEs. In particular, we have the following result (a proof of which can be found in Appendix A):

Proposition 2.4

Assume that

• (i) There is a ${\mathbb {P}}$-a.s. non-negative, ${\mathcal {P}}_{\mathbb {F}}$-measurable, continuous process $(L_t:t\in [0,T])$ (our stochastic Lipschitz coefficient) such that for all ${\mathcal {P}}_{\mathbb {F}}$-measurable processes $(\zeta _t:t\in [0,T])$ with $|\zeta _t|\le L_t$ for all $t\in [0,T]$ (outside of a ${\mathbb {P}}$-null set) we have ${\mathbb {E}}[|{\mathcal {E}}(\zeta *W)_T|^{q'}]<\infty$ and ${\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}[|{\mathcal {E}}(-\zeta *W^{\zeta })_T|^{q'}]<\infty$.

• (ii) The terminal value $\xi \in L^{p}(\Omega ,{\mathcal {F}}_T,{\mathbb {P}})$.

• (iii) The driver $(t,\omega ,y,z)\mapsto f(t,y,z): [0,T]\times \Omega \times {\mathbb {R}}\times {\mathbb {R}}^{d}\rightarrow {\mathbb {R}}$ is ${\mathcal {P}}_{{\mathbb {F}}}\otimes {\mathcal {B}}({\mathbb {R}})\otimes {\mathcal {B}}({\mathbb {R}}^{d})$-measurable. Furthermore,

  • (a) For each $p\ge 1$, we have

    $$\begin{aligned} {\mathbb {E}}\Big [\int _0^T|f(s,0,0)|^{p}ds\Big ]<\infty . \end{aligned}$$

    (2.1)

  • (b) There is a constant $k_f>0$ such that

    $$\begin{aligned} |f(t,y',z')-f(t,y,z)|\le k_f|y'-y|+L_t|z'-z|, \end{aligned}$$

    (2.2)

    for all $(t,y,y',z,z')\in [0,T]\times {\mathbb {R}}^{2(1+d)}$, ${\mathbb {P}}$-a.s.

• (iv) The barrier S is real-valued, ${\mathcal {P}}_{\mathbb {F}}$-measurable and continuous with $S^+\in {\mathcal {S}}^p$ for each $p\ge 1$ and $S_T\le \xi$, ${\mathbb {P}}$-a.s.

Then, there exists a unique triple $(Y,Z,K):=(Y_t,Z_t,K_t)_{0\le t\le T}$ with $Y,K\in {\mathcal {S}}^2$ and $Z\in {\mathcal {H}}^2$, where K is non-decreasing with $K_0=0$, such that

$$\begin{aligned} {\left\{ \begin{array}{ll} Y_t=\xi +\int _t^T f(s,Y_s,Z_s)ds-\int _t^T Z_s dW_s+K_T-K_t,\\ Y_t\ge S_t,\, \forall t\in [0,T] \text{ and } \int _0^T \left( Y_t-S_t\right) dK_t=0. \end{array}\right. } \end{aligned}$$

(2.3)

Furthermore²,

$$\begin{aligned} \Vert Y\Vert _{{\mathcal {S}}^p}^p+\Vert Z\Vert _{{\mathcal {H}}^p}^p+\Vert K\Vert _{{\mathcal {S}}^p}^p \le C\left( \Vert S^+\Vert _{{\mathcal {S}}^{q^2p}}^p+{\mathbb {E}}\Big [|\xi |^{q^2p} +\big (\int _0^T|f(s,0,0)|ds\big )^{q^2p}\Big ]^{1/q^2}\right) \end{aligned}$$

(2.4)

and if $(\tilde{Y},\tilde{Z},\tilde{K})$ is a solution to (2.3) with parameters $(\tilde{f},\tilde{\xi },\tilde{S})$ then

$$\begin{aligned}&\Vert \tilde{Y}-Y\Vert _{{\mathcal {S}}^p}^p+\Vert \tilde{Z}-Z\Vert _{{\mathcal {H}}^p}^p +\Vert \tilde{K}-K\Vert _{{\mathcal {S}}^p}^p\nonumber \\&\le C\left( \Vert \tilde{S}-S\Vert _{{\mathcal {S}}^{q^2p}}^{p/2} \Psi _T^{1/2}+{\mathbb {E}}\Big [|\tilde{\xi }-\xi |^{q^2p} +\big (\int _0^T |\tilde{f}(s,Y_s,Z_s)-f(s,Y_s,Z_s)|ds \big )^{q^2p}\Big ]^{1/q^2}\right) , \end{aligned}$$

(2.5)

where

$$\begin{aligned} \Psi _T&:={\mathbb {E}}\Big [|\tilde{\xi }|^{q^4p}+|\xi |^{q^4p} +\big (\int _0^T (|\tilde{f}(s,0,0)|+|f(s,0,0)|)ds \big )^{q^4p}\\&\qquad \qquad +\sup _{t\in [0,T]}|(\tilde{S}_{t})^+ + (S_{t})^+|^{q^4p}\Big ]^{1/q^4}. \end{aligned}$$

In addition, Y can be interpreted as the Snell envelope in the following way

$$\begin{aligned} Y_t=\mathop {\mathrm{ess}\,\sup }_{\tau \in {\mathcal {T}}_t}{\mathbb {E}}\bigg [\int _t^\tau f(s,Y_s,Z_s)ds+S_\tau {\mathbbm {1}}_{[\tau <T]}+\xi {\mathbbm {1}}_{[\tau =T]}\Big |{\mathcal {F}}_t\bigg ]. \end{aligned}$$

(2.6)

In particular, with $D_t:=\inf \{r\ge t: Y_r=S_r\}\wedge T$ we have the representation

$$\begin{aligned} Y_t={\mathbb {E}}\bigg [\int _t^{D_t} f(s,Y_s,Z_s)ds+S_{D_t} {\mathbbm {1}}_{[D_t<T]}+\xi {\mathbbm {1}}_{[D_t=T]}\Big |{\mathcal {F}}_t\bigg ] \end{aligned}$$

(2.7)

and $K_{D_t}-K_t=0$, ${\mathbb {P}}$-a.s.

3 Sequential Systems of Reflected BSDEs in Finite Horizon

In this section we move on to the sequential system of reflected BSDEs in (1.1). To be able to use our results for impulse control we must allow the stochastic Lipschitz coefficient in (1.1) to depend on the control parameter $v\in {\mathcal {U}}^f$, rendering us a family of stochastic Lipschitz coefficients $(L^v:v\in {\mathcal {U}}^f)$ with $L^v\in {\mathcal {S}}^p$ for all $v\in {\mathcal {U}}^f$ and $p\ge 0$.

3.1 Assumptions

We introduce the following sets of probability measures on $(\Omega ,{\mathcal {F}})$.

Definition 3.1

We let ${\mathfrak {P}}^v:={\mathfrak {P}}^{L^v}$ and define ${\mathcal {K}}^v$ to be the set of all ${\mathcal {P}}_{\mathbb {F}}$-measurable processes $\zeta$ with $|\zeta _t|\le L^{v}_t$ for all $t\in [0,T]$ (outside of a ${\mathbb {P}}$-null set). Moreover, for all $t\in [0,T]$, we let ${\mathfrak {P}}^v_t:=\cup _{u\in {\mathcal {U}}^f_t}{\mathfrak {P}}^{v(t)\circ u}$ and ${\mathcal {K}}^v_t:=\cup _{u\in {\mathcal {U}}^f_t}{\mathcal {K}}^{v(t)\circ u}$. We also use the shorthands ${\mathfrak {P}}_0:={\mathfrak {P}}^{\emptyset }_0$ and ${\mathcal {K}}_0:={\mathcal {K}}^{\emptyset }_0$.

To streamline presentation we will formulate our assumptions on the coefficients in terms of the existence of a family of bounding processes:

Definition 3.2

We say that a family of processes $(L^v,\Lambda ^{\mathbf {v},\mathbf {v}',v},\bar{K}^{v,p},\bar{K}^{\mathbf {v},\mathbf {v}',k,p}: (\mathbf {v},\mathbf {v}')\in \cup _{\kappa \ge 1}{\mathcal {D}}^\kappa \times {\mathcal {D}}^\kappa ,\,v\in {\mathcal {U}}^f,\,k\ge 0, p\ge 1)$ is a bounding family if for each $k\ge 0$ and $p,\kappa \ge 1$, there is a $C>0$ and a $p'\ge 1$ such that for all $v\in {\mathcal {U}}^f$ and $\mathbf {v},\mathbf {v}'\in {\mathcal {D}}^\kappa$ and some $q'>1$, we have:

• (i) $L^v\in {\mathcal {S}}^p$ is non-decreasing, the map $\mathbf {v}\mapsto L^{\mathbf {v}}$ is continuous from ${\mathcal {D}}^f$ to ${\mathcal {H}}^2$. Moreover, for all $\zeta \in {\mathcal {K}}^v$ and ${\mathbb {Q}}\in {\mathfrak {P}}^v$ we have ${\mathbb {E}}^{{\mathbb {Q}}}[|{\mathcal {E}}(\zeta *W^{{\mathbb {Q}}})_T|^{q'}]\le C$ (where $W^{\mathbb {Q}}$ is a Brownian motion under ${\mathbb {Q}}$). Moreover, ${\mathbb {E}}\big [e^{q'\int _0^T|L^v_t|^2dt}\big ]\le C$.

• (ii) $\Lambda ^{\mathbf {v},\mathbf {v}',v}\in {\mathcal {H}}^{p}$ is a càdlàg process and the map $(\mathbf {v},\mathbf {v}')\mapsto \int _0^T|\Lambda ^{\mathbf {v},\mathbf {v}',v}_s|^2ds$ is ${\mathbb {P}}$-a.s. continuous with $\int _0^T|\Lambda ^{\mathbf {v},\mathbf {v},v}_s|^2ds=0$.

• (iii) $\bar{K}^{v,p}\in {\mathcal {S}}^2$ with $\Vert \bar{K}^{v,p}\Vert _{{\mathcal {S}}^2}^2\le C$ and $(\bar{K}^{v,p})^r\le (\bar{K}^{v,rp})$ for $r\ge 1$.

• (iv) $\bar{K}^{\mathbf {v},\mathbf {v}',k,p}\in {\mathcal {S}}^1$ with $\Vert \bar{K}^{\mathbf {v},\mathbf {v}',k,p'}\Vert _{{\mathcal {S}}^1}\le C\Vert \mathbf {v}'-\mathbf {v}\Vert ^p_{{\mathcal {D}}^f}$.

Moreover, for each $r>1$, there is a $C>0$ such that

$$\begin{aligned} \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t}{\mathbb {E}}\big [|L^{v\circ u}_T|^p\big |{\mathcal {F}}_t\big ]&\le \bar{K}^{v,p}_t, \end{aligned}$$

(3.1)

$$\begin{aligned} \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^k_t}{\mathbb {E}}\Big [\big (\int _0^T|\Lambda ^{\mathbf {v},\mathbf {v}', u}_s|^{2}ds\big )^{p/2}\Big |{\mathcal {F}}_t\Big ]&\le \bar{K}^{\mathbf {v},\mathbf {v}',k,p}_t \end{aligned}$$

(3.2)

$$\begin{aligned} \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t}{\mathbb {E}}\Big [\sup _{s\in [t,T]}|\bar{K}^{v\circ (u(s)),p}_s|^{r}\Big |{\mathcal {F}}_t\Big ]&\le C\bar{K}^{v,pr}_t, \end{aligned}$$

(3.3)

for all ${\mathbb {Q}}\in {\mathfrak {P}}^{v}_t$.

We will make the following assumptions on the involved coefficients:

Assumption 3.3

There is a bounding family $(L^v,\Lambda ^{\mathbf {v},\mathbf {v}',v},\bar{K}^{v,p},\bar{K}^{\mathbf {v},\mathbf {v}',k,p}: (\mathbf {v},\mathbf {v}')\in \cup _{\kappa \ge 1}{\mathcal {D}}^\kappa \times {\mathcal {D}}^\kappa ,\,v\in {\mathcal {U}}^f,\,k\ge 0,\,p\ge 1)$ such that for each $k\ge 0$, $p,\kappa \ge 1$, $v\in {\mathcal {U}}^f$ and $\mathbf {v},\mathbf {v}'\in {\mathcal {D}}^\kappa$ we have:

• (i) The map $(\omega ,\mathbf {v})\mapsto \xi ^\mathbf {v}:\Omega \times {\mathcal {D}}^f\rightarrow {\mathbb {R}}$ is ${\mathcal {F}}_T\otimes {\mathcal {B}}({\mathcal {D}}^f)$-measurable, ${\mathbb {P}}$-a.s. continuous in $\mathbf {v}$ and satisfies

  $$\begin{aligned} \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t}{\mathbb {E}}\big [|\xi ^{v\circ u}|^p \big |{\mathcal {F}}_t\big ]\le \bar{K}^{v,p}_t \end{aligned}$$

  (3.4)

  and

  $$\begin{aligned} \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^k_t}{\mathbb {E}}\big [|\xi ^{\mathbf {v}'\circ u} -\xi ^{\mathbf {v}\circ u}|^{p}\big |{\mathcal {F}}_t\big ] \le \bar{K}^{\mathbf {v},\mathbf {v}',k,p}_t. \end{aligned}$$

  (3.5)

• (ii) The intervention cost $c^v$ is such that $(t,b)\mapsto -c^v(t,b)\in {\mathcal {O}}_c$ and satisfies

  $$\begin{aligned} \inf _{(t,b)\in [0,T]\times U}c^v(t,b)\ge \delta , \end{aligned}$$

  (3.6)

  for some $\delta >0$. Moreover,

  $$\begin{aligned} \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^k_t}{\mathbb {E}}\big [|c^{\mathbf {v}'\circ [u]_{N-1}} (\tau _N,\beta _N)-c^{\mathbf {v}\circ [u]_{N-1}}(\tau _N,\beta _N)|^{p} \big |{\mathcal {F}}_t\big ]\le \bar{K}^{\mathbf {v},\mathbf {v}',k,p}_t. \end{aligned}$$

  (3.7)

• (iii) We have $\xi ^{v}\ge \sup _{b\in U}\{\xi ^{v\circ (T,b)}-c^v(T,b)\}$, ${\mathbb {P}}$-a.s.

• (iv) The map $(\mathbf {v},t,\omega ,y,z)\mapsto f^{\mathbf {v}}(t,y,z):{\mathcal {D}}^f\times [0,T]\times \Omega \times {\mathbb {R}}\times {\mathbb {R}}^{d}\rightarrow {\mathbb {R}}$ is ${\mathcal {B}}({\mathcal {D}}^f)\otimes {\mathcal {P}}_{{\mathbb {F}}} \otimes {\mathcal {B}}({\mathbb {R}})\otimes {\mathcal {B}}({\mathbb {R}}^{d})$-measurable and for each $(y,z)\in {\mathbb {R}}^{1+d}$ the map $\mathbf {v}\mapsto f^\mathbf {v}(\cdot ,y,z)$ is a continuous map from ${\mathcal {D}}^f$ to ${\mathcal {H}}^2$. Furthermore, we have

  • (a) the bound

    $$\begin{aligned} \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t}{\mathbb {E}}\Big [\int _t^{T}|f^{v\circ u} (s,0,0)|^p ds\big |{\mathcal {F}}_t\Big ]\le \bar{K}^{v,p}_t, \end{aligned}$$

    (3.8)

  • (b) the Lipschitz condition

    $$\begin{aligned} |f^{\mathbf {v}'\circ u}(t,y',z')-f^{\mathbf {v}\circ u}(t,y,z)|&\le k_f|y'-y|+(L^{\mathbf {v}\circ u}_t\vee L^{\mathbf {v}'\circ u}_t)|z' -z|\nonumber \\&\quad +(1+|z|+|z'|)\Lambda _t^{\mathbf {v}',\mathbf {v},u}, \end{aligned}$$

    (3.9)

    for all $(t,\mathbf {v},\mathbf {v}',y,y',z,z')\in [0,T]\times \cup _{\kappa \ge 1}({\mathcal {D}}^\kappa \times {\mathcal {D}}^\kappa )\times {\mathbb {R}}^{2(1+d)}$, ${\mathbb {P}}$-a.s., for all $u\in {\mathcal {U}}^f$; and

  • (c) for $u:=(\tau _1,\ldots ,\tau _N;\beta _1,\ldots ,\beta _N) \in {\mathcal {U}}^f$ we have the causality property

    $$\begin{aligned} f^u(t,y,z)=\sum _{j=0}^N{\mathbbm {1}}_{[\tau _j,\tau _{j+1})}(t)f^{[u]_j}(t,y,z), \end{aligned}$$

    where $\tau _0:=0$ and $\tau _{N+1}=\infty$.

Before moving on to show existence and uniqueness of solutions to (1.1) under Assumption 3.3 we give the following auxiliary result:

Lemma 3.4

For each $p\ge 1$ there is a $r'>1$ such that $R^v$ defined as

$$\begin{aligned} R^v_t:=\mathop {\mathrm{ess}\,\sup }_{\zeta \in {\mathcal {K}}^v_t}{\mathbb {E}}\big [|{\mathcal {E}}(\zeta *W)_{t,T}|^{r'} \big |{\mathcal {F}}_t\big ], \end{aligned}$$

and $\tilde{R}^v$ defined as

$$\begin{aligned} \tilde{R}^v_t:=\mathop {\mathrm{ess}\,\sup }_{\zeta \in {\mathcal {K}}^v_t}{\mathbb {E}}^{{\mathbb {Q}}^\zeta } \big [|{\mathcal {E}}(-\zeta *W^{\zeta })_{t,T}|^{r'}\big |{\mathcal {F}}_t\big ], \end{aligned}$$

where $W^{\zeta }$ is a ${\mathbb {Q}}^\zeta$-Brownian motion, are both ${\mathcal {P}}_{\mathbb {F}}$-measurable, càdlàg processes and $\Vert R^v\Vert _{{\mathcal {S}}^p}$ and $\Vert \tilde{R}^v\Vert _{{\mathcal {S}}^p}$ are uniformly bounded in $v\in {\mathcal {U}}^f$.

Proof

By continuity of the map $\mathbf {v}\mapsto L^\mathbf {v}:{\mathcal {D}}^f\rightarrow {\mathcal {H}}^2$ it follows that $R^u$ is continuous on $[\tau _j,\tau _{j+1})$ for $j=0,\ldots ,N+1$. For $x\ge 0$, we let $\tau ^x:=\inf \{s\ge 0: R^v_s\ge x\}\wedge T$ and note that for $\theta > 1$, we have by right-continuity that

$$\begin{aligned} {\mathbb {P}}[\sup _{t\in [0,T]}|R^v_t|^{p\theta }\ge x^\theta ] ={\mathbb {P}}[|R^v_{\tau ^x}|^{p\theta }\ge x^\theta ] \le \frac{{\mathbb {E}}\big [|R^v_{\tau ^x}|^{p\theta }\big ]}{x^\theta }. \end{aligned}$$

However, for each $\epsilon >0$, there is a $u^\epsilon \in {\mathcal {U}}^f_{\tau ^x}$ and a $\zeta ^\epsilon$, with $|\zeta ^\epsilon _t|\le {\mathbbm {1}}_{[\tau ^x,T]}(t){L^{v(\tau ^x)\circ u^\epsilon }_t}$ such that $R^v_{\tau ^x}<{\mathbb {E}}\big [|{\mathcal {E}}(\zeta ^{\epsilon }*W)_{\tau ^x,T}|^{r'}\big |{\mathcal {F}}_{\tau ^x}\big ]+\epsilon$. In particular, it follows that

$$\begin{aligned} |R^v_{\tau ^x}|^{p\theta }&\le 2^{{p\theta }-1}\left( {\mathbb {E}}\Big [{\mathcal {E}}(r'\zeta ^{\epsilon }*W)_{\tau ^x,T}e^{\frac{(r')^2 -r'}{2}\int _{\tau ^x}^T|\zeta ^{\epsilon }_s|^2ds} \Big |{\mathcal {F}}_{\tau ^x}\Big ]^{p\theta }+\epsilon ^{p\theta }\right) \\&\le 2^{{p\theta }-1}\left( {\mathbb {E}}^{{\mathbb {Q}}^{r'\zeta ^{\epsilon }}} \Big [e^{p\theta \frac{(r')^2 -r'}{2}\int _{\tau ^x}^T| \zeta ^{\epsilon }_s|^2ds}\Big |{\mathcal {F}}_{\tau ^x}\Big ] +\epsilon ^{p\theta }\right) \end{aligned}$$

which implies that

$$\begin{aligned} {\mathbb {E}}\big [|R^v_{\tau ^x}|^{p\theta }\big ]&\le 2^{{p\theta }-1} \left( {\mathbb {E}}\Big [{\mathcal {E}}(r'\zeta ^{\epsilon }*W)_{\tau ^x,T}e^{p\theta \frac{(r')^2 -r'}{2}\int _{\tau ^x}^T|\zeta ^{\epsilon }_s|^2ds}\Big ] +\epsilon ^{p\theta }\right) \\&=2^{{p\theta }-1}\left( {\mathbb {E}}^{{\mathbb {Q}}^{r'\zeta ^{\epsilon }}}\Big [e^{{p\theta } \frac{(r')^2 -r'}{2}\int _{\tau ^x}^T|\zeta ^{\epsilon }_s|^2ds}\Big ] +\epsilon ^{p\theta }\right) \end{aligned}$$

Since $\epsilon >0$ was arbitrary we find that

$$\begin{aligned} {\mathbb {P}}\big [\sup _{t\in [0,T]}|R^v_t|^{p}\ge x\big ]&={\mathbb {P}}\big [\sup _{t\in [0,T]}|R^v_t|^{p\theta }\ge x^{\theta }\big ]\\&\le 2^{p\theta -1}\sup _{\zeta \in {\mathcal {K}}_0}\frac{{\mathbb {E}}^{{\mathbb {Q}}^{r'\zeta }} \big [e^{{p\theta }\frac{(r')^2 -r'}{2}\int _{0}^T| \zeta _s|^2ds}\big ]}{x^\theta }. \end{aligned}$$

Now, with $\tilde{q}'=q'/r'$ and $\tilde{q}=\tilde{q}'/(\tilde{q}'-1)=q'/(q'-r')$ we have

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {Q}}^{r'\zeta }}\big [e^{{p\theta }\frac{(r')^2 -r'}{2} \int _{0}^T|\zeta _s|^2ds}\big ]\le {\mathbb {E}}\big [|{\mathcal {E}}(r'\zeta *W)|^{\tilde{q}'} \big ]^{1/\tilde{q}'}{\mathbb {E}}\big [e^{{p\theta \tilde{q}}\frac{(r')^2 -r'}{2} \int _{0}^T|\zeta _s|^2ds}\big ]^{1/\tilde{q}}. \end{aligned}$$

For any $\theta >1$ and arbitrary $p\ge 1$ the coefficient ${p\theta \tilde{q}(r')}\frac{(r')^2 -r'}{2}$ can be made arbitrarily small by choosing $r'>1$ sufficiently small and since there is a $C\ge 0$ such that

$$\begin{aligned} {\mathbb {E}}\big [e^{q'\int _{0}^T|\zeta _s|^2ds}\big ]\le \sup _{u\in {\mathcal {U}}^f} {\mathbb {E}}\big [e^{q'\int _{0}^T|L^u_s|^2ds}\big ]\le C \end{aligned}$$

by Definition 3.2. (i) we conclude that

$$\begin{aligned} {\mathbb {P}}\big [\sup _{t\in [0,T]}|R^v_t|^{p}\ge x\big ]\le \frac{C}{x^\theta }. \end{aligned}$$

In particular, using integration by parts, we find that

$$\begin{aligned} {\mathbb {E}}\big [\sup _{t\in [0,T]}|R^v_t|^p\big ]=\int _0^\infty \left( \frac{C}{x^\theta }\wedge 1\right) dx<\infty , \end{aligned}$$

showing that $R^v\in {\mathcal {S}}^p$ with norm uniformly bounded in v. The result for $\tilde{R}^v$ follows similarly. $\square$

Lemma 3.5

For each $p\ge 1$ there is a $r'>1$ such that $\bar{R}^v$ defined as

$$\begin{aligned} \bar{R}^v_t:=\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t}{\mathbb {E}}\Big [\sup _{s\in [t,T]} \mathop {\mathrm{ess}\,\sup }_{\zeta \in {\mathcal {K}}^{v(t)\circ u}_s} {\mathbb {E}}\big [|{\mathcal {E}}(\zeta *W)_{s,T}|^{r'}\big |{\mathcal {F}}_s\big ]\Big |{\mathcal {F}}_t\Big ] \end{aligned}$$

is a ${\mathcal {P}}_{\mathbb {F}}$-measurable, càdlàg process and $\Vert \bar{R}^v\Vert _{{\mathcal {S}}^p}$ is uniformly bounded in $v\in {\mathcal {U}}^f$.

Proof

Let $r'>1$ be such that $\Vert R^v\Vert _{{\mathcal {S}}^{p\theta }}$ is uniformly bounded in $v\in {\mathcal {U}}^f$ for some $\theta >1$. For $x\ge 0$, we let $\tau ^x:=\inf \{s\ge 0: \bar{R}^v_s\ge x\}\wedge T$ and note that for each $\epsilon >0$, there is a $u^\epsilon \in {\mathcal {U}}^f_{\tau ^x}$ such that $\bar{R}^v_{\tau ^x}<{\mathbb {E}}\Big [\sup _{s\in [\tau ^x,T]}R^{v\circ u_\epsilon }_s\big |{\mathcal {F}}_{\tau ^x}\Big ]+\epsilon$. Jensen’s inequality now gives

$$\begin{aligned} {\mathbb {E}}\big [|\bar{R}^v_{\tau ^x}|^{p\theta }\big ]\le 2^{p\theta -1} \left( {\mathbb {E}}\Big [\sup _{s\in [\tau ^x,T]}|R^{v(\tau ^x)\circ u_\epsilon }_s|^{p\theta } \Big ]+\epsilon ^{p\theta }\right) . \end{aligned}$$

Since $\epsilon >0$ was arbitrary and the first term is bounded by $2^{p\theta -1}\sup _{u\in {\mathcal {U}}^f}\Vert R^u\Vert _{{\mathcal {S}}^{p\theta }}$ the result follows by repeating the last steps in the proof of Lemma 3.4. $\square$

Throughout this section, we assume that $r'>1$ is small enough that $\Vert R^v\Vert _{{\mathcal {S}}^3}$, $\Vert \tilde{R}^v\Vert _{{\mathcal {S}}^3}$ and $\Vert \bar{R}^v\Vert _{{\mathcal {S}}^3}$ are bounded uniformly in $v\in {\mathcal {U}}^f$ and let r be such that $\frac{1}{r'}+\frac{1}{r}=1$.

3.2 An Approximating Sequence

In this section we outline a Picard type approximation scheme, that will ultimately lead us to the conclusion that (1.1) has a solution under Assumption 3.3. We note that for all $v\in {\mathcal {U}}^f$,

$$\begin{aligned} Y^{v,0}_t=\xi ^{v}+\int _t^T f^v(s,Y^{v,0}_s,Z^{v,0}_s)ds -\int _t^T Z^{v,0}_sdW_s \end{aligned}$$

(3.10)

admits a unique solution by Proposition 2.4 (note that (3.10) can be seen as a reflected BSDE with barrier $S\equiv -\infty$). We, thus, consider the following sequence of families of reflected BSDEs

$$\begin{aligned} {\left\{ \begin{array}{ll} Y^{v,k}_t=\xi ^{v}+\int _t^T f^v(s,Y^{v,k}_s,Z^{v,k}_s)ds -\int _t^T Z^{v,k}_sdW_s+ K^{v,k}_T-K^{v,k}_t \\ Y^{v,k}_t\ge \sup _{b\in U}\{Y^{v\circ (t,b),k-1}_t-c^v(t,b)\}\\ \int _0^T(Y^{v,k}_t-\sup _{b\in U}\{Y^{v\circ (t,b),k-1}_t-c^v(t,b)\}) dK^{v,k}_t=0. \end{array}\right. } \end{aligned}$$

(3.11)

for $k\ge 1$. Hypothesis We will make use of the following induction hypothesis:

Hypothesis

(RBSDE. l) There is a family of pairs $((Y^{v,0},Z^{v,0}): v\in {\mathcal {U}}^f)$ and a sequence of families of triples $((Y^{v,k},Z^{v,k},K^{v,k}): v\in {\mathcal {U}}^f)_{1\le k\le l}$ such that:

• i) For each $v\in {\mathcal {U}}^f$, the pair $(Y^{v,0},Z^{v,0}) \in {\mathcal {S}}^2\times {\mathcal {H}}^2$ solves (3.10) and the triple $(Y^{v,k},Z^{v,k},K^{v,k})\in {\mathcal {S}}^2\times {\mathcal {H}}^2\times {\mathcal {S}}^2$ solves (3.11) for $k=1,\ldots ,l$.

• ii) For all $k\in \{0,\ldots ,l\}$ and each $j\ge 0$, the map $h^{k,j}:[0,T]\times {\mathcal {D}}^j\rightarrow L^2(\Omega ,{\mathcal {F}},{\mathbb {P}})\,((t,\mathbf {v}) \mapsto Y^{\mathbf {v},k}_t)$ is jointly continuous (outside of a ${\mathbb {P}}$-null set) and, moreover, for each $v\in {\mathcal {U}}^f$ we have $Y^{v,k}_t=\sum _{j=0}^\infty {\mathbbm {1}}_{[N=j]}h^{k,j}(t,v)$ for all $t\in [0,T]$ outside of a ${\mathbb {P}}$-null set.

Recall Definition 2.3 specifying what we mean by a solution to (2.3). We extend this definition to incorporate solutions to (3.10) and (3.11) as well. Through this definition the first condition in Hypothesis RBSDE.l implies that $\sup _{u\in {\mathcal {U}}^f}\Vert Y^{v,k}\Vert <\infty$ and also dictates the regularity of the map $(t,b)\mapsto Y^{v\circ (t,b)}_t$. The second statement, on the other hand, is a stronger version of the consistency property for families of processes introduced in Definition 2.2. To simplify presentation we will refer to the second property as strong consistency.

Proposition 3.6

Hypothesis RBSDE.0 holds.

Proof

Existence of solutions to (3.10) with $Y^v\in {\mathcal {S}}^2$ and $Z^v\in {\mathcal {H}}^2$ and uniform boundedness (specified in (i) of Definition 2.3) follows from Proposition 2.4 with barrier $S\equiv -\infty$.

For $\kappa \ge 1$ and $\mathbf {v},\mathbf {v}'\in {\mathcal {D}}^\kappa$ we have by Proposition 2.4, Assumption 3.3 and Definition 3.2.(iv) that for each $p\ge 1$, there is a $p'\ge 1$ such that

$$\begin{aligned}&\Vert Y^{\mathbf {v}',0}-Y^{\mathbf {v},0}\Vert _{{\mathcal {S}}^{p'}}^{p'}+\Vert Z^{\mathbf {v}',0} -Z^{\mathbf {v},0}\Vert _{{\mathcal {H}}^{p'}}^{p'}\\&\le C{\mathbb {E}}\Big [|\xi ^{\mathbf {v}'}-\xi ^{\mathbf {v}}|^{q^2p'} +\big (\int _0^T |f^{\mathbf {v}'}(s,Y^{\mathbf {v},0}_s,Z^{\mathbf {v},0}_s) -f^{\mathbf {v}}(s,Y^{\mathbf {v},0}_s,Z^{\mathbf {v},0}_s)|ds\big )^{q^2p'}\Big ]^{1/q^2}\\&\le C{\mathbb {E}}\Big [|\xi ^{\mathbf {v}'}-\xi ^{\mathbf {v}}|^{q^2p'} +\big (\int _0^T (1+|Z^{\mathbf {v},0}_s|)\Lambda ^{\mathbf {v},\mathbf {v}', \emptyset }_sds\big )^{q^2p'}\Big ]^{1/q^2}\\&\le C{\mathbb {E}}\Big [|\xi ^{\mathbf {v}'}-\xi ^{\mathbf {v}}|^{q^2p'} +\big (\int _0^T (1+|Z^{\mathbf {v},0}_s|^2)ds\int _0^T | \Lambda ^{\mathbf {v},\mathbf {v}',\emptyset }_s|^2ds\big )^{q^2p'/2}\Big ]^{1/q^2}\\&\le C({\mathbb {E}}\Big [|\xi ^{\mathbf {v}'}-\xi ^{\mathbf {v}}|^{q^2p'}\Big ]^{1/q^2}\\&\quad +\left( 1+{\mathbb {E}}\Big [\big (\int _0^T |Z^{\mathbf {v},0}_s|^{2}ds\big )^{q^2p'} \Big ]^{1/2q^2}\right) {\mathbb {E}}\Big [\big (\int _0^T|\Lambda ^{\mathbf {v},\mathbf {v}', \emptyset }_s|^2ds\big )^{q^2p'}ds\Big ]^{1/2q^2}\\&\le C\Vert \mathbf {v}'-\mathbf {v}\Vert _{{\mathcal {D}}^f}^{p}, \end{aligned}$$

where $C>0$ does not depend on $\mathbf {v},\mathbf {v}'$. By picking $p=(m+1)\kappa +1$, Kolmogorov’s continuity theorem (see e.g. Theorem 72 in Chapter IV of [28]) guarantees the existence of a family of processes $(\hat{Y}^{\mathbf {v}}, \hat{Z}^{\mathbf {v}}:\mathbf {v}\in {\mathcal {D}}^\kappa )$, where for each $\mathbf {v}\in {\mathcal {D}}^\kappa$, $\hat{Y}^{\mathbf {v}}\in {\mathcal {S}}^2$ and $\hat{Z}^{\mathbf {v}}\in {\mathcal {H}}^2$, such that, outside of a ${\mathbb {P}}$-null set, $(s,\mathbf {v})\mapsto \hat{Y}^{\mathbf {v}}_s$ is continuous in $\mathbf {v}$ uniformly in s and $\mathbf {v}\mapsto \hat{Z}^{\mathbf {v}}_\cdot$ is $L^2([0,T])$-continuous (and in particular that $(s,\mathbf {v})\mapsto \int _s^TZ^{\mathbf {v}}_rdW_r$ is continuous in $\mathbf {v}$ uniformly in s) and moreover $(\hat{Y}^{\mathbf {v}}_s,\hat{Z}^{\mathbf {v}}_s) =(Y^{\mathbf {v},0}_s,Z^{\mathbf {v},0}_s)$, ${\mathbb {P}}$-a.s.

Now, since this holds for all $\kappa \ge 0$ it is clear that for each $\mathbf {v}\in {\mathcal {D}}^f$, the pair $(\hat{Y}^{\mathbf {v}}_s, \hat{Z}^{\mathbf {v}}_s:s\in [0,T])$ solves the corresponding BSDE (3.10) and that the map $\mathbf {v}\mapsto (\hat{Y}^{\mathbf {v}},\hat{Z}^{\mathbf {v}}):{\mathcal {D}}^f\rightarrow {\mathcal {S}}^2\times {\mathcal {H}}^2$ is continuous.

To establish the strong consistency in Hypothesis RBSDE.0-(ii), we need to show that for any $v\in {\mathcal {U}}^f$, the pair $(\hat{Y}^{v},\hat{Z}^{v})\in {\mathcal {S}}^2\times {\mathcal {H}}^2$ solves the BSDE corresponding to the control v. We let $(v_j)_{j\ge 0}$ be an approximating sequence in ${\mathcal {U}}^f$ (i.e. $v_j\in {\mathcal {U}}^f$ and $v_j\rightarrow v$, ${\mathbb {P}}$-a.s. as $j\rightarrow \infty$) taking values in a countable subset of ${\mathcal {D}}^f$. By continuity we have

$$\begin{aligned} \sup _{s\in [0,T]}|\hat{Y}^{v_j,0}_s-\hat{Y}^{v,0}_s|\rightarrow 0, \end{aligned}$$

${\mathbb {P}}$-a.s., as $j\rightarrow \infty$. Moreover, due to continuity of the map $\mathbf {v}\mapsto \xi ^{\mathbf {v}}$ we have

$$\begin{aligned} \xi ^{v_j}\rightarrow \xi ^{v} \end{aligned}$$

and by Assumption 3.3 we get

$$\begin{aligned}&\int _0^T |f^{v_j}(s,\hat{Y}^{v_j,0}_s,\hat{Z}^{v_j,0}_s) -f^{v}(s,\hat{Y}^{v,0}_s,\hat{Z}^{v,0}_s)|ds\\&\quad \le \int _0^T |f^{v_j}(s,\hat{Y}^{v_j,0}_s, \hat{Z}^{v_j,0}_s)-f^{v}(s,\hat{Y}^{v_j,0}_s,\hat{Z}^{v_j,0}_s)|ds\\&\qquad +\int _0^T |f^{v}(s,\hat{Y}^{v_j,0}_s,\hat{Z}^{v_j,0}_s) -f^{v}(s,\hat{Y}^{v,0}_s,\hat{Z}^{v,0}_s)|ds\\&\quad \le \int _0^T (1+2|\hat{Z}^{v_j,0}_s|)\Lambda _s^{v_j,v,\emptyset }ds\\&\qquad + \int _0^T (k_f|\hat{Y}^{v_j,0}_s-\hat{Y}^{v,0}_s|+L^{v}_s| \hat{Z}^{v_j,0}_s-\hat{Z}^{v,0}_s|)ds \end{aligned}$$

which tends to 0, ${\mathbb {P}}$-a.s., as $j\rightarrow \infty$. Moreover, for each $j\ge 0$, the pair $(\hat{Y}^{v_j},\hat{Z}^{v_j})$ solves (3.10) with control $v_j$ and we conclude that

$$\begin{aligned} \hat{Y}^{v,0}_\eta&=\lim _{j\rightarrow \infty }\Big \{\xi ^{v_j} +\int _\eta ^T f^{v_j}(s,\hat{Y}^{v_j,0}_s,\hat{Z}^{v_j,0}_s)ds -\int _\eta ^T \hat{Z}^{v_j,0}_sdW_s\Big \}\\&=\xi ^{v}+\int _\eta ^T f^{v}(s,\hat{Y}^{v,0}_s,\hat{Z}^{v,0}_s)ds -\int _\eta ^T \hat{Z}^{v,0}_sdW_s, \end{aligned}$$

for each $\eta \in {\mathcal {T}}$ and strong consistency follows. $\square$

We now turn to the reflected BSDEs (3.11). To obtain estimates for the triple $(Y^{v,k},Z^{v,k},K^{v,k})$ we rely on Proposition 2.4 to reduce the system of reflected BSDEs to a single non-reflected BSDE with jumps. We, thus, introduce the following BSDE:

Definition 3.7

For $v,u\in {\mathcal {U}}^f$, let the pair $(U^{v,u}, V^{v,u})\in {\mathcal {S}}^2_l\times {\mathcal {H}}^2$ (recall that ${\mathcal {S}}^2_l$ is the set of ${\mathcal {P}}_{\mathbb {F}}$-measurable càglàd processes with finite ${\mathcal {S}}^2$-norm) be the unique solution to the BSDE

$$\begin{aligned} U_t^{v,u}=\xi ^{v\circ u}&+\int _t^{T}f^{v\circ u} (s, U^{v,u}_s, V^{v,u}_s)ds -\int _t^{T} V^{v,u}_sdW_s \nonumber \\&-\sum _{j=1}^N {\mathbbm {1}}_{[\tau _j\ge t]}c^{v\circ [u]_{j-1}}(\tau _j,\beta _j), \end{aligned}$$

(3.12)

whenever a unique solution exists and let $U^{v,u}\equiv -\infty$, otherwise.

Proposition 3.8

For each $k\ge 0$, $v\in {\mathcal {U}}^f$ and $u\in {\mathcal {U}}^k$ the BSDE (3.12) admits a unique solution and $V^{v,u}\in {\mathcal {H}}^2_{\mathbb {Q}}$ for all ${\mathbb {Q}}\in {\mathfrak {P}}^v$.

Proof

Existence of a unique solution to (3.12) follows from repeated use of Proposition 2.4 since the intervention costs belong to $L^p(\Omega ,{\mathcal {F}},{\mathbb {P}})$ for all $p\ge 1$. Moreover, a similar argument gives that

$$\begin{aligned} \Vert V^{v,u}\Vert _{{\mathcal {H}}^p}^p&\le C{\mathbb {E}}\Big [|\xi ^{v\circ u}|^{q^2 p} +\int _0^T |f^{v\circ u}(s,0,0)|^{q^2 p}ds \\&+\sum _{j=1}^N |c^{v\circ [u]_{j-1}} (\tau _j,\beta _j)|^{q^2 p}\Big ]^{1/q^2} \le C. \end{aligned}$$

Now,

$$\begin{aligned} \Vert V^{u,v}\Vert ^2_{{\mathcal {H}}^2_{\mathbb {Q}}}&= {\mathbb {E}}\Big [{\mathcal {E}}(\zeta *W)_T \int _0^T |V^{u,v}_s|^2ds\Big ] \\&\le {\mathbb {E}}\big [|{\mathcal {E}}(\zeta *W)_T|^{q'}\big ]^{1/q'}\Vert V^{v,u}\Vert _{{\mathcal {H}}^q}^2 \end{aligned}$$

and the assertion follows. $\square$

In addition, we introduce the following notation:

Definition 3.9

For $v,u\in {\mathcal {U}}^f$ such that (3.12) admits a unique solution, we define

$$\begin{aligned} \gamma ^{v,u}_s:=\frac{f^{v\circ u}(s,U^{v,u}_s,V^{v,u}) -f^{v\circ u}(s,0,V^{v,u})}{U^{v,u}_s}{\mathbbm {1}}_{[U^{v,u}_s\ne 0]} \end{aligned}$$

and for $0\le s\le t\le T$, we set $e^{v,u}_{s,t}:=e^{\int _s^t\gamma ^{v,u}_rdr}$ and $e^{v,u}_t:=e^{v,u}_{0,t}$. Moreover, we define

$$\begin{aligned} \zeta ^{v,u}:=\frac{f^{v\circ u}(s,0,V^{v,u}) -f^{v\circ u}(s,0,0)}{|V^{v,u}_s|^2}(V^{v,u}_s)^\top {\mathbbm {1}}_{[V^{v,u}_s\ne 0]} \end{aligned}$$

and let ${\mathbb {Q}}^{v,u}:={\mathbb {Q}}^{\zeta ^{v,u}}$, the probability measure, equivalent to ${\mathbb {P}}$, under which $W^{v,u}:=W-\int _0^\cdot \zeta ^{v,u}_sds$ is a Brownian motion.

Before we move on to show that Hypothesis RBSDE.l holds for all $l\ge 0$ we give three helpful lemmas.

Lemma 3.10

Assume that Hypothesis RBSDE.l holds for some $l\ge 0$, then for each $p\ge 1$, there is a $C>0$ (that does not depend on l) such that

$$\begin{aligned} \Vert (\sup _{b\in U}|Y^{v\circ (t,b),l}_{t}|: t\in [0,T])\Vert _{{\mathcal {S}}^p}\le C. \end{aligned}$$

(3.13)

Proof

We let $\tilde{v}:=v\circ (t,b)$ and set $\tau ^*_1:=\inf \{s\ge t:Y^{\tilde{v},l}_s=\sup _{b\in U}\{Y^{\tilde{v}\circ (s,b),l-1}_s -c^{\tilde{v}}(s,b)\}\}\wedge T$ and have by Proposition 2.4 and consistency that

$$\begin{aligned} Y^{\tilde{v},l}_t&={\mathbbm {1}}_{[\tau ^*_1<T]}\sup _{b'\in U}\{Y^{\tilde{v} \circ (\tau ^*_1,b'),l-1}_{\tau ^*_1}-c^{\tilde{v}}(\tau ^*_1,b')\} +{\mathbbm {1}}_{[\tau ^*_1=T]}\xi ^{\tilde{v}} \\&\quad +\int _t^{\tau ^*_1}f^{\tilde{v}}(s,Y^{\tilde{v},l}_s, Z^{\tilde{v},l}_s)ds-\int _t^{\tau ^*_1}Z^{\tilde{v},l}_sdW_s \\&={\mathbbm {1}}_{[\tau ^*_1<T]}\{Y^{\tilde{v}\circ (\tau ^*_1, \beta ^*_1),l-1}_{\tau ^*_1}-c^{\tilde{v}}(\tau ^*_1, \beta ^*_1)\}+{\mathbbm {1}}_{[\tau ^*_1=T]}\xi ^{\tilde{v}} \\&\quad +\int _t^{\tau ^*_1}f^{\tilde{v}}(s,Y^{\tilde{v},l}_s, Z^{\tilde{v},l}_s)ds-\int _t^{\tau ^*_1}Z^{\tilde{v},l}_sdW_s, \end{aligned}$$

where $\beta ^*_1$ can be chosen to be ${\mathcal {F}}_{\tau ^*_1}$-measurable by continuity of the map $b'\mapsto Y^{\tilde{v}\circ (\tau ^*_1,b'), l-1}_{\tau ^*_1}-c^{\tilde{v}}(\tau ^*_1,b')$ and the measurable selection theorem (see e.g. Chapter 7 in [3] or [11]).

Now, we can continue and inductively define $\tau ^*_{j}:=\inf \{s\ge \tau _{j-1}^*:Y^{\tilde{v}\circ (\tau ^*_1,\ldots , \tau ^*_{j-1},\beta ^*_1,\ldots ,\beta ^*_{j-1}),l+1-j}_s=\sup _{b\in U}\{Y^{\tilde{v}\circ (\tau ^*_1,\ldots ,\tau ^*_{j-1},\beta ^*_1, \ldots ,\beta ^*_{j-1})\circ (s,b),l-j}_s-c^{\tilde{v}\circ (\tau ^*_1,\ldots ,\tau ^*_{j-1},\beta ^*_1,\ldots , \beta ^*_{j-1})}(s,b)\}\}\wedge T$ for $j=1,\ldots , l$, and take $\beta ^*_j$ to be the corresponding ${\mathcal {F}}_{\tau _j^*}$-measurable maximizer. By induction we get that

$$\begin{aligned} Y^{\tilde{v},l}_t&=\xi ^{\tilde{v}\circ u^*} +\int _t^{T}\sum _{j=0}^{N^*}{\mathbbm {1}}_{[\tau ^*_j,\tau ^*_{j+1})} (s)f^{\tilde{v}\circ [u^*]_j}(s,Y^{\tilde{v}\circ [u^*]_j,l-j}_s, Z^{\tilde{v}\circ [u^*]_j,l-j}_s)ds\nonumber \\&\quad -\int _t^{T}\sum _{j=0}^{N^*}{\mathbbm {1}}_{[\tau ^*_j,\tau ^*_{j+1})} (s)Z^{\tilde{v}\circ [u^*]_j,l-j}_sdW_s-\sum _{j=1}^{N^*} c^{\tilde{v}\circ [u^*]_{j-1}}(\tau ^*_j,\beta ^*_j), \end{aligned}$$

(3.14)

where $u^*:=(\tau _1^*,\ldots ,\tau ^*_{N^*};\beta ^*_1, \ldots ,\beta ^*_{N^*})$ with $N^*:=\max \{j\in \{0,\ldots ,l\}: \tau ^*_j<T\}$ and using the convention that $\tau ^*_0=0$ and $\tau ^*_{N^*+1}=T$.

In particular, (3.14) implies by comparison and positivity of the intervention cost that $U^{\tilde{v},\emptyset }_t\le Y^{\tilde{v},l}_t\le \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^{l}_t}U^{\tilde{v}\circ u,\emptyset }_t$, ${\mathbb {P}}$-a.s., and we find that $|Y^{\tilde{v},l}_t|\le \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^{l}_t}|U^{\tilde{v}\circ u,\emptyset }_t|$. Furthermore, by continuity of the map $b\mapsto Y^{v\circ (t,b)}_t$ we can find a $\beta ^\diamond \in {\mathcal {I}}(t)$ such that $\sup _{b\in U}|Y^{v\circ (t,b)}_t|=|Y^{v\circ (t,\beta ^\diamond )}_t|$, ${\mathbb {P}}$-a.s., and we conclude that $\sup _{b\in U}|Y^{v\circ (t,b)}_t|\le \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^{l+1}_t}|U^{v\circ u,\emptyset }_t|$. Now, for arbitrary $u\in {\mathcal {U}}^f$ we have that

$$\begin{aligned} U^{v\circ u,\emptyset }_t =e^{v\circ u,\emptyset }_{t,T} \xi ^{v\circ u}+\int _t^{T}e^{v\circ u,\emptyset }_{t,s} f^{v\circ u}(s,0,0)ds-\int _t^{T}e^{v\circ u,\emptyset }_{t,s} V^{v\circ u,\emptyset }_sdW^{v\circ u,\emptyset }_s. \end{aligned}$$

This gives, since $V^{v\circ u,\emptyset }\in {\mathcal {H}}^2_{{\mathbb {Q}}^{v\circ u,\emptyset }}$ (see Proposition 3.8), that

$$\begin{aligned} |U^{v\circ u,\emptyset }_t|^p&\le C{\mathbb {E}}^{{\mathbb {Q}}^{v\circ u,\emptyset }} \Big [|\xi ^{v\circ u}|^p+\int _t^{T}|f^{v\circ u} (s,0,0)|^pds\big |{\mathcal {F}}_t\Big ]\nonumber \\&\le C{\mathbb {E}}\big [|{\mathcal {E}}(\zeta ^{v\circ u,\emptyset }*W)_{t,T}|^{r'} \big |{\mathcal {F}}_t\big ]^{1/r'}{\mathbb {E}}\Big [\big (|\xi ^{v\circ u}|^p\nonumber \\&\quad +\int _t^{T}|f^{v\circ u}(s,0,0)|^pds\big )^{r}\big | {\mathcal {F}}_t\Big ]^{1/r}\le C (R^v_t)^{1/r'}(\bar{K}^{v,pr}_t)^{1/r}, \end{aligned}$$

(3.15)

where the last inequality follows by Assumption 3.3 since $|\zeta ^{v\circ u,\emptyset }|\le L^{v\circ u}$. In particular, continuity of $(\sup _{b\in U}Y^{v\circ (t,b),l}_{t}:t\in [0,T])$ implies that, outside of a ${\mathbb {P}}$-null set, we have

$$\begin{aligned} \sup _{b\in U}|Y^{v\circ (t,b),l}_{t}|^p \le C (R^v_t)^{1/r'}(\bar{K}^{v,pr}_t)^{1/r} \end{aligned}$$

for all $t\in [0,T]$. We can thus apply Hölder’s inequality to find that

$$\begin{aligned} \Vert (\sup _{b\in U}Y^{v\circ (t,b),l}_{t}: t\in [0,T])\Vert _{{\mathcal {S}}^p}^p&\le C\Vert (R^v)^{1/r'}(\bar{K}^{v,pr})^{1/r}\Vert _{{\mathcal {S}}^1}\\&\le C\Vert R^v\Vert _{{\mathcal {S}}^1}^{1/r'}\Vert \bar{K}^{v,pr}\Vert _{{\mathcal {S}}^1}^{1/r}\le C \end{aligned}$$

by Lemma 3.4 and since $\bar{K}^{v,p}_t\in {\mathcal {S}}^1$ for all $p\ge 1$. $\square$

Lemma 3.11

Assume that Hypothesis RBSDE.l holds for some $l\ge 0$, then for each $p\ge 1$ there is a $C>0$, that does not depend on l, such that

$$\begin{aligned} \Vert Y^{v,l+1}\Vert _{{\mathcal {S}}^p}+\Vert Z^{v,l+1}\Vert _{{\mathcal {H}}^p} +\Vert K^{v,l+1}\Vert _{{\mathcal {S}}^p}\le C, \end{aligned}$$

(3.16)

for all $v\in {\mathcal {U}}^f$.

Proof

This is immediate from Proposition 2.4 and Lemma 3.10. $\square$

Lemma 3.12

Assume that Hypothesis RBSDE.l holds for some $l\ge 0$, then for each $\kappa \ge 0$ and $p\ge 1$, there is a $C>0$ and a $p'\ge 1$ such that for any $\mathbf {v},\mathbf {v}'\in {\mathcal {D}}^\kappa$, we have

$$\begin{aligned} \Vert (\sup _{b\in U}|Y^{\mathbf {v}'\circ (t,b),l}_t-Y^{\mathbf {v}\circ (t,b),l}_t|: t\in [0,T])\Vert _{{\mathcal {S}}^{p'}}^{p'}\le C\Vert \mathbf {v}'-\mathbf {v}\Vert _{{\mathcal {D}}^f}^{p}. \end{aligned}$$

(3.17)

Proof

We let $u^*$ and $u':=(\tau '_1,\ldots ,\tau '_{N'};\beta '_1, \ldots ,\beta '_{N'})$ be the controls obtained by repeating the construction in the proof of Lemma 3.11, starting from $Y^{\mathbf {v}\circ (t,b),l}_t$ and $Y^{\mathbf {v}'\circ (t,b),l}_t$, respectively, instead of $Y^{v\circ (t,b),l}_t$. By Proposition 3.8 it follows that for each $v\in {\mathcal {U}}^f$ and $u\in {\mathcal {U}}^{l+1}$, there is a unique pair $(U^{v,u},V^{v,u})\in {\mathcal {S}}^2\times {\mathcal {H}}^2$ that solves (3.12), i.e.

$$\begin{aligned} U_t^{v,u}&=\xi ^{v\circ u}+\int _t^{T}f^{v\circ u}(s,U^{v,u}_s,V^{v,u}_s)ds -\int _t^{T}V^{v,u}_sdW_s \\&\qquad -\sum _{j=1}^N {\mathbbm {1}}_{[\tau _j\ge t]}c^{v\circ [u]_{j-1}}(\tau _j,\beta _j). \end{aligned}$$

By a trivial argument we find that $Y^{\mathbf {v}\circ (t,b),l}_t=U_t^{\mathbf {v}\circ (t,b),u^*} =\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^l}U_t^{\mathbf {v}\circ (t,b),u}$ and conclude that

$$\begin{aligned} |Y^{\mathbf {v}\circ (t,b),l}_t-Y^{\mathbf {v}'\circ (t,b),l}_t|&\le |U_t^{\mathbf {v}\circ (t,b),u^*}-U_t^{\mathbf {v}'\circ (t,b),u^*}| +|U_t^{\mathbf {v}\circ (t,b),u'} - U_t^{\mathbf {v}'\circ (t,b),u'}| \\&\le 2\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^{l+1}} |U_t^{\mathbf {v},u}-U_t^{\mathbf {v}',u}| \end{aligned}$$

Letting $\delta U^u:=U^{\mathbf {v}',u}-U^{\mathbf {v},u}$ and $\delta V^u:=V^{\mathbf {v}',u}-V^{\mathbf {v},u}$ and writing $\delta \Box ^u:=\Box ^{\mathbf {v}'\circ u}-\Box ^{\mathbf {v}\circ u}$ (for $\Box =\xi ,f$ and c) gives

$$\begin{aligned} e_t\delta U^u_t=e_T\delta \xi ^{u}&+\int _t^{T}e_s\delta f^{u} (s,U^{\mathbf {v},u}_s,V^{\mathbf {v},u}_s)ds \nonumber \\&-\int _t^{T}e_s\delta V^{u} dW^\zeta _s-\sum _{j=1}^N e_{\tau _j}\delta c^{[u]_{j-1}} (\tau _j,\beta _j), \end{aligned}$$

(3.18)

where $e_t:=e^{\int _0^t \gamma _sds}$ with

and $W^\zeta :=W-\int _0^\cdot \zeta _sds$, with

is a Brownian motion under the measure ${\mathbb {Q}}^\zeta \in {\mathfrak {P}}^{\mathbf {v}'}_t$ given by $d{\mathbb {Q}}^\zeta ={\mathcal {E}}(\zeta *W)_T d{\mathbb {P}}$.

For $u\in {\mathcal {U}}^{l+1}_t$, by again appealing to Proposition 3.8, we have that $\delta V^{u}\in {\mathcal {H}}^2_{{\mathbb {Q}}^\zeta }$ and since $e^{-k_fT}\le e_t\le e^{k_fT}$, taking conditional expectation in (3.18) gives

$$\begin{aligned} |\delta U^u_t|&\le C{\mathbb {E}}^{{\mathbb {Q}}^\zeta }\Big [|\delta \xi ^u| +\int _0^T|\delta f^u(s,U^{\mathbf {v},u}_s,V^{\mathbf {v},u}_s)| ds +\sum _{j=1}^N |\delta c^{[u]_{j-1}}(\tau _j,\beta _j)|\Big |{\mathcal {F}}_t\Big ]\\&\le C{\mathbb {E}}^{{\mathbb {Q}}^\zeta }\Big [|\delta \xi ^u|+\left( \int _t^T|\Lambda _s^{\mathbf {v}', \mathbf {v},u}|^2ds\right) ^{1/2}\left( 1+\int _t^T|V^{\mathbf {v},u}_s|^2ds\right) ^{1/2}\\&\qquad \qquad \quad +\sum _{j=1}^N |\delta c^{[u]_{j-1}}(\tau _j,\beta _j)|\Big |{\mathcal {F}}_t\Big ], \end{aligned}$$

where we have used the Lipschitz condition on f to arrive at the last inequality. In particular, since $N\le l+1$, ${\mathbb {P}}$-a.s., this gives that

$$\begin{aligned} |\delta U^u_t|^p&\le C{\mathbb {E}}^{{\mathbb {Q}}^\zeta }\Big [|\delta \xi ^u|^p +\left( \int _t^T|\Lambda _s^{\mathbf {v}',\mathbf {v},u}|^2ds\right) ^{p/2} \left( 1+\left( \int _t^T|V^{\mathbf {v},u}_s|^2ds\right) ^{p/2}\right) \\&\qquad \qquad \quad +\sum _{j=1}^N |\delta c^{[u]_{j-1}} (\tau _j,\beta _j)|^p\Big |{\mathcal {F}}_t\Big ]\\&\le C(R^{\mathbf {v}'}_t)^{1/r'}\left( (\bar{K}^{\mathbf {v},\mathbf {v}',l+1,pr}_t)^{1/r} +(\bar{K}^{\mathbf {v},\mathbf {v}',l+1,2pr}_t)^{1/2r}\right. \\&\qquad \qquad \quad \left. \left( 1+{\mathbb {E}}\Big [\big (\int _t^T|V^{\mathbf {v},u}_s|^2ds\big )^{pr} \Big |{\mathcal {F}}_t\Big ]^{1/2r}\right) \right) , \end{aligned}$$

as $|\zeta _t|\le L^{\mathbf {v}'\circ u}_t$. Applying the standard change of measure approach it follows that

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {Q}}^{\mathbf {v},u}}\left[ \left( \int _t^T|V^{\mathbf {v},u}_s|^2 ds\right) ^{p} \big |{\mathcal {F}}_t\right]&\le C{\mathbb {E}}^{{\mathbb {Q}}^{\mathbf {v},u}}\left[ |\xi ^{\mathbf {v}\circ u}|^{2p} +\left( \int _t^T|f^{\mathbf {v}\circ u}(s,0,0)|ds\right) ^{2p}\right. \\&\quad \left. +\sum _{j=1}^N |c^{\mathbf {v}\circ [u]_{j-1}}(\tau _j,\beta _j)|^{2p} \Big |{\mathcal {F}}_t\right] . \end{aligned}$$

Changing back to ${\mathbb {P}}$-expectation we get by the Girsanov theorem that

$$\begin{aligned} {\mathbb {E}}\Big [\big (\int _t^T|V^{\mathbf {v},u}_s|^2\big )^{pr} ds\big |{\mathcal {F}}_t\Big ]&={\mathbb {E}}^{{\mathbb {Q}}^{\mathbf {v},u}}\Big [{\mathcal {E}}(-\gamma ^{\mathbf {v},u}*W^{\mathbf {v},u})_{t,T} \big (\int _t^T|V^{\mathbf {v},u}_s|^2 ds\big )^{pr}\Big |{\mathcal {F}}_t\Big ]\\&\le C(\tilde{R}^{\mathbf {v}}_t)^{1/r'}{\mathbb {E}}^{{\mathbb {Q}}^{\mathbf {v},u}} \Big [|\xi ^{\mathbf {v}\circ u}|^{2pr^2}\\&\quad +\left( \int _t^T|f^{\mathbf {v}\circ u}(s,0,0)|ds\right) ^{2pr^2}\\&\quad +\sum _{j=1}^N |c^{\mathbf {v}\circ [u]_{j-1}}(\tau _j,\beta _j)|^{2pr^2} \Big |{\mathcal {F}}_t\Big ]^{1/r}\\&\le C(\tilde{R}^{\mathbf {v}}_t)^{1/r'}(R^{\mathbf {v}}_t)^{1/r'} (\bar{K}^{\mathbf {v},2pr^3}_t)^{1/r^2}. \end{aligned}$$

Combined, this gives that

$$\begin{aligned} |\delta U^u_t|^p&\le C(R^{\mathbf {v}'}_t)^{1/r'} ((\bar{K}^{\mathbf {v},\mathbf {v}',l+1,pr}_t)^{1/r}\\&\quad +(\bar{K}^{\mathbf {v},\mathbf {v}',l+1,2pr}_t)^{1/2r} (1+(\tilde{R}^{\mathbf {v}}_t)^{1/2rr'}(R^{\mathbf {v}}_t)^{1/2rr'} (\bar{K}^{\mathbf {v},2pr^3}_t)^{1/2r^3})). \end{aligned}$$

Hence, repeated application of Hölder’s inequality gives, with $\delta Y_t:=\sup _{b\in U}|Y^{\mathbf {v}'\circ (t,b),l}_t -Y^{\mathbf {v}\circ (t,b),l}_t|$, that

$$\begin{aligned} \Vert \delta Y\Vert _{{\mathcal {S}}^{p}}^{p}&\le C(\Vert R^{\mathbf {v}'}\Vert _{{\mathcal {S}}^1}^{1/r'} (\Vert \bar{K}^{\mathbf {v},\mathbf {v}',l+1,pr}\Vert _{{\mathcal {S}}^1}^{1/r} +\Vert (\bar{K}^{\mathbf {v},\mathbf {v}',l+1,2pr})^{1/2}\\&\qquad (1+(\tilde{R}^{\mathbf {v}})^{1/2r'} (R^{\mathbf {v}})^{1/2r'}(\bar{K}^{\mathbf {v},2pr^3})^{1/2r^2})\Vert _{{\mathcal {S}}^1}^{1/r})\\&\le C(\Vert R^{\mathbf {v}'}\Vert _{{\mathcal {S}}^1}^{1/r'}(\Vert \bar{K}^{\mathbf {v},\mathbf {v}',l+1,pr} \Vert _{{\mathcal {S}}^1}^{1/r}+\Vert \bar{K}^{\mathbf {v},\mathbf {v}',l+1,2pr}\Vert _{{\mathcal {S}}^1}^{1/2}\\&\qquad (1+\Vert (\tilde{R}^{\mathbf {v}})^{1/r'}(R^{\mathbf {v}})^{1/r'} (\bar{K}^{\mathbf {v},2pr^3})^{1/r^2})\Vert _{{\mathcal {S}}^1}^{1/2r})\\&\le C(\Vert R^{\mathbf {v}'}\Vert _{{\mathcal {S}}^1}^{1/r'}(\Vert \bar{K}^{\mathbf {v}, \mathbf {v}',l+1,pr}\Vert _{{\mathcal {S}}^1}^{1/r}+\Vert \bar{K}^{\mathbf {v},\mathbf {v}',l +1,2pr}\Vert _{{\mathcal {S}}^1}^{1/2}\\&\qquad (1+\Vert \tilde{R}^{\mathbf {v}}R^{\mathbf {v}} \Vert _{{\mathcal {S}}^1}^{1/2rr'}\Vert \bar{K}^{\mathbf {v},2pr^3}\Vert _{{\mathcal {S}}^1}^{1/2r^3}) \end{aligned}$$

Hence, as $\Vert \tilde{R}^{\mathbf {v}}R^{\mathbf {v}}\Vert _{{\mathcal {S}}^1} \le \Vert \tilde{R}^{\mathbf {v}}\Vert _{{\mathcal {S}}^2}\Vert R^{\mathbf {v}}\Vert _{{\mathcal {S}}^2}\le C$ it follows that

$$\begin{aligned} \Vert \delta Y\Vert _{{\mathcal {S}}^{p'}}^{p'}&\le C(\Vert \bar{K}^{\mathbf {v}, \mathbf {v}',l+1,p'r}\Vert _{{\mathcal {S}}^1}^{1/r}+\Vert \bar{K}^{\mathbf {v}, \mathbf {v}',l+1,2p'r}\Vert _{{\mathcal {S}}^1}^{1/2r}) \end{aligned}$$

and the result follows by Definition 3.2.iv). $\square$

Proposition 3.13

Hypothesis RBSDE.l holds for all $l\ge 0$.

Proof

We note that the triple $(Y^{v,l+1},Z^{v,l+1},K^{v,l+1})$ solves a reflected BSDE with barrier $(\sup _{b\in U}\{-c^{v}(t,b)+Y^{v\circ (t,b),l}_t\}:t\in [0,T])$. By Lemma 3.10 we find that $((\sup _{b\in U}\{-c^{v}(t,b)+Y^{v\circ (t,b),l}_t\})^+:t\in [0,T])\in {\mathcal {S}}^p$ for all $p\ge 1$ uniformly in v. Whenever the statement in Hypothesis RBSDE.l holds for some $l\ge 0$, then Proposition 2.4 guarantees the existence of a unique triple $(Y^{v,l+1},Z^{v,l+1},K^{v,l+1})$ solving (3.11) with $k=l+1$. Moreover, we have

$$\begin{aligned}&{\mathbb {E}}\Big [\sup _{t\in [0,T]}|\sup _{b\in U}\{-c^{v}(t,b) +Y^{v\circ (t,b),l}_{t}\}-\sup _{b\in U}\{-c^{v'}(t,b) +Y^{v'\circ (t,b),l}_t\}|^p\Big ]\\&\le C{\mathbb {E}}\Big [\sup _{t\in [0,T]}\sup _{b\in U}|c^{v}(t,b) -c^{v'}(t,b)|^p+\sup _{t\in [0,T]}\sup _{b\in U}|Y^{v\circ (t,b),l}_{t} -Y^{v'\circ (t,b),l}_t|^p\Big ]. \end{aligned}$$

By (2.5) of Proposition 2.4, Assumption 3.3 and Lemma 3.12 it, thus, follows by repeating the argument in the proof of Proposition 3.6 that for each $p\ge 1$, there is a $p'\ge 1$ such that

$$\begin{aligned}&\Vert Y^{\mathbf {v}',l+1}-Y^{\mathbf {v},l+1}\Vert _{{\mathcal {S}}^{p'}}^{p'}+\Vert Z^{\mathbf {v}',l+1} -Z^{\mathbf {v},l+1}\Vert _{{\mathcal {H}}^{p'}}^{p'} + \Vert K^{\mathbf {v}',l+1}-K^{\mathbf {v},l+1} \Vert _{{\mathcal {S}}^{p'}}^{p'} \\&\le C\Vert \mathbf {v}'-\mathbf {v}\Vert _{{\mathcal {D}}^f}^p. \end{aligned}$$

For $\mathbf {v}',\mathbf {v}\in {\mathcal {D}}^{\kappa }$ we let $p=(m+1){\kappa }+1$ and Kolmogorov’s continuity theorem implies the existence of a family of processes $(\hat{Y}^{\mathbf {v}},\hat{Z}^{\mathbf {v}},\hat{K}^{\mathbf {v}}:\mathbf {v}\in {\mathcal {D}}^{\kappa })$, with $(\hat{Y}^{\mathbf {v}},\hat{Z}^{\mathbf {v}}, \hat{K}^{\mathbf {v}})\in {\mathcal {S}}^2\times {\mathcal {H}}^2\times {\mathcal {S}}^2$, such that (outside of a ${\mathbb {P}}$-null set) $\mathbf {v}\mapsto \hat{Y}^{\mathbf {v}}_\cdot$, $\mathbf {v}\mapsto \hat{Z}^{\mathbf {v}}_\cdot$ and $\mathbf {v}\mapsto \hat{K}^{\mathbf {v}}_\cdot$ are uniformly continuous, $L^2([0,T])$ continuous (and that $\mathbf {v}\mapsto \int _s^T Z^{\mathbf {v}}_rdW_r$ is continuous in $\mathbf {v}$ uniformly in s) and uniformly continuous, respectively, and moreover $(\hat{Y}^{\mathbf {v}}_s,\hat{Z}^{\mathbf {v}}_s,\hat{K}^{\mathbf {v}}_s) =(Y^{\mathbf {v},l+1}_s,Z^{\mathbf {v},l+1}_s,K^{\mathbf {v},l+1}_s)$, ${\mathbb {P}}$-a.s. for all $\mathbf {v}\in {\mathcal {D}}^{\kappa }$ and $s\in [0,T]$.

Now, taking countable unions this extends to ${\mathcal {D}}^f$, and for all $\mathbf {v}\in {\mathcal {D}}^f$, the triple $(\hat{Y}^{\mathbf {v}},\hat{Z}^{\mathbf {v}}, \hat{K}^{\mathbf {v}})$ solves the BSDE (3.11) for $Y^{\mathbf {v},l+1}$.

Furthermore, for any $v\in {\mathcal {U}}^f$ and any approximating sequence $(v_j)_{j\ge 0}$ taking values in a countable dense subset of ${\mathcal {D}}^f$ with $v_j\rightarrow v$, ${\mathbb {P}}$-a.s., and $v_j\in {\mathcal {U}}^f$, we have by repeating the argument in the proof of Proposition 3.6 that

$$\begin{aligned} \hat{Y}^{v}_\eta&=\lim _{j\rightarrow \infty }\Big \{\xi ^{v_j}+\int _\eta ^T f^{v_j}(s,\hat{Y}^{v_j}_s,\hat{Z}^{v_j}_s)ds-\int _\eta ^T \hat{Z}^{v_j}_sdW_s+\hat{K}^{v_j}_T-\hat{K}^{v_j}_\eta \Big \}\\&=\xi ^{v}+\int _\eta ^T f^{v}(s,\hat{Y}^{v}_s,\hat{Z}^{v}_s)ds -\int _\eta ^T \hat{Z}^{v}_sdW_s+\hat{K}^{v}_T- \hat{K}^{v}_\eta , \end{aligned}$$

for all $\eta \in {\mathcal {T}}$. Finally, by Helly’s convergence theorem (see e.g. [23], p. 370) we have

$$\begin{aligned}&\lim _{j\rightarrow \infty }\int _0^T\left( \hat{Y}^{v}_s-\sup _{b\in U} \{-c^{v}(s,b)+\hat{Y}^{v\circ (s,b),l}_s\}\right) d\hat{K}^{v_j}_s\\&\quad =\int _0^T(\hat{Y}^{v}_s-\sup _{b\in U}\{-c^{v}(s,b) +\hat{Y}^{v\circ (s,b),l}_s\})d\hat{K}^{v}_s \end{aligned}$$

and since

$$\begin{aligned}&\int _0^T|\left( \hat{Y}^{v}_s-\sup _{b\in U}\{-c^{v}(s,b) +\hat{Y}^{v\circ (s,b),l}_s\}\right) -\left( \hat{Y}^{v_j}_s -\sup _{b\in U}\{-c^{v_j}(s,b)+\hat{Y}^{v_j \circ (s,b),l}_s\}\right) |d\hat{K}^{v_j}_s\\&\le \hat{K}^{v_j}_T\left( \sup _{s\in [0,T]\times U}| \hat{Y}^{v}_s-\hat{Y}^{v_j}_s|+\sup _{(s,b)\in [0,T] \times U}|c^{v}(s,b)-c^{v_j}(s,b)|\right. \\&\quad \left. +\sup _{(s,b)\in [0,T]\times U}|\hat{Y}^{v\circ (s,b),l}_s -\hat{Y}^{v_j\circ (s,b),l}_s|\right) \end{aligned}$$

which tends to zero, ${\mathbb {P}}$-a.s., as $j\rightarrow \infty$ we get that

$$\begin{aligned}&\int _0^T\left( \hat{Y}^{v}_s-\sup _{b\in U}\{-c^{v}(s,b) +\hat{Y}^{v\circ (s,b),l}_s\}\right) d\hat{K}^{v}_s\\&\quad =\lim _{j\rightarrow \infty }\int _0^T\left( \hat{Y}^{v_j}_s -\sup _{b\in U}\{-c^{v_j}(s,b)+\hat{Y}^{v_j\circ (s,b),l}_s\}\right) d\hat{K}^{v_j}_s \\&\quad =0 \end{aligned}$$

and we conclude that Hypothesis RBSDE.$l+1$ holds as well. The statement of the proposition now follows by an induction argument. $\square$

3.3 Convergence of the Scheme

We now show that there exists a limit family of triples $(\bar{Y}^v,\bar{Z}^v,\bar{K}^v:v\in {\mathcal {U}}^f):=\lim _{k\rightarrow \infty } (Y^{v,k},Z^{v,k},K^{v,k}:v\in {\mathcal {U}}^f)$ that solves the sequential system of reflected BSDEs (1.1). This result relies heavily upon the following two lemmas and their corollaries.

Lemma 3.14

For $v\in {\mathcal {U}}^f$ and $k\ge 0$, assume that $u^*\in {\mathcal {U}}^k_t$ is such that $U_t^{v,u^*}=\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^k_t}U_t^{v,u}$. Then, for each $p\ge 1$, there is a $C>0$, that does not depend on v or k, such that

$$\begin{aligned} {\mathbb {E}}\Big [\sup _{s\in [t,T]}|U_s^{v,u^*}|^p+\big (\int _t^T|V_s^{v, u^*}|^2ds\big )^{p/2}\Big |{\mathcal {F}}_t\Big ]\le C(\tilde{R}^v_t R^v_t \bar{R}^v_t)^{1/r'}(1+\bar{K}^{v,2pr^3}_t)^{1/r^3}. \end{aligned}$$

Proof

For the bound on $U^{v,u^*}$ we note that by (3.15) we have

$$\begin{aligned} |U^{v,u^*}_s|^p&= |Y^{v\circ (u^*(s-)),k-N^*(s-)}_s|^p \\&\le C (R^{v\circ u^*(s-)}_s)^{1/r'} (\bar{K}^{v\circ u^*(s-),pr}_s)^{1/r} \end{aligned}$$

and since by definition we have $R^{v\circ (u^*(s))}_s=R^{v\circ u^*}_s$, we find that

$$\begin{aligned} {\mathbb {E}}\Big [\sup _{s\in [t,T]}|U^{v,u^*}_s|^{p}\Big |{\mathcal {F}}_t\Big ]&\le {\mathbb {E}}\Big [\sup _{s\in [t,T]}(R^{v\circ u^*}_s)^{1/r'} (\bar{K}^{v\circ (u^*(s)),pr}_s)^{1/r}\Big |{\mathcal {F}}_t\Big ]\nonumber \\&\le \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t}{\mathbb {E}}\Big [\sup _{s\in [t,T]} R^{v\circ u}_s\Big |{\mathcal {F}}_t\Big ]^{1/r'}{\mathbb {E}}\Big [\sup _{s\in [t,T]}\bar{K}^{v\circ (u^*(s)),pr}_s \Big |{\mathcal {F}}_t\Big ]^{1/r}\nonumber \\&\le C(\bar{R}^v_t)^{1/r'}(1+\bar{K}^{v,2pr}_t)^{1/2r} \end{aligned}$$

(3.19)

where we have used Jensen’s inequality and (3.3) to reach the last inequality. The first bound then follows by Jensen’s inequality since $\tilde{R}^v_t R^v_t\ge 1$, ${\mathbb {P}}$-a.s.

We apply Ito’s formula to $(U^{v,u^*})^2$ and get that

$$\begin{aligned}&|U^{v,u^*}_t|^2+\int _t^T|V^{v,u^*}_s|^2ds\\&\quad =|\xi ^{v\circ u^*}|^2 +2\int _t^{T}U^{v,u^*}_sf^{v\circ u^*}(s, U^{v,u^*}_s, V^{v,u^*}_s)ds -2\int _t^{T} U^{v,u^*}_sV^{v,u^*}_sdW_s\\&\qquad +\sum _{j=1}^{N^*} \left( -2U^{v,[u^*]_{j-1}}_{\tau ^*_j} c^{v\circ [u^*]_{j-1}}(\tau ^*_j,\beta ^*_j) +|c^{v\circ [u^*]_{j-1}}(\tau ^*_j,\beta ^*_j)|^2\right) \\&\quad \le |\xi ^{v\circ u^*}|^2+2\int _t^{T}\left( \gamma ^{v,u^*}_s| U^{v,u^*}_s|^2+U^{v,u^*}_s f^{v\circ u^*}(s,0, 0)ds\right. \\&\qquad -2\int _t^{T} U^{v,u^*}_sV^{v,u^*}_sdW^{v,u^*}_s +4\sup _{s\in [t,T]}|U^{v,u^*}_s|\sum _{j=1}^{N^*}| c^{v\circ [u^*]_{j-1}}(\tau ^*_j,\beta ^*_j)|, \end{aligned}$$

where the last term appears after applying the relation $|c^{v\circ [u^*]_{j-1}}(\tau ^*_j,\beta ^*_j)|\le 2\sup _{s\in [t,T]}|U^{v,u^*}_s|$. Using the relation $ab\le \frac{1}{2}(\kappa a^2+\frac{1}{\kappa }b^2)$ for $\kappa >0$ we get

$$\begin{aligned} |U^{v,u^*}_t|^2+\int _t^T|V^{v,u^*}_s|^2ds&\le |\xi ^{v\circ u^*}|^2 +(C+2\kappa )\sup _{s\in [t,T]}|U^{v,u^*}_s|^2+\int _t^{T}|f^{v\circ u^*} (s,0, 0)|^2ds\\&\quad -2\int _t^{T} U^{v,u^*}_sV^{v,u^*}_sdW^{v,u^*}_s+\frac{2}{\kappa } \Big (\sum _{j=1}^{N^*} c^{v\circ [u^*]_{j-1}}(\tau ^*_j,\beta ^*_j)\Big )^2. \end{aligned}$$

On the other hand, applying the usual manipulations to (3.12) we get

$$\begin{aligned} U_t^{v,u^*}&=e^{v,u^*}_{t,T}\xi ^{v\circ u^*} +\int _t^{T}e^{v,u^*}_{t,s}f^{v\circ u^*}(s, 0, 0)ds -\int _t^{T} e^{v,u^*}_{t,s}V^{v,u^*}_sdW^{v,u^*}_s\\&\quad -\sum _{j=1}^{N^*} e^{v,u^*}_{t,\tau _j}c^{v\circ [u^*]_{j-1}} (\tau ^*_j,\beta ^*_j). \end{aligned}$$

Rearranging terms now gives us (with $e_{t,\cdot }=e_{t,\cdot }^{v,u^*}$)

$$\begin{aligned} \sum _{j=1}^N e_{t,\tau _j^*}c^{v\circ [u^*]_{j-1}} (\tau _j^*,\beta _j^*)&= e_{t,T}\xi ^{v\circ u^*} +\int _t^{T}e_{t,s}f^{v\circ u^*}(s, 0, 0)ds\nonumber \\&\quad -\int _t^{T} e_{t,s}V^{v,u^*}_sdW^{v,u^*}_s-U_t^{v,u^*} \end{aligned}$$

(3.20)

From (3.20) we have that

$$\begin{aligned} \Big (\sum _{j=1}^{N^*} c^{v\circ [u^*]_{j-1}} (\tau ^*_j,\beta ^*_j)\Big )^2&\le C(|\xi ^{v\circ u^*}|^2 +\int _t^{T}|f^{v\circ u^*}(s, 0, 0)|^2ds \\&\quad +\Big |\int _t^{T} e_{t,s}V^{v,u^*}_sdW^{v,u^*}_s \Big |^2+|U_t^{v,u^*}|^2). \end{aligned}$$

Put together this gives

$$\begin{aligned} |U^{v,u^*}_t|^2+\int _t^T|V^{v,u^*}_s|^2ds&\le C\left( 1+\kappa +\frac{1}{\kappa }\right) \\&\quad \left( |\xi ^{v\circ u^*}|^2+\int _t^{T}|f^{v\circ u^*} (s,0, 0)|^2ds+\sup _{s\in [t,T]}|U^{v,u^*}_s|^2\right) \\&\quad -2\int _t^{T} U^{v,u^*}_sV^{v,u^*}_sdW^{v,u^*}_s+\frac{C}{\kappa } \Big |\int _t^{T} e_{t,s}V^{v,u^*}_sdW^{v,u^*}_s\Big |^2. \end{aligned}$$

Raising both sides to p/2 and taking the conditional expectation we find that

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {Q}}^{v,u^*}}\Big [\big (\int _t^T|V^{v,u^*}_s|^2ds\big )^{p/2} \Big |{\mathcal {F}}_t\Big ]\\&\quad \le C{\mathbb {E}}^{{\mathbb {Q}}^{v,u^*}}\Big [\left( 1+\kappa ^{p/2} +\frac{1}{\kappa ^{p/2}}\right) \left( |\xi ^{v\circ u^*}|^p+\int _t^{T}|f^{v\circ u^*}(s,0, 0)|^pds\right. \\&\qquad \left. +\sup _{s\in [t,T]}|U^{v,u^*}_s|^p\right) +\big (\int _t^{T} |U^{v,u^*}_sV^{v,u^*}_s|^2ds\big )^{p/4} +\frac{1}{\kappa ^{p/2}}\big (\int _t^{T} |V^{v,u^*}_s|^2ds\big )^{p/2} \Big |{\mathcal {F}}_t\Big ] \end{aligned}$$

and since

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {Q}}^{v,u^*}}\Big [\big (\int _t^{T} |U^{v,u^*}_s V^{v,u^*}_s|^2ds\big )^{p/4}\Big |{\mathcal {F}}_t\Big ]\\&\quad \le \frac{1}{2}{\mathbb {E}}^{{\mathbb {Q}}^{v,u^*}}\Big [\kappa \sup _{s\in [t,T]}| U^{v,u^*}_s|^p+\frac{1}{\kappa }\big (\int _t^{T} |V^{v,u^*}_s |^2ds\big )^{p/2}\Big |{\mathcal {F}}_t\Big ] \end{aligned}$$

we arrive at the inequality

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {Q}}^{v,u^*}}\Big [\big (\int _t^T|V^{v,u^*}_s|^2ds\big )^{p/2} \Big |{\mathcal {F}}_t\Big ]\nonumber \\&\quad \le C{\mathbb {E}}^{{\mathbb {Q}}^{v,u^*}}\Big [|\xi ^{v\circ u^*}|^p +\int _t^{T}|f^{v\circ u^*}(s,0, 0)|^pds+\sup _{s\in [t,T]}| U^{v,u^*}_s|^p\Big |{\mathcal {F}}_t\Big ] \end{aligned}$$

(3.21)

by choosing $\kappa >0$ sufficiently large. Under ${\mathbb {P}}$ this rewrites as

$$\begin{aligned} {\mathbb {E}}\Big [\big (\int _t^T|V^{v,u^*}_s|^2ds\big )^{p/2}\Big | {\mathcal {F}}_t\Big ]&\le C(\tilde{R}^v_t R^v_t)^{1/r'}{\mathbb {E}}\Big [|\xi ^{v\circ u^*}|^{pr^2}\nonumber \\&\quad +\int _t^{T}|f^{v\circ u^*}(s,0, 0)|^{pr^2}ds +\sup _{s\in [t,T]}|U^{v,u^*}_s|^{pr^2}\Big |{\mathcal {F}}_t\Big ]^{1/r^2} \end{aligned}$$

(3.22)

The desired result now follows by setting $p\leftarrow pr^2$ in (3.19) and using Jensen’s inequality while noting that $\bar{R}^v_t\ge 1$, ${\mathbb {P}}$-a.s. $\square$

Corollary 3.15

For $v\in {\mathcal {U}}^f$, $t\in [0,T]$, $\beta \in {\mathcal {I}}(t)$ and $k\ge 0$, assume that $u^*\in {\mathcal {U}}^k_t$ is such that $U_t^{v\circ (t,\beta ),u^*}=\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^k_t}U_t^{v\circ (t,\beta ),u}$. Then, for each $p\ge 1$, there is a $C>0$ that does not depend on v or k, such that

$$\begin{aligned}&{\mathbb {E}}\Big [\sup _{s\in [t,T]}|U_s^{v\circ (t,\beta ),u^*}|^p +\big (\int _t^T|V_s^{v\circ (t,\beta ),u^*}|^2ds\big )^{p/2} \Big |{\mathcal {F}}_t\Big ]\\&\quad \le C(\tilde{R}^v_t R^v_t\bar{R}^v_t)^{1/r'} (1+\bar{K}^{v,2pr^3}_t)^{1/r^3}. \end{aligned}$$

Proof

This follows immediately by making suitable manipulations, i.e. setting $v\leftarrow v\circ (t,\beta )$, in the proof of Lemma 3.14. $\square$

Lemma 3.16

For $v\in {\mathcal {U}}^f$ and $k\ge 0$, assume that $u^*\in {\mathcal {U}}^k_t$ is such that $U_t^{v,u^*}=\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^k_t}U_t^{v,u}$. Then, for each $p\ge 1$, there is a $C>0$ that does not depend on v or k, such that

$$\begin{aligned} {\mathbb {E}}\big [(N^*)^p\big |{\mathcal {F}}_t\big ]&\le C{\mathbb {E}}\Big [\Big (\sum _{j=1}^N c^{v\circ [u^*]_{j-1}}(\tau _j^*,\beta _j^*)\Big )^p\Big |{\mathcal {F}}_t\Big ] \\&\le C(\tilde{R}^v_t R^v_t\bar{R}^v_t)^{1/r'} (1+\bar{K}^{v,2pr^3}_t)^{1/r^3}. \end{aligned}$$

Proof

Since the intervention costs are bounded from below by $\delta >0$, we have

$$\begin{aligned} \delta N^*\le \sum _{j=1}^N c^{v\circ [u^*]_{j-1}}(\tau _j^*,\beta _j^*) \end{aligned}$$

from which the first inequality follows. Now, from (3.20) we have

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {Q}}^{v, u^*}}\Big [\Big (\sum _{j=1}^N c^{v\circ [u^*]_{j-1}} (\tau _j^*,\beta _j^*)\Big )^p\Big |{\mathcal {F}}_t\Big ]\\&\le C{\mathbb {E}}^{{\mathbb {Q}}^{v,u^*}}\Big [|\xi ^{v\circ u^*}|^p +\int _t^{T}|f^{v\circ u^*}(s,0, 0)|^pds +\big (\int _t^{T}|V^{v, u^*}_s|^2ds\big )^{p/2} +|U^{v,u^*}_t|^p\Big |{\mathcal {F}}_t\Big ]\\&\le C{\mathbb {E}}^{{\mathbb {Q}}^{v,u^*}}\Big [|\xi ^{v\circ u^*}|^p +\int _t^{T}|f^{v\circ u^*}(s,0, 0)|^pds +\sup _{s\in [t,T]}|U^{v,u^*}_s|^p\Big |{\mathcal {F}}_t\Big ] \end{aligned}$$

where we have used (3.21) to get the last inequality. Now the result is immediate from the last part in the proof of Lemma 3.14. $\square$

Corollary 3.17

For $v\in {\mathcal {U}}^f$, $t\in [0,T]$, $\beta \in {\mathcal {I}}(t)$ and $k\ge 0$, assume that $u^*\in {\mathcal {U}}^k_t$ is such that $U_t^{v\circ (t,\beta ),u^*} =\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^k_t}U_t^{v\circ (t,\beta ),u}$. Then, for each $p\ge 1$, there is a $C>0$ that does not depend on v or k, such that

$$\begin{aligned} {\mathbb {E}}\big [(N^*)^p\big |{\mathcal {F}}_t\big ]&\le C{\mathbb {E}}\Big [\Big (\sum _{j=1}^N c^{v\circ (t,\beta )\circ [u^*]_{j-1}}(\tau _j^*,\beta _j^*)\Big )^p \Big |{\mathcal {F}}_t\Big ]\\&\le C(\tilde{R}^v_t R^v_t\bar{R}^v_t)^{1/r'} (1+\bar{K}^{v,2pr^3}_t)^{1/r^3}. \end{aligned}$$

Proof

This follows by repeating the argument in the proof of Lemma 3.16 after making the swap $v\leftarrow v\circ (t,\beta )$. $\square$

We are now ready to tackle the convergence of the sequence $Y^{v,k}$, this is done in the following proposition

Proposition 3.18

There exists a limit family $(\bar{Y}^v:v\in {\mathcal {U}}^f)$ such that for all $v\in {\mathcal {U}}^f$ (outside of a ${\mathbb {P}}$-null set) and each $p\ge 2$, we have

• (i) $Y^{v\circ (\cdot ,\cdot ),k}_\cdot \nearrow \bar{Y}^{v\circ (\cdot ,\cdot )}_\cdot$ pointwisely, and

• (ii) $\Vert \sup _{b\in U}|\bar{Y}^{v\circ (\cdot ,b)}_\cdot -Y^{v\circ (\cdot ,b),k}_\cdot |\Vert _{{\mathcal {S}}^p}\rightarrow 0$ as $k\rightarrow \infty$.

Proof

The sequence $(Y^{v\circ (\cdot ,\cdot ),k}_\cdot )_{k\ge 0}$ is non-decreasing and ${\mathbb {P}}$-a.s. bounded by Lemma 3.10. Thus it converges pointwisely, ${\mathbb {P}}$-a.s., and i) follows.

We now turn our focus to the second claim and note that for each $t\in [0,T]$, continuity and measurable selection implies that there is a $\beta \in {\mathcal {I}}(t)$ such that

$$\begin{aligned} \sup _{b\in U}|Y^{v\circ (t,b),k}_t-Y^{v\circ (t,b),k'}_t|= |Y^{v\circ (t,\beta ),k}_t-Y^{v\circ (t,\beta ),k'}_t| \end{aligned}$$

To simplify notation we set $\tilde{v}\leftarrow v\circ (t,\beta )$ and have by Corollary 3.17 that if $u^*=(\tau _1^*,\ldots ,\tau ^*_{N^*};\beta _1^*, \ldots ,\beta ^*_{N^*})\in {\mathcal {U}}^k_t$ is such that $U^{\tilde{v},u^*}_t=Y^{\tilde{v},k}_t$, then ${\mathbb {E}}\big [N^*\big |{\mathcal {F}}_t\big ]\le C(\tilde{R}^v_t R^v_t\bar{R}^v_t)^{1/r'}(1+\bar{K}^{v,2r^3}_t)^{1/r^3}$ and, in particular, we find that ${\mathbb {E}}\big [{\mathbbm {1}}_{[N^*> k']}\big |{\mathcal {F}}_t\big ]\le C(\tilde{R}^v_t R^v_t\bar{R}^v_t)^{1/r'}(1+\bar{K}^{v,2r^3}_t)^{1/r^3}/k'$ for all $k'\ge 1$.

For any $k'$ with $0\le k'\le k$, the truncation $[u^*]_{k'}$ belongs to ${\mathcal {U}}^{k'}_t$. We, thus, have

$$\begin{aligned} U^{\tilde{v},[u^*]_{k'}}_t\le Y^{\tilde{v},k'}_t\le Y^{\tilde{v},k}_t. \end{aligned}$$

Since $Y^{\tilde{v},k}_t=U^{\tilde{v},u^*}_t$, this gives

$$\begin{aligned} |Y^{\tilde{v},k}_t-Y^{\tilde{v},k'}_t| \le U^{\tilde{v},u^*}_t-U^{\tilde{v},[u^*]_{k'}}_t \end{aligned}$$

Moreover, since the intervention costs are positive, we have that

$$\begin{aligned} U^{\tilde{v},u^*}_t-U^{\tilde{v},[u^*]_{k'}}_t&\le \xi ^{\tilde{v}\circ u^*}-\xi ^{\tilde{v} \circ [u^*]_{k'}}+\int _t^T(f^{\tilde{v}\circ u^*} (s,U^{\tilde{v},u^*}_s,V^{\tilde{v},u^*}_s)\\&\quad -f^{\tilde{v}\circ [u^*]_{k'}}(s,U^{\tilde{v},[u^*]_{k'}}_s, V^{\tilde{v},[u^*]_{k'}}_s))ds \\&\quad -\int _t^T(V^{\tilde{v},u^*}_s -V^{\tilde{v},[u^*]_{k'}}_s)dW_s. \end{aligned}$$

Setting $e_{t,s}:=e^{\int _t^s\gamma _s}$ with

$$\begin{aligned} \gamma _s:=\frac{f^{\tilde{v}\circ [u^*]_{k'}} (s,U^{\tilde{v},u^*}_s,V^{\tilde{v},[u^*]_{k'}}_s) -f^{\tilde{v}\circ [u^*]_{k'}}(s,U^{\tilde{v},[u^*]_{k'}}_s, V^{\tilde{v},[u^*]_{k'}}_s)}{U^{\tilde{v},u^*}_s -U^{\tilde{v},[u^*]_{k'}}_s}{\mathbbm {1}}_{[U^{\tilde{v},u^*}_s \ne U^{\tilde{v},[u^*]_{k'}}_s]} \end{aligned}$$

gives

$$\begin{aligned} U^{\tilde{v},u^*}_t-U^{\tilde{v},[u^*]_{k'}}_t&\le e_{t,T}(\xi ^{\tilde{v}\circ u^*}-\xi ^{\tilde{v} \circ [u^*]_{k'}})+\int _t^Te_{t,s}(f^{\tilde{v}\circ u^*} (s,U^{\tilde{v},u^*}_s,V^{\tilde{v},u^*}_s)\\&\quad -f^{\tilde{v}\circ [u^*]_{k'}}(s,U^{\tilde{v},u^*}_s, V^{\tilde{v},[u^*]_{k'}}_s))ds-\int _t^Te_{t,s}(V^{\tilde{v},u^*}_s -V^{\tilde{v},[u^*]_{k'}}_s)dW_s\\&=e_{t,T}(\xi ^{\tilde{v}\circ u^*}-\xi ^{\tilde{v} \circ [u^*]_{k'}})+\int _t^Te_{t,s}(f^{\tilde{v}\circ u^*} -f^{\tilde{v}\circ [u^*]_{k'}})(s,U^{\tilde{v},u^*}_s, V^{\tilde{v},u^*}_s)ds\\&\quad -\int _t^Te_{t,s}(V^{\tilde{v},u^*}_s-V^{\tilde{v},[u^*]_{k'}}_s) dW^{\zeta }_s, \end{aligned}$$

where $W^{\zeta }_t:=W_t-\int _0^t\zeta _sds$, with

$$\begin{aligned} \zeta _s&:=\frac{f^{\tilde{v}\circ [u^*]_{k'}} (s,U^{\tilde{v},u^*}_s,V^{\tilde{v},u^*}_s) -f^{\tilde{v}\circ [u^*]_{k'}}(s,U^{\tilde{v},u^*}_s, V^{\tilde{v},[u^*]_{k'}}_s)}{|V^{\tilde{v},u^*}_s -V^{\tilde{v},[u^*]_{k'}}_s|^2}\\&\qquad (V^{\tilde{v},u^*}_s -V^{\tilde{v},[u^*]_{k'}}_s)^{\top } {\mathbbm {1}}_{[V^{\tilde{v},u^*}_s \ne V^{\tilde{v},[u^*]_{k'}}_s]}. \end{aligned}$$

Taking the conditional expectation, using that $V^{\tilde{v},u^*},V^{\tilde{v},[u^*]_{k'}}\in {\mathcal {H}}^2_{{\mathbb {Q}}^{\zeta }}$ by Proposition 3.8 and noting that the right-hand side is non-zero only when $N^*>k'$ gives

$$\begin{aligned}&U^{\tilde{v},u^*}_t-U^{\tilde{v},[u^*]_{k'}}_t\\&\quad \le C{\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [{\mathbbm {1}}_{[N^*>k']} \big (|\xi ^{\tilde{v}\circ u^*}-\xi ^{\tilde{v} \circ [u^*]_{k'}}|\\&\qquad +\int _t^T|(f^{\tilde{v}\circ u^*} -f^{\tilde{v}\circ [u^*]_{k'}})(s,U^{\tilde{v},u^*}_s, V^{\tilde{v},u^*}_s)|ds\big )\Big |{\mathcal {F}}_t\Big ]. \end{aligned}$$

By Hölder’s inequality we find that

$$\begin{aligned} |Y^{\tilde{v},k}_t-Y^{\tilde{v},k'}_t|^p&\le C {\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\big [{\mathbbm {1}}_{[N^*>k']} \big |{\mathcal {F}}_t\big ]^{1/2}{\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\\&\quad \Big [|\xi ^{\tilde{v}\circ u^*}-\xi ^{\tilde{v} \circ [u^*]_{k'}}|^{2p}+\big (\int _t^T|(f^{\tilde{v} \circ u^*}-f^{\tilde{v}\circ [u^*]_{k'}})(s,0,0)|ds\big )^{2p}\\&\quad +\sup _{s\in [t,T]}|U^{\tilde{v},u^*}_s|^{2p}+|L^{\tilde{v} \circ u^*}_T|^{4p}+|L^{\tilde{v}\circ [u^*]_{k'}}_T|^{4p} +\big (\int _t^T|V^{\tilde{v},u^*}_s|^{2}ds\big )^{2p}\Big |{\mathcal {F}}_t\Big ]^{1/2}\\&\le C (R^v_t)^{1/r'}{\mathbb {E}}\big [{\mathbbm {1}}_{[N^*>k']}\big |{\mathcal {F}}_t\big ]^{1/2r} {\mathbb {E}}\Big [|\xi ^{\tilde{v}\circ u^*}-\xi ^{\tilde{v}\circ [u^*]_{k'}}|^{2pr}\\&\quad +\big (\int _t^T|(f^{\tilde{v}\circ u^*}-f^{\tilde{v} \circ [u^*]_{k'}})(s,0,0)|ds\big )^{2pr}\\&\quad +\sup _{s\in [t,T]}|U^{\tilde{v},u^*}_s|^{2pr} +|L^{\tilde{v}\circ u^*}_T|^{4pr}+|L^{\tilde{v} \circ [u^*]_{k'}}_T|^{4pr}\\&\quad +\big (\int _t^T|V^{\tilde{v},u^*}_s|^2 ds\big )^{2pr}\Big |{\mathcal {F}}_t\Big ]^{1/2r}\\&\le C(R^v_t)^{1/r'}(\tilde{R}^v_t R^v_t\bar{R}^v_t)^{1/rr'} (1+\bar{K}^{v,2r^3}_t)^{1/2r^4}(1+\bar{K}^{v,8pr^4}_t)^{1/2r^4}/ (k')^{1/2r},\\&\le C(R^v_t)^{1/r'}(\tilde{R}^v_t R^v_t\bar{R}^v_t)^{1/rr'} (1+\bar{K}^{v,8pr^4}_t)^{1/r^4}/(k')^{1/2r}, \end{aligned}$$

where we have used (3.1) and Corollary 3.15 to arrive at the last inequality. Since $\sup _{b\in U}|\bar{Y}^{v\circ (\cdot ,b),k'}_\cdot -Y^{v\circ (\cdot ,b),k}_\cdot$ is continuous and the right hand side of the above equation is a càdlàg process this extends to all $t\in [0,T]$ (outside of a ${\mathbb {P}}$-null set) and we can take the ${\mathcal {S}}^1$-norm followed by Hölder’s inequality to get that

$$\begin{aligned}&\Vert \sup _{b\in U}|\bar{Y}^{v\circ (\cdot ,b),k'}_\cdot -Y^{v\circ (\cdot ,b), k}_\cdot |\Vert _{{\mathcal {S}}^p}^p\\&\quad \le C\Vert (R^v)^{1/r'}(\tilde{R}^v R^v\bar{R}^v)^{1/rr'}(1+\bar{K}^{v,8pr^4})^{1/r^4}\Vert _{{\mathcal {S}}^1}/(k')^{1/2r}\\&\quad \le C\Vert R^v\Vert _{{\mathcal {S}}^1}^{1/r'}\Vert (\tilde{R}^v R^v\bar{R}^v)^{1/r'}(1+\bar{K}^{v,8pr^4})^{1/r^3}\Vert _{{\mathcal {S}}^1}^{1/r}/(k')^{1/2r}\\&\quad \le C\Vert R^v\Vert _{{\mathcal {S}}^1}^{1/r'}\Vert \tilde{R}^v R^v\bar{R}^v\Vert _{{\mathcal {S}}^1}^{1/rr'}\Vert (1+\bar{K}^{v,8pr^4})^{1/r^2}\Vert _{{\mathcal {S}}^1}^{1/r^2}/(k')^{1/2r}\\&\quad \le C/(k')^{1/2r}, \end{aligned}$$

where $C>0$ is independent of $k,k'$. The last inequality holds since there is a $C>0$ such that

$$\begin{aligned} \Vert \tilde{R}^v R^v\bar{R}^v\Vert _{{\mathcal {S}}^1}\le \Vert \tilde{R}^v\Vert _{{\mathcal {S}}^3}\Vert R^v\Vert _{{\mathcal {S}}^3}\Vert \bar{R}^v\Vert _{{\mathcal {S}}^3}\le C, \end{aligned}$$

for all $v\in {\mathcal {U}}^f$. Finally, taking the limit as $k\rightarrow \infty$, (i) and Fatou’s lemma gives that $\Vert \sup _{b\in U}|\bar{Y}^{v\circ (\cdot ,b),k'}_\cdot -Y^{v\circ (\cdot ,b),k}_\cdot |\Vert _{{\mathcal {S}}^p} = C/(k')^{1/2r}$. $\square$

Proposition 3.19

There is a family $(\bar{Z}^{v},\bar{K}^v:v\in {\mathcal {U}}^f)$ such that $(\bar{Y}^v,\bar{Z}^{v},\bar{K}^v:v\in {\mathcal {U}}^f)$ is a solution to (1.1).

Proof

Having established that $\Vert \sup _{b\in U}|Y^{v\circ (\cdot ,b),k}_\cdot -Y^{v\circ (\cdot ,b),k'}_\cdot |\Vert _{{\mathcal {S}}^p}\rightarrow 0$ as $k,k'\rightarrow \infty$ in the previous proposition it follows by Proposition 2.4 that $\Vert Z^{v,k}-Z^{v,k'}\Vert _{{\mathcal {H}}^2}\rightarrow 0$ as $k,k'\rightarrow \infty$. In particular, $(Z^{v,k})_{k\ge 0}$ is a Cauchy sequence in the Hilbert space ${\mathcal {H}}^2$ and we conclude that there is a $\bar{Z}^v\in {\mathcal {H}}^2$ such that $Z^{v,k}\rightarrow \bar{Z}^v$ in ${\mathcal {H}}^2$.

Now, letting $\bar{K}^v$ be defined by $\bar{K}^v=0$ and

$$\begin{aligned} \bar{K}^{v}_T-\bar{K}^{v}_t&=\sup _{r\in [t,T]}\Big (\xi ^v +\int _r^Tf^{v}(s,\bar{Y}^{v}_s,\bar{Z}^{v}_s)ds-\int _r^T \bar{Z}^{v}_sdW_s \\&\quad -\sup _{b\in U}\{\bar{Y}^{v\circ (r,b)}_r -c^{v}(r,b)\}\Big )^- \end{aligned}$$

we note that $\Vert (\bar{K}^v-K^{v,k}){\mathbbm {1}}_{[0,\eta _l]}\Vert _{{\mathcal {S}}^2}\rightarrow 0$ as $k\rightarrow \infty$ where $\eta _l:=\inf \{s\ge 0:L_s^v\ge l\}\wedge T$ and by Lemma 3.11 we have that $\bar{K}^v\in {\mathcal {S}}^2$. Since $L^v$ is continuous, and thus has ${\mathbb {P}}$-a.s. bounded trajectories, we find that

$$\begin{aligned} {\left\{ \begin{array}{ll} \bar{Y}^{v}_t=\xi ^v+\int _t^T f^v(s,\bar{Y}^{v}_s,\bar{Z}^{v}_s)ds -\int _t^T \bar{Z}^{v}_sdW_s + \bar{K}^{v}_T-\bar{K}^{v}_t,\quad \forall t\in [0,T], \\ \bar{Y}^{v}_t\ge \sup _{b\in U}\{\bar{Y}^{v\circ (t,b)}_t -c^v(t,b)\},\quad \forall t\in [0,T],\\ \int _0^T(\bar{Y}^{v}_t-\sup _{b\in U}\{\bar{Y}^{v\circ (t,b)}_t -c^v(t,b))\})dK^{v}_t=0. \end{array}\right. } \end{aligned}$$

Finally, the map $(t,b)\mapsto \bar{Y}^{v\circ (t,b)}_t\in {\mathcal {O}}_c$ by uniform convergence. $\square$

3.4 Uniqueness by a Verification Argument

Theorem 3.20

The finite horizon sequential system of reflected BSDEs (1.1) admits a unique solution $(Y^v,Z^v,K^v:v\in {\mathcal {U}}^f)$ and $Y^v_t=\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t}U^{v,u}_t=U^{v,u^*}_t$ with $u^*=(\tau ^*_1,\ldots ,\tau ^*_{N^*};\beta ^*_1,\ldots , \beta ^*_{N^*})\in {\mathcal {U}}^f_t$ defined as:

• $\tau ^*_{j}:=\inf \Big \{s \ge \tau ^*_{j-1}:\,Y_s^{v\circ [u^*]_{j-1}}=\sup _{b\in U} \{Y^{v\circ [u^*]_{j-1}\circ (s,b)}_s-c^{v\circ [u^*]_{j-1}} (s,b)\}\Big \}\wedge T$,

• $\beta ^*_j\in \mathop {\arg \max }_{b\in U}\{Y^{v\circ [u^*]_{j-1}\circ (\tau ^*_j,b)}_{\tau ^*_j}-c^{v\circ [u^*]_{j-1}}(\tau ^*_j,b)\}$

and $N^*=\sup \{j:\tau ^*_j<T\}$, with $\tau _0^*:=t$.

Proof

Assume that $(Y^v,Z^v,K^v:v\in {\mathcal {U}}^f)$ is a solution to (1.1) (i.e. $(Y^v:v\in {\mathcal {U}})$ is a consistent family such that $(t,b)\mapsto Y^{v\circ (t,b)}_t$ is continuous and it satisfies equation (1.1)). Using Proposition 2.4 together with consistency k times, gives that

$$\begin{aligned} Y^{v}_t&=Y^{v\circ [u^*]_{N^*\wedge k}}_{\tau ^*_k} +\int _t^{\tau ^*_k} \sum _{j=0}^{k-1}{\mathbbm {1}}_{[\tau ^*_j, \tau ^*_{j+1})}(s)f^{v\circ u^*}(s,Y^{v\circ [u^*]_j}_s, Z^{v\circ [u^*]_j}_s)ds \nonumber \\&\quad - \int _t^{\tau ^*_k}\sum _{j=0}^{k-1} {\mathbbm {1}}_{[\tau ^*_j,\tau ^*_{j+1})}(s)Z^{v\circ [u^*]_j}_s dW_s-\sum _{j=1}^{N^*\wedge k}c^{v\circ [u^*]_{j-1}} (\tau ^*_j,\beta ^*_j). \end{aligned}$$

(3.23)

Now, by the definition of a solution to (1.1) the sequence $Y^{v\circ [u^*]_{N^*\wedge k}}_{\tau ^*_k}$ is uniformly bounded in $L^2(\Omega ,{\mathbb {Q}}^{v,u^*})$ and repeating the argument in the proof of Lemma 3.16 implies that ${\mathbb {P}}[N^*<\infty ]=1$ and thus that $u^*\in {\mathcal {U}}^f$. Letting $k\rightarrow \infty$ in (3.23) then gives

$$\begin{aligned} Y^{v}_t=\xi ^{v\circ u^*}+\int _t^{T} f^{v\circ u^*}(s,U^{v, u^*}_s, V^{v, u^*}_s)ds - \int _t^{T}V^{v, u^*}_sdW_s-\sum _{j=1}^{N^*} c^{v\circ [u^*]_{j-1}}(\tau ^*_j,\beta ^*_j) \end{aligned}$$

and uniqueness follows.

Concerning optimality let $\tilde{u}=(\tilde{\tau }_1,\ldots , \tilde{\tau }_{\tilde{N}};\tilde{\beta }_1,\ldots , \tilde{\beta }_{\tilde{N}})\in {\mathcal {U}}^f$ and note that if $\tilde{N}\ge 1$, then

$$\begin{aligned} Y^{v}_t\ge U^{v\circ (\tilde{\tau }_1, \tilde{\beta }_1),[\tilde{u}]_{2:}}_{\tilde{\tau }_1} +\int _t^{\tilde{\tau }_1}f^{v\circ \tilde{u}} (s,U^{v,\tilde{u}}_s,V^{v,\tilde{u}}_s)ds -\int _t^{\tilde{\tau }_1} V^{v,\tilde{u}}_sdW_s -c^{v}(\tilde{\tau }_1,\tilde{\beta }_1), \end{aligned}$$

where $[u]_{2:}:=(\tilde{\tau }_2,\ldots ,\tilde{\tau }_{\tilde{N}}; \tilde{\beta }_2,\ldots ,\tilde{\beta }_{\tilde{N}})$. Successively repeating this process while considering the fact that $\tilde{u}\in {\mathcal {U}}^f$ eventually leads us to the conclusion that $Y^v_t\ge U^{v,\tilde{u}}_t$.

$\square$

4 Application to SDGs Involving Impulse Control

We now apply the above results to find solutions to stochastic differential games of impulse versus continuous control under weak formulation. In particular, we are interested in finding a saddle point for the game, i.e. a pair $(u^*,\alpha ^*)\in {\mathcal {U}}^f\times {\mathcal {A}}$ such that

$$\begin{aligned} J(u,\alpha ^*)\le J(u^*,\alpha ^*)\le J(u^*,\alpha ) \end{aligned}$$

(4.1)

for all $(u,\alpha )\in {\mathcal {U}}^f\times {\mathcal {A}}$. Since we consider a weak formulation, each pair $(u,\alpha )\in {\mathcal {U}}^f\times {\mathcal {A}}$ gives rise to a probability measure ${\mathbb {Q}}$ and a corresponding Brownian motion $W^{\mathbb {Q}}$ such that (1.3) and (1.4) admits a solution on $(\Omega ,{\mathcal {F}},{\mathbb {F}},{\mathbb {Q}},W^{\mathbb {Q}})$ and $J(u,\alpha )$ is to be interpreted as the corresponding cost functional under the expectation induced by ${\mathbb {Q}}$.

Throughout, we assume the following forms on the drift and volatility terms in the forward SDE (1.3) and (1.4),

$$\begin{aligned} a(t,x,\alpha )=\left[ \begin{array}{c} a_1(t,x)\\ a_2(t,x,\alpha ) \end{array}\right] \quad \mathrm{and}\quad \sigma (t,x) =\left[ \begin{array}{cc} \sigma _{1,1}(t,x)&{} 0\\ \sigma _{2,1}(t,x) &{} \sigma _{2,2}(t,x) \end{array}\right] , \end{aligned}$$

where a is of at most linear growth in the data x and $\sigma$ is uniformly bounded. The drift is split into two terms $a_1:[0,T]\times {\mathbb {D}}\rightarrow {\mathbb {R}}^{d_1}$ (we let ${\mathbb {D}}$ denote the set of all càdlàg functions $x:[0,T]\rightarrow {\mathbb {R}}^d$) and $a_2:[0,T]\times {\mathbb {D}}\times A\rightarrow {\mathbb {R}}^{d_2}$, with $d=d_1+d_2$ the total dimension. The diffusion coefficient has a component $\sigma _{2,2}:[0,T]\times {\mathbb {D}}\rightarrow {\mathbb {R}}^{d_2\times d_2}$ that has an inverse, $\sigma _{2,2}^{-1}$, which is uniformly bounded on $[0,T]\times {\mathbb {D}}$.

For the purpose of solving (4.1) we let $f^u$ be given by

$$\begin{aligned} f^u(t,\omega ,y,z):=\inf _{\alpha \in A}H^u(t,\omega ,z,\alpha ) =:H^{*,u}(t,\omega ,z), \end{aligned}$$

(4.2)

where³

$$\begin{aligned} H^u(t,\omega ,z,\alpha ):=z\breve{a}(t,(X^{u}_s)_{s\le t},\alpha ) +\phi (t,X^{u}_t,\alpha ), \end{aligned}$$

with

$$\begin{aligned} \breve{a}(t,x,\alpha ):=\left[ \begin{array}{c} 0 \\ \sigma _{2,2}^{-1}(t,x)a_2(t,x,\alpha ) \end{array}\right] \end{aligned}$$

and $X^u$ is the unique solution to the impulsively controlled forward SDE

$$\begin{aligned} X^{u}_t&=x_0+\int _0^t\tilde{a}(s,(X^{u}_r)_{ r\le s})dt +\int _0^t\sigma (s,(X^{u}_r)_{ r\le s})dW_s,\quad \mathrm{for} \, t\in [0,\tau _{1}), \end{aligned}$$

(4.3)

$$\begin{aligned} X^{u}_{t}&=\Gamma (\tau _j,X^{[u]_{j-1}}_{\tau _j},\beta _j) +\int _{\tau _j}^t\tilde{a}(s,(X^{u}_r)_{ r\le s})ds\nonumber \\&\quad +\int _0^t\sigma (s,(X^{u}_r)_{ r\le s})dW_s, \quad \mathrm{for}\, t\in [{\tau _j},\tau _{j+1}), \end{aligned}$$

(4.4)

with

$$\begin{aligned} \tilde{a}(t,x):=\left[ \begin{array}{c} a_1(t,x)\\ 0 \end{array}\right] . \end{aligned}$$

Our approach to solving the above optimization problem is to define a measure ${\mathbb {Q}}^{u,\alpha }$ under which $W^{u,\alpha }_t=W_t-\int _0^t\breve{a}(s,(X^{u}_r)_{ r\le s},\alpha ^*_s)ds$ is a Brownian motion, where $\alpha ^*$ is a measurable selection of a minimizer in (4.2). In particular, we note that for any $(u,\alpha )\in {\mathcal {U}}^f\times {\mathcal {A}}$, the 6-tuple $(\Omega ,{\mathcal {F}},{\mathbb {F}},{\mathbb {Q}}^{u,\alpha },X^u,W^{u,\alpha })$ is a weak solution to (1.3) and (1.4) with impulse control u and continuous control $\alpha$.

Before we move on to show optimality of the above scheme, we give assumptions on $a,\sigma$ and $\Gamma$ and $\phi ,\psi$ and $\ell$ under which the sequential system (1.1) with driver given by (4.2) attains a unique solution.

Assumption 4.1

For any $t,t'\ge 0$, $b,b'\in U$, $\xi ,\xi '\in {\mathbb {R}}^d$, $x,x'\in {\mathbb {D}}$ and $\alpha \in A$ and for some $\rho \ge 0$ we have:

• (i) The function $\Gamma :[0,T]\times {\mathbb {D}}\times U\rightarrow {\mathbb {R}}^d$ satisfies the Lipschitz condition

  $$\begin{aligned}&|\Gamma (t,(x_s)_{s\le t},b)-\Gamma (t',(x'_s)_{s\le t'},b')|\\&\le C\left( \int _{0}^{t\wedge t'}|x'_s-x_s|ds+|x'_{t'}-x_t|\right. \\&\quad \left. +(|t'-t|+|b'-b|) \left( 1+\sup _{s\le t}|x_s|+\sup _{s\le t'}|x'_s|\right) \right) \end{aligned}$$

  and the growth condition

  $$\begin{aligned} |\Gamma (t,(x_s)_{s\le t},b)|\le K_\Gamma \vee |x_t|. \end{aligned}$$

  for some constant $K_\Gamma >0$.

• (ii) The coefficients $a:[0,T]\times {\mathbb {D}}\times A\rightarrow {\mathbb {R}}^{d}$ and $\sigma :[0,T]\times {\mathbb {D}}\rightarrow {\mathbb {R}}^{d\times d}$ are continuous in t (and $\alpha$ when applicable) and satisfy the growth conditions

  $$\begin{aligned} |a(t,(x_s)_{s\le t},\alpha )|&\le C(1+\sup _{s\le t}|x_s|), \\ |\sigma (t,(x_s)_{s\le t})|&\le C \end{aligned}$$

  and the Lipschitz continuity

  $$\begin{aligned} |\tilde{a}(t,(x_s)_{s\le t})-\tilde{a}(t,(x'_s)_{s\le t})| +|\sigma (t,(x_s)_{s\le t})-\sigma (t,(x'_s)_{s\le t})|&\le C\sup _{s\le t}|x'_s-x_s|, \\ \int _0^t |\tilde{a}(s,(x_r)_{r\le s})-\tilde{a}(s,(x'_r)_{r\le s})|ds&\le C\int _{0}^t|x'_s-x_s|ds\\ \int _0^t |\sigma (s,(x_r)_{r\le s})-\sigma (s,(x'_r)_{r\le s})|^2ds&\le C\int _{0}^t|x'_s-x_s|^2ds. \end{aligned}$$

  Moreover, for each $(t,x)\in [0,T]\times {\mathbb {D}}$, the matrix $\sigma _{2,2}(t,(x_s)_{s\le t})$ has an inverse, $\sigma _{2,2}^{-1}(t,(x_s)_{s\le t})$, that is uniformly bounded on $[0,T]\times {\mathbb {D}}$ and

  $$\begin{aligned} \int _0^t |\breve{a}(s,(x_r)_{r\le s},\hat{\alpha }(s)) -\breve{a}(s,(x'_r)_{r\le s},\hat{\alpha }(s))|^2 ds&\le C\int _{0}^t|x'_s-x_s|^2ds, \end{aligned}$$

  for all measurable functions $\hat{\alpha }:[0,T]\rightarrow A$.

• (iii) The running reward $\phi :[0,T]\times {\mathbb {R}}^d\times A\rightarrow {\mathbb {R}}$ is ${\mathcal {B}}([0,T]\times {\mathbb {R}}^d\times A)$-measurable, continuous in $\alpha$ and the terminal reward $\psi :{\mathbb {R}}^d\rightarrow {\mathbb {R}}$ is ${\mathcal {B}}({\mathbb {R}}^d)$-measurable. Moreover, we have the growth condition

  $$\begin{aligned} |\phi (t,\xi ,\alpha )|+|\psi (\xi )|\le C^g(1+|\xi |^\rho ) \end{aligned}$$

  for some $C^g>0$ and all $\xi \in {\mathbb {R}}^d$, and there is a nondecreasing function $C^L:{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+$ such that for each $K>0$,

  $$\begin{aligned} |\phi (t,\xi ,\alpha )-\phi (t,\xi ',\alpha )|+|\psi (\xi ) -\psi (\xi ')|\le C^L(K)|\xi -\xi '|, \end{aligned}$$

  whenever $|\xi |\vee |\xi '|\le K$.

• (iv) The intervention cost $\ell :[0,T]\times {\mathbb {R}}^d\times U\rightarrow {\mathbb {R}}_+$ is jointly continuous in $(t,\xi ,b)$, bounded from below, i.e.

  $$\begin{aligned} \ell (t,\xi ,b)\ge \delta >0, \end{aligned}$$

  and locally Lipschitz in $\xi$ and Hölder continuous in t, i.e. there is a nondecreasing function $C^L_\ell :{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+$ such that for each $K>0$,

  $$\begin{aligned} |\ell (t,\xi ,b)-\ell (t',\xi ',b)|\le C^L_\ell (K)|\xi -\xi '|+C^H_\ell |t'-t|^\varsigma , \end{aligned}$$

  whenever $|\xi |\vee |\xi '|\le K$ for some $C^H_\ell ,\varsigma >0$.

Under these assumptions, we note that $f^u$ as defined in (4.2) is stochastic Lipschitz with Lipschitz coefficient

$$\begin{aligned} L^{v}_t:=\sup _{s\in [0,t]}\sup _{\alpha \in A}|\breve{a} (\cdot ,(X^v_s)_{ s\le \cdot },\alpha )|\vee C, \end{aligned}$$

(4.5)

where $C>0$ is chosen to eliminate jumps. Moreover, for some $k_L>0$, we have $L^v_t\le k_L(1+\sup _{s\in [0,t]}|X^v_s|)$.

4.1 Some Preliminary Estimates

We now present some preliminary estimates of moments and stability of solutions to (4.3) and (4.4). Towards the end of the section we will prove that any necessary changes of measure are feasible.

Proposition 4.2

Under Assumption 4.1, the path-dependent SDE (4.3) and (4.4) admits a unique solution for each $u\in {\mathcal {U}}$. Furthermore, the solution has moments of all orders, in particular we have for $p\ge 0$, that

$$\begin{aligned} \sup _{u\in {\mathcal {U}}^f}{\mathbb {E}}\Big [\sup _{t\in [0,T]}|X^{u}_t|^{p}\Big ]\le C, \end{aligned}$$

(4.6)

where $C=C(p)$ and

$$\begin{aligned} \sup _{v\in {\mathcal {U}}^f}{\mathbb {E}}\Big [\sup _{t\in [0,T]} \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t}|{\mathbb {E}}\big [\sup _{s\in [t,T]}|X^{v(t) \circ u}_s|^{\rho }\big |{\mathcal {F}}_t\big ]|^p\Big ]\le C, \end{aligned}$$

(4.7)

where $C=C(\rho ,p)$.

Proof

See Proposition 5.4 in [26]. In particular, both moment estimates follows by noting that

$$\begin{aligned} {\mathbb {E}}\Big [\sup _{s\in [t,T]}|X^{u}_s|^{p}\big |{\mathcal {F}}_t\Big ] \le C\left( 1+ \sup _{s\in [0,t]}|X^{u}_s|^{p}\right) . \end{aligned}$$

(4.8)

$\square$

For any $\kappa \ge 1$ and all $\mathbf {v}:=(t_1,\ldots ,t_\kappa ;b_1,\ldots ,b_\kappa )\in {\mathcal {D}}^\kappa$ and $\mathbf {v}':=(t'_1,\ldots ,t'_\kappa ;b'_1,\ldots ,b'_\kappa )\in {\mathcal {D}}^\kappa$ we define the set $\Upsilon _{\mathbf {v},\mathbf {v}'} :=\cup _{j=1}^\kappa [\underline{t}_j,\bar{t}_j)$, with $\underline{t}_j:=t_j\wedge t'_j$ and $\bar{t}_j:=t_j\vee t'_j$.

Lemma 4.3

For each $k,\kappa \ge 0$ and $p\ge 1$, there is a $C\ge 0$ such that

$$\begin{aligned} {\mathbb {E}}\Big [\sup _{t\in [0,T]}\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^k_t}{\mathbb {E}}\Big [\sup _{s\in [0,T] \setminus \Upsilon _{\mathbf {v},\mathbf {v}'}}|X^{v\circ \mathbf {v}'\circ u}_{s} -X^{v\circ \mathbf {v}\circ u}_{s}|^{2p}\Big |{\mathcal {F}}_t\Big ]\Big ] \le C\Vert \mathbf {v}'-\mathbf {v}\Vert ^p_{{\mathcal {D}}^f}, \end{aligned}$$

for all $v\in {\mathcal {U}}^f$ and all $\mathbf {v},\mathbf {v}'\in {\mathcal {D}}^\kappa$.

Proof

See the proof of Lemma 5.5 in [26]. $\square$

A fundamental assumption in Sect. 3 is the existence of a $q'>1$ and a $C>0$ such that $\sup _{\zeta \in {\mathcal {K}}_0}{\mathbb {E}}^{\mathbb {Q}}\big [|{\mathcal {E}}(\zeta *W^{\mathbb {Q}})_T|^{q'}\big ]\le C$ for all ${\mathbb {Q}}\in {\mathfrak {P}}_0$. In the following two lemmas we show that since $L^v\le k_L(1+\sup _{s\in [0,\cdot ]}|X^v_s|)$, this statement is true.

Lemma 4.4

For $v\in {\mathcal {U}}^f$ and $\varsigma >1$, let $\Upsilon ^{v,\varsigma }$ be the set of all ${\mathcal {P}}_{\mathbb {F}}$-measurable processes $\zeta$ with $|\zeta _t|\le L^v_t$ for all $t\in [0,T]$ (outside of a ${\mathbb {P}}$-null set) such that ${\mathbb {E}}\big [{\mathcal {E}}(r\zeta *W)_T\big ]=1$ for all $r\in [1,\varsigma ]$. Then, there is a $q'>1$ such that $\sup _{u\in {\mathcal {U}}^f}\sup _{\zeta \in \Upsilon ^{u,\varsigma }}{\mathbb {E}}\big [|{\mathcal {E}}(\zeta *W)_T|^{q'}\big ]<\infty$.

Proof

For $q'\in [1,\varsigma ]$ we have

$$\begin{aligned} {\mathbb {E}}\big [|{\mathcal {E}}(\zeta *W)_T|^{q'}\big ]&={\mathbb {E}}\big [{\mathcal {E}}(q'\zeta *W)_T e^{\frac{(q')^2-q'}{2}\int _0^T|\zeta _s|^2ds}\big ]\\&={\mathbb {E}}^{{\mathbb {Q}}^{q'\zeta }}\big [e^{\frac{(q')^2-q'}{2} \int _0^T|\zeta _s|^2ds}\big ]. \end{aligned}$$

Now, since $|\zeta |\le k_L(1+\sup _{s\in [0,\cdot ]}|X^v_s|)$ we have (see Lemma 1 in [2])

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {Q}}^{q'\zeta }}\big [e^{\frac{(q')^2-q'}{2}\int _0^T| \zeta _s|^2ds}\big ]\le {\mathbb {E}}\big [e^{\frac{(q')^2-q'}{2}h(q') (1+\sup _{t\in [0,T]}|W_t|^2)}\big ], \end{aligned}$$

where $h:{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+$ is bounded on compacts and we conclude that the left hand side is finite for $q'>1$ sufficiently small. $\square$

Lemma 4.5

Let $\zeta$ be a ${\mathcal {P}}_{\mathbb {F}}$-measurable process with trajectories in ${\mathbb {D}}$ such that for some $v\in {\mathcal {U}}^f$, we have $|\zeta _t|\le k_L(1+\sup _{s\in [0,t]}|X^v_s|)$ for all $t\in [0,T]$, then ${\mathbb {E}}\big [{\mathcal {E}}(\zeta *W)_T\big ]=1$.

Proof

We will reach the result by adapting the proof of Lemma 7 in [14] to solutions of impulsively controlled path-dependent SDEs (see also Lemma 0 of Sect. 5 in [2]). Since $({\mathcal {E}}(\zeta *W)_t:0\le t\le T)$ is a ${\mathbb {P}}$-a.s. non-negative local martingale it is a supermartingale and we only need to show that ${\mathbb {E}}\big [{\mathcal {E}}(\zeta *W)_T\big ]\ge 1$. For $M\ge 0$ and $t\in [0,T]$, we define the sets

$$\begin{aligned} C_M(t):=\left\{ x\in {\mathbb {D}}: k_L \left( 1+\sup _{s\in [0,t]}|x_s|\right) <M\right\} \end{aligned}$$

Then for each $M\ge 0$, $(C_M(t))_{t\in [0,T]}$ is a non-increasing collection of open subsets of ${\mathbb {D}}$ and:

• (a) If for some $x\in {\mathbb {D}}$ we have $x\in C_M(s)$ and $x\notin C_M(t)$ for some $0\le s<t\le T$, then there is a $t'\in (s,t]$ such that $x\in C_M(r)$ for all $r\in [0,t')$ and $x\notin C_M(t')$.

• (b) For each $\epsilon >0$, there is a $M\ge 0$ such that ${\mathbb {P}}[X^v\in C_M(T)]>1-\epsilon$ for all $v\in {\mathcal {U}}^f$.

Here, the second property follows from Proposition 4.2. Moreover, let

$$\begin{aligned} D_M(t):=\{\omega :X^{v,\zeta }\in C_M(t)\}\in {\mathcal {F}}_t, \end{aligned}$$

where $X^{u,\zeta }$ solves (1.3) and (1.4) with drift $a=\tilde{a}+\sigma \zeta$.

We first restrict our attention to the situation when $v\in {\mathcal {U}}^k$ for some $k\ge 0$, and note that (by arguing as in the proof of Lemma 4.3) we have

$$\begin{aligned} |X^{v,\zeta }_t-X^{v}_t|&\le C\big (\int _{0}^t\big (|\tilde{a} (s,(X^{v,\zeta }_r)_{ r\le s})-\tilde{a}(s,(X^{v}_r)_{ r\le s})|\\&\qquad +|\sigma (s,(X^{v,\zeta }_r)_{ r\le s})\zeta _s| +|X^{v,\zeta }_s -X^{v}_s|\big )ds\\&\quad +\Big |\int _0^{t}(\sigma (s,(X^{v,\zeta }_r)_{ r\le s}) -\sigma (s,(X^{v}_r)_{ r\le s}))dW_s\Big |\big )\\&\le C\big (\int _{0}^t\big (|X^{v,\zeta }_s-X^{v}_s| +|\zeta _s|\big )ds+\Xi ^{v,\zeta }_t\big ), \end{aligned}$$

where

$$\begin{aligned} \Xi ^{v,\zeta }_{t}:=\sup _{t'\in [0,t]} \Big |\int _0^{t'}(\sigma (s,(X^{v,\zeta }_r)_{ r\le s}) -\sigma (s,(X^{v}_r)_{ r\le s}))dW_s\Big |. \end{aligned}$$

Applying Grönwall’s inequality together with the fact that $|\zeta _t|\le k_L(1+\sup _{s\in [0,t]}|X^{v}_s|)$ gives that for any $T'\in [0,T]$, we have

$$\begin{aligned} \sup _{t\in [0,T']}|X^{v,\zeta }_t-X^{v}_t|&\le C\big (1+\int _{0}^{T'} \sup _{r\in [0,s]}|X^{v}_r|ds+\Xi ^{v,\zeta }_{T'}\big ), \end{aligned}$$

Now, for all $\omega \in D_M(T)$ we have $\sup _{t\in [0,T]}|X^{v,\zeta }_t|<M$ and we can apply Grönwall’s inequality once more to obtain

$$\begin{aligned} \sup _{t\in [0,T]}|X^{v}_t|&\le C\big (1+M+\Xi ^{v,\zeta }_T\big ) \end{aligned}$$

Letting $E_M(t):=\{\omega \in D_M(t): \Xi ^{v,\zeta }_t\le M\}\in {\mathcal {F}}_t$ we note that

• (c) For $\omega \in E_M(t)$ we have $\zeta _t\le C(1+M)$, where C does not depend on t or M.

Now, set

$$\begin{aligned} \zeta ^{M}_t:={\mathbbm {1}}_{E_M(t)}\zeta _t \end{aligned}$$

and let $X^{v,\zeta ,M}:=X^{v,\zeta ^{M}}$.

Since $|\zeta ^{M}_t|\le C(1+M)$, the Novikov condition holds for any constant multiple of $\zeta ^{M}$. In particular, we conclude that the ${\mathbb {Q}}_M$ defined by $d{\mathbb {Q}}_M:={\mathcal {E}}(\zeta ^{M}*W)_Td{\mathbb {P}}$ is a probability measure. Moreover, for some $q'>1$ we have

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {Q}}_M}\Big [(\Xi ^{v,\zeta }_T)^2\Big ]&\le C{\mathbb {E}}\big [|{\mathcal {E}}(\zeta ^{M}*W)_T|^{q'}\big ]^{1/q'}\\&\quad {\mathbb {E}}\Big [\big (\int _0^T(|\sigma (s,(X^{v,\zeta }_r)_{ r\le s})|^2 +|\sigma (s,(X^{v}_r)_{ r\le s})|^2)ds\big )^{q}\Big ]^{1/q}\\&\le C \end{aligned}$$

by Assumption 4.1.ii, where $\frac{1}{q'}+\frac{1}{q}=1$ and, by Lemma 4.4, $C>0$ can be chosen independently from M. This gives that

• (d) For each $\epsilon >0$, there is a $M\ge 0$ such that ${\mathbb {Q}}_{M'}(\{\omega :\Xi ^{v,\zeta }_T\le M'\})>1-\epsilon$ for all $M'\ge M$.

Making use of b) and d) we find that for each $\epsilon >0$, there is a $M\ge 0$ such that

$$\begin{aligned} 1-\epsilon <{\mathbb {P}}(\{\omega :X^v\in C_M(T)\}) ={\mathbb {Q}}_M(\{\omega :X^{v,\zeta ,M} \in C_M(T)\}) \end{aligned}$$

and

$$\begin{aligned} {\mathbb {Q}}_M( \{\omega :\Xi ^{v,\zeta }_T> M\})<\epsilon . \end{aligned}$$

Combining these gives that

$$\begin{aligned} 1-\epsilon&<{\mathbb {Q}}_M(\{\omega :X^{v,\zeta ,M}\in C_M(T)\} \cap \{\omega :\Xi ^{v,\zeta }_T\le M\})+{\mathbb {Q}}_M(\{\omega : \Xi ^{v,\zeta }_T> M\}) \\&\le {\mathbb {Q}}_M(\{\omega :X^{v,\zeta ,M}\in C_M(T)\} \cap \{\omega :\Xi ^{v,\zeta }_T\le M\})+\epsilon \end{aligned}$$

Moreover, by property (a) above and right-continuity we have that

$$\begin{aligned} \{\omega : X^{v,\zeta ,M}\in C_M(t)\}\cap \{\omega :\Xi ^{v,\zeta }_t\le M\}&=\{\omega : X^{v,\zeta }\in C_M(t)\}\cap \{\omega :\Xi ^{v,\zeta }_t\le M\} \\&=E_M(t) \end{aligned}$$

so that

$$\begin{aligned} 1-2\epsilon <{\mathbb {Q}}_M\big (E_M(T))={\mathbb {E}}\big [{\mathcal {E}}(\zeta ^{M}*W)_T {\mathbbm {1}}_{E_M(T)} \big ]\le {\mathbb {E}}\big [{\mathcal {E}}(\zeta *W)_T\big ]. \end{aligned}$$

Since $\epsilon >0$ was arbitrary, this proves the assertion whenever $v\in {\mathcal {U}}^k$ for some finite $k\ge 0$. To get the result for arbitrary $v:=(\tau _1,\ldots ,\tau _N;\beta _1,\ldots ,\beta _N) \in {\mathcal {U}}^f$, we define the sets

$$\begin{aligned} F_k(t):=\left\{ \omega :|\zeta _t|\le k_L \left( 1+\sup _{s\in [0,t]}|X^{[v]_k}_s|\right) \right\} \supset \{\omega : N\le k\} \end{aligned}$$

for all $k\ge 0$ and let

$$\begin{aligned} \zeta ^{[v]_k}_t={\mathbbm {1}}_{F_k(t)}\zeta _t. \end{aligned}$$

Now, for any $v\in {\mathcal {U}}^f$ we have by definition that ${\mathbb {P}}(\{\omega :N>k,\,\forall k\ge 0\})=0$ and so we can by again appealing to Lemma 4.4 find a $k\ge 0$ such that

$$\begin{aligned} {\mathbb {Q}}^{[v]_k}(N> k):&={\mathbb {E}}\big [{\mathcal {E}}(\zeta ^{[v]_k}*W)_T {\mathbbm {1}}_{[N> k]}\big ] \\&\le {\mathbb {E}}\big [|{\mathcal {E}}(\zeta ^{[v]_k}*W)_T|^{q'}\big ]^{1/q'}({\mathbb {P}}(N> k))^{1/q} <\epsilon \end{aligned}$$

implying that

$$\begin{aligned} 1-\epsilon <{\mathbb {Q}}^{[v]_k}(N \le k)&={\mathbb {E}}\big [{\mathcal {E}}(\zeta ^{[v]_k}*W)_T{\mathbbm {1}}_{[N\le k]}\big ]\\&={\mathbb {E}}\big [{\mathcal {E}}(\zeta ^{v}*W)_T{\mathbbm {1}}_{[N\le k]}\big ] \le {\mathbb {E}}\big [{\mathcal {E}}(\zeta ^{v}*W)_T\big ] \end{aligned}$$

and the assertion follows as $\epsilon >0$ was arbitrary. $\square$

Corollary 4.6

There is a $q'>1$ such that $\sup _{\zeta \in {\mathcal {K}}_0}{\mathbb {E}}\big [|{\mathcal {E}}(\zeta *W)_T|^{q'}\big ]<\infty$.

Proof

Lemma 4.5 shows that for each $\varsigma >1$ the $\Upsilon ^{v,\varsigma }$ in Lemma 4.4 is in fact all ${\mathcal {P}}_{\mathbb {F}}$-measurable processes $\zeta$ with $|\zeta _t|\le k_L(1+\sup _{s\in [0,t]}|X^v_s|)$ for all $t\in [0,T]$ (outside of a ${\mathbb {P}}$-null set). $\square$

The above corollary gives the following:

Proposition 4.7

Under Assumption 4.1, the path-dependent SDE (1.3) and (1.4) admits a weak solution for each $(u,\alpha )\in {\mathcal {U}}\times {\mathcal {A}}$. Furthermore, the solution has moments of all orders, in particular we have for $p\ge 0$, that

$$\begin{aligned} \sup _{u\in {\mathcal {U}}^f,{\mathbb {Q}}\in {\mathfrak {P}}^u}{\mathbb {E}}^{{\mathbb {Q}}}\Big [\sup _{t\in [0,T]}| X^{u}_t|^{p}\Big ]\le C, \end{aligned}$$

(4.9)

where $C=C(p)$ and

$$\begin{aligned} \sup _{v\in {\mathcal {U}}^f}{\mathbb {E}}\Big [\sup _{t\in [0,T]} \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t,{\mathbb {Q}}\in {\mathfrak {P}}^{v(t)\circ u}}| {\mathbb {E}}^{{\mathbb {Q}}}\big [\sup _{s\in [t,T]}|X^{v(t)\circ u}_s|^{\rho } \big |{\mathcal {F}}_t\big ]|^p\Big ]\le C, \end{aligned}$$

(4.10)

where $C=C(\rho ,p)$.

Moreover, there is a $q'>1$ and a $C>0$ such that for each $v\in {\mathcal {U}}^f$ and all $\zeta \in {\mathcal {K}}^v$ and ${\mathbb {Q}}\in {\mathfrak {P}}^v$ we have ${\mathbb {E}}^{{\mathbb {Q}}}[|{\mathcal {E}}(\zeta *W^{{\mathbb {Q}}})_T|^{q'}]\le C$ (where $W^{\mathbb {Q}}$ is a Brownian motion under ${\mathbb {Q}}$).

Proof

Existence of a weak solution to (1.3) and (1.4) follows by taking $\zeta _t=\breve{a}(t,(X^u_s)_{ s\le t},\alpha _t)$ and using Lemma 4.5 to conclude that under ${\mathbb {Q}}^{u,\alpha }$, the process $W^{u,\alpha }:=W-\int _0^\cdot \breve{a}(t,(X_s)_{ s\le t},\alpha _t)dt$ is a Brownian motion.

The moment estimates (4.9) and (4.10) now follow by repeating the steps in the proof of Proposition 5.4 in [26] and the last assertion follows by repeating the steps in Lemmas 4.4 and 4.5 while referring to the bound (4.9) rather than (4.6). $\square$

4.2 The Corresponding Sequential System of Reflected BSDEs

In the present section we show that there is a unique family $(Y^{v},Z^{v},K^{v})$ that solves the sequential system of reflected BSDEs

$$\begin{aligned} {\left\{ \begin{array}{ll} Y^{v}_t=\psi (X^v_T)+\int _t^T H^{*,v}(s,Z^{v}_s)ds -\int _t^T Z^{v}_sdW_s+ K^{v}_T-K^{v}_t,\quad \forall t\in [0,T], \\ Y^{v}_t\ge \sup _{b\in U}\{Y^{v\circ (t,b)}_t-\ell ^v (t,X^{v}_t,b))\},\quad \forall t\in [0,T],\\ \int _0^T(Y^{v}_t-\sup _{b\in U}\{Y^{v\circ (t,b)}_t -\ell ^v(t,X^{v}_t,b))\})dK^{v}_t=0, \end{array}\right. } \end{aligned}$$

(4.11)

where $\ell ^u(t,X^{u}_t,b)):=\infty {\mathbbm {1}}_{[0,\tau _N)}(t) +\ell (t,X^{u}_t,b))$ making (4.11) a non-reflected BSDE on $[0,\pi (u))$. In the remainder of the article we will drop the superscript v in $\ell ^v$ but remind ourselves that no reflection can occur before the time of the last intervention in v. Then, we will leverage the result in Theorem 3.20 to find a weak solution to the SDG in finite horizon.

Letting

$$\begin{aligned} \bar{K}^{v,p}&:=2^{p-1}\big (((1+T)(C^g)^p+(k_L)^p)\\&\quad \left( 1+\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t}{\mathbb {E}}\Big [\sup _{s\in [0,T]}| X^{v(t)\circ u}_s|^{\rho p}\Big |{\mathcal {F}}_t\Big ]\right) :0\le t\le T\big ), \end{aligned}$$

we note that

$$\begin{aligned} \mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}_f}{\mathbb {E}}\Big [|L^{v\circ u}_T|^p +|\psi (X^{v\circ u}_T)|^p+\int _t^T|H^{*,v\circ u}(s,0)|^p ds\Big |{\mathcal {F}}_t\Big ]\le \bar{K}^{v,p}_t. \end{aligned}$$

Moreover, by Proposition 4.2 we have that $\sup _{v\in {\mathcal {U}}^f}\Vert \bar{K}^{v,p}\Vert _{{\mathcal {S}}^2}<\infty$, for all $p\ge 1$ and by (4.8) it follows that $(\bar{K}^{v,p}:v\in {\mathcal {U}}^f,p\ge 1)$ satisfies the relation in (3.3).

On the other hand $H^{*,v'}(t,z')-H^{*,v}(t,z)$ contains the term $\phi (t,X^{v'}_t)-\phi (t,X^{v}_t)$ and so $H^{*,v}$ generally fails to satisfy the conditions in Assumption 3.3 since $\phi$ is only locally Lipschitz in x. The same thing applies to $\psi$ and $\ell$ and we will rely on a localization argument leading us to introduce

$$\begin{aligned} H^{*,u,m,n}(t,\omega ,z):=\inf _{\alpha \in A}H^{u,m,n}(t,\omega ,z,\alpha ), \end{aligned}$$

where

$$\begin{aligned} H^{u,m,n}(t,\omega ,z,\alpha ):=z\breve{a} (t,(X^{u}_s)_{ s\le t},\alpha )+\phi ^{m,n}(t,X^{u}_t,\alpha ), \end{aligned}$$

with $\phi ^{m,n}:[0,T]\times {\mathbb {R}}^d\times A\rightarrow {\mathbb {R}}$ given by $\phi ^{m,n}(t,x,\alpha ):=\phi ^{+,m}(t,x,\alpha )-\phi ^{-,n}(t,x,\alpha )$, where $(\phi ^{+,m})_{m\ge 0}$ and $(\phi ^{-,n})_{n\ge 0}$ are both non-decreasing sequences of Borel-measurable, non-negative functions that are Lipschitz continuous in x and continuous in $\alpha$ such that $\phi ^{+,m}=\phi ^+(x)$ on $|x|\le m$ and $\phi ^{-,n}(x)= \phi ^-(x)$ on $|x|\le n$.

Similarly, for $m,n\ge 0$, we let $\psi ^{m,n}:{\mathbb {R}}^d\rightarrow {\mathbb {R}}$ be given by $\psi ^{m,n}(x):=\psi ^{+,m}(x)-\psi ^{-,n}(x)$, where $(\psi ^{+,m})_{m\ge 0}$ and $(\psi ^{-,n})_{n\ge 0}$ are both non-decreasing sequences of non-negative, Lipschitz continuous functions such that $\psi ^{+,m}=\psi ^+(x)$ on $|x|\le m$ and $\psi ^{-,n}(x)= \psi ^-(x)$ on $|x|\le n$ and let $\ell ^{n}:[0,T]\times {\mathbb {R}}^d\times U\rightarrow [\delta ,\infty )$ be a non-decreasing sequence of jointly continuous functions that are Lipschitz continuous in x and Hölder continuous in t (uniformly in the other variables) and satisfy $\ell ^{n}(t,x,b)=\ell (t,x,b)$ on $|x|\le n$ and $\ell ^{n}(t,x,b)\ge \delta$ for all $(t,x,b) \in [0,T]\times {\mathbb {R}}^d\times U$.

We now consider the following localized form of (4.11)

$$\begin{aligned} {\left\{ \begin{array}{ll} Y^{v,m,n}_t=\psi ^{m,n}(X^v_T)+\int _t^T H^{*,v,m,n}(s,Z^{v,m,n}_s)ds\\ \qquad \qquad \qquad -\int _t^T Z^{v,m,n}_sdW_s+ K^{v,m,n}_T-K^{v,m,n}_t, \quad \forall t\in [0,T], \\ Y^{v,m,n}_t\ge \sup _{b\in U}\{Y^{v\circ (t,b),m,n}_t-\ell ^{n} (t,X^{v}_{t},b))\},\quad \forall t\in [0,T],\\ \int _0^T(Y^{v,m,n}_t-\sup _{b\in U}\{Y^{v\circ (t,b),m,n}_t -\ell ^{n}(t,X^{v}_{t},b))\})dK^{v,m,n}_t=0. \end{array}\right. } \end{aligned}$$

(4.12)

Since

$$\begin{aligned} |H^{*,v',m,n}(t,z')-H^{*,v,m,n}(t,z)|&\le C((1+|L^{v'}_t| \vee |L^{v}_t|)|z'-z| \\&\quad +(|z|+|z'|)\sup _{\alpha \in A}|\breve{a}(t,(X^{v'}_s)_{s\le t}, \alpha )\\&\quad -\breve{a}(t,(X^{v}_s)_{ s\le t},\alpha )|)\\&\quad +|\phi ^{m,n}(t,X^{v'}_t)-\phi ^{m,n}(t,X^{v}_t)|, \end{aligned}$$

Lemma 4.3 and Assumption 4.1 implies the existence of a family $(\Lambda ^{\mathbf {v},\mathbf {v}',u}:(\mathbf {v},\mathbf {v}')\in \cup _{\kappa \ge 1}{\mathcal {D}}^\kappa \times {\mathcal {D}}^\kappa ,\,v\in {\mathcal {U}}^f)$ and a family $(\bar{K}^{\mathbf {v},\mathbf {v}',k,p}:(\mathbf {v},\mathbf {v}')\in \cup _{\kappa \ge 1}{\mathcal {D}}^\kappa \times {\mathcal {D}}^\kappa ,k,p\ge 0)$ satisfying the conditions in Definition 3.2 and Assumption 3.3 and it follows by Proposition 3.19 and Theorem 3.20 that there is a unique family $(Y^{v,m,n},Z^{v,m,n},K^{v,m,n})$ that solves the sequential system of reflected BSDEs in (4.12).

We have,

Lemma 4.8

For $v,u\in {\mathcal {U}}^f$, let $(U^{v,u,m,n},V^{v,u,m,n})\in {\mathcal {S}}^2_l\times {\mathcal {H}}^2$ solve

$$\begin{aligned} U_t^{v,u,m,n}&=\psi ^{m,n}(X^{v\circ u}_T)+\int _t^{T} H^{*,v\circ u,m,n}(s,V^{v,u,m,n}_s)ds -\int _t^{T} V^{v,u,m,n}_sdW_s\nonumber \\&\quad -\sum _{j=1}^N {\mathbbm {1}}_{[t\le \tau _j]}\ell ^{n} (\tau _j,X^{v\circ [u]_{j-1}}_{\tau _j},\beta _j), \end{aligned}$$

(4.13)

whenever it has a unique solution and set $U^{v,u,m,n}\equiv -\infty$, otherwise. Then, there is a $C>0$ (that does not depend on m, n) such that, whenever $u^*\in {\mathcal {U}}^f$ is such that $\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f}U_t^{v,u,m,n}=U_t^{v,u^*,m,n}$, we have

$$\begin{aligned}&|U^{v,u^*,m,n}_t|^2+{\mathbb {E}}^{{\mathbb {Q}}}\Big [\int _t^T |V^{v,u^*,m,n}_s|^2ds +(N^*)^2\Big |{\mathcal {F}}_t\Big ] \nonumber \\&\quad \le C(1+\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t,\zeta \in {\mathcal {K}}^{v\circ u}}{\mathbb {E}}^{{\mathbb {Q}}^\zeta }\Big [\sup _{s\in [t,T]}| X^{v\circ u}_s|^{2\rho }\Big |{\mathcal {F}}_t\Big ]). \end{aligned}$$

(4.14)

for all ${\mathbb {Q}}\in {\mathfrak {P}}^{v\circ u^*}$.

Proof

First note that whenever $u^*\in {\mathcal {U}}^f$ is a maximizer then (4.13) admits a unique solution. The bounds on $|U^{v,u^*,m,n}_t|^2$ and ${\mathbb {E}}^{\mathbb {Q}}\big [\int _t^T |V^{v,u^*,m,n}_s|^2ds\big |{\mathcal {F}}_t\big ]$ now follow by repeating the argument in the proof of Lemma 3.14 while noting that

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {Q}}}\Big [|\psi ^{m,n}(X^{v\circ u}_T)|^p&+\int _t^{T}| H^{*,v\circ u,m,n}(s,0)|^pds\Big |{\mathcal {F}}_t\Big ] \\&\le C\big (1+{\mathbb {E}}^{\mathbb {Q}}\Big [\sup _{s\in [0,T]}| X^{v\circ u}_s|^{p \rho }\Big |{\mathcal {F}}_t\Big ]\big ). \end{aligned}$$

From this, the bound on $N^*$ is immediate from (3.20). $\square$

The statement of Lemma 4.8 holds for all ${\mathbb {Q}}\in {\mathfrak {P}}^{v\circ u^*}$. Here it is notable that, since for any $m,n,m',n>0$ the drivers $H^{*,m,n}$ and $H^{*,m',n'}$ have the same stochastic Lipschitz coefficients, the set ${\mathfrak {P}}^{v\circ u^*}$ is not parameterized by m, n. This is a key property when deriving the following stability result:

Lemma 4.9

For each $m\ge 0$ and $p\ge 1$ we have

$$\begin{aligned} \Vert \sup _{b\in U}|Y^{v\circ (\cdot ,b),m,n'}_\cdot -Y^{v\circ (\cdot ,b),m,n}_\cdot |\Vert _{{\mathcal {S}}^p}\rightarrow 0 \end{aligned}$$

(4.15)

as $n,n'\rightarrow \infty$.

Proof

We have,

$$\begin{aligned} |Y^{v\circ (t,b),m,n'}_t-Y^{v\circ (t,b),m,n}_t|&\le |U_t^{v\circ (t,b),u^*,m,n}-U_t^{v\circ (t,b), u^*,m,n'}|\\&\quad +|U_t^{v\circ (t,b),u',m,n}-U_t^{v\circ (t,b),u',m,n'}|, \end{aligned}$$

where $u^*$ and $u'$ are elements of ${\mathcal {U}}^f_t$ such that $\sup _{u\in {\mathcal {U}}^f}U_t^{v\circ (t,b),u,m,n}=U_t^{v\circ (t,b),u^*,m,n}$ and $\sup _{u\in {\mathcal {U}}^f}U_t^{v\circ (t,b),u,m,n'}=U_t^{v\circ (t,b),u',m,n'}$. We now consider the first term and suppress the references to the control strategies (i.e. $v\circ (t,b)$ and $u^*$) in the superscript, we have

$$\begin{aligned} U_t^{m,n}-U_t^{m,n'}&=(\psi ^{m,n}(X_T)-\psi ^{m,n'}(X_T)) \\&\quad +\int _t^T (H^{*,m,n}(s,V^{m,n}_s)-H^{*,m,n'}(s,V^{m,n'}_s)ds\\&\quad -\int _t^{T} (V^{m,n}_s-V^{m,n'}_s)dW_s\\&\quad -\sum _{j=1}^{N^*} (\ell ^{n}(\tau _j,X_{\tau _j},\beta _j)-\ell ^{n'}(\tau _j, X_{\tau _j},\beta _j)). \end{aligned}$$

Taking the conditional expectation under the measure ${\mathbb {Q}}^{n,n'}$ where $W_t-\int _0^t\zeta ^{n,n'}_sds$ is a martingale, with

$$\begin{aligned} \zeta ^{n,n'}_s=\frac{H^{*,m,n'}(s,V^{m,n}_s)-H^{*,m,n'} (s,V^{m,n'}_s)}{|V^{m,n}_s -V^{m,n'}_s|^2} (V^{m,n}_s -V^{m,n'}_s)^\top {\mathbbm {1}}_{[V^{m,n}_s\ne V^{m,n'}_s]}, \end{aligned}$$

gives

$$\begin{aligned}&|U_t^{m,n}-U_t^{m,n'}|\\&\quad \le C{\mathbb {E}}^{{\mathbb {Q}}^{n,n'}} \Big [|\psi ^{m,n}(X_T)-\psi ^{m,n'}(X_T)|+\int _t^T | H^{*,m,n}(s,V^{m,n}_s)\\&\qquad -H^{*,m,n'}(s,V^{m,n}_s)|ds+\sum _{j=1}^{N^*} |\ell ^{n}(\tau _j,X_{\tau _j},\beta _j) \\&\qquad -\ell ^{n'}(\tau _j,X_{\tau _j},\beta _j)|\Big |{\mathcal {F}}_t\Big ]\\&\quad \le C{\mathbb {E}}^{{\mathbb {Q}}^{n,n'}}\Big [{\mathbbm {1}}_{[|X_{T}|>\underline{n}]}(1+|X_T|^\rho )+\int _t^T {\mathbbm {1}}_{[|X_{s}|>\underline{n}]}(1+|X_s|^\rho )ds\\&\qquad +\sum _{j=1}^{N^*} (1+|X_{\tau _j}|^\rho ) {\mathbbm {1}}_{[|X_{\tau _j}|>\underline{n}]}\Big |{\mathcal {F}}_t\Big ], \end{aligned}$$

where $\underline{n}:=n\wedge n'$. As $H^{*,m,n}$ and $H^{*,m,n'}$ have the same z-coefficient and thus also the same stochastic Lipschitz coefficient we find that ${\mathbb {Q}}^{n,n'}\in {\mathfrak {P}}^{v\circ u^*}$. In particular, we have

$$\begin{aligned} |U_t^{m,n}-U_t^{m,n'}|&\le C{\mathbb {E}}^{{\mathbb {Q}}^{n,n'}} \Big [{\mathbbm {1}}_{[\sup _{s\in [t,T]}|X_{s}|>\underline{n}]} (1+\sup _{s\in [t,T]}|X_{s}|^\rho )(1+N^*)\Big |{\mathcal {F}}_t\Big ]\\&\le C{\mathbb {E}}^{{\mathbb {Q}}^{n,n'}}\Big [{\mathbbm {1}}_{[\sup _{s\in [t,T]}|X_{s}|>\underline{n}]}(1+\sup _{s\in [t,T]}|X_{s}|^{2\rho })\Big |{\mathcal {F}}_t\Big ]^{1/2}\\&\quad {\mathbb {E}}^{{\mathbb {Q}}^{n,n'}}\Big [1+(N^*)^2\Big |{\mathcal {F}}_t\Big ]^{1/2}\\&\le C\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t,{\mathbb {Q}}\in {\mathfrak {P}}^{v\circ u}} {\mathbb {E}}^{\mathbb {Q}}\Big [{\mathbbm {1}}_{[\sup _{s\in [t,T]}|X^{v\circ u}_{s}| >\underline{n}]}\Big |{\mathcal {F}}_t\Big ]^{1/4}\\&\quad \left( 1+\mathop {\mathrm{ess}\,\sup }_{u\in {\mathcal {U}}^f_t, {\mathbb {Q}}\in {\mathfrak {P}}^{v\circ u}}{\mathbb {E}}^{\mathbb {Q}}\Big [\sup _{s\in [t,T]}| X^{v\circ u}_{s}|^{4\rho })\Big |{\mathcal {F}}_t\Big ]\right) . \end{aligned}$$

The result now follows by Proposition 4.7. $\square$

Now, as clearly $\Vert \sup _{b\in U}|\ell ^{n'}(\cdot ,X^v_\cdot , b)-\ell ^{n}(\cdot ,X^v_\cdot ,b)|\Vert _{{\mathcal {S}}^p}\rightarrow 0$ as $n,n'\rightarrow \infty$ for all $p\ge 1$, Lemma 4.9 and Proposition 2.4 implies that

$$\begin{aligned} \lim _{n,n'\rightarrow \infty }\{\Vert Y^{v,m,n'}-Y^{v,m,n} \Vert _{{\mathcal {S}}^2}+\Vert Z^{v,m,n'}-Z^{v,m,n}\Vert _{{\mathcal {H}}^2} +\Vert K^{v,m,n'}-K^{v,m,n}\Vert _{{\mathcal {S}}^2}\}=0. \end{aligned}$$

For each $m\ge 0$ we note that $(Y^{v,m,n})_{n\ge 0}$ is a non-increasing sequence of continuous processes that is ${\mathbb {P}}$-a.s. bounded and we have that $Y^{v,m,n}$ converges pointwisely to a progressively measurable process $Y^{v,m}:=\lim _{n\rightarrow \infty }Y^{v,m,n}$. Furthermore, by Lemma 4.9 we find that $Y^{v,m}$ is continuous and thus belongs to ${\mathcal {S}}^2$.

Proposition 4.10

For each $m\ge 0$, there is a family of pairs $(Z^{v,m},K^{v,m}:v\in {\mathcal {U}}^f)$ such that $(Y^{v,m},Z^{v,m},K^{v,m}:v\in {\mathcal {U}}^f)$ is the unique solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} Y^{v,m}_t=\psi ^{m}(X^v_T)+\int _t^T H^{*,v,m}(s,Z^{v,m}_s)ds -\int _t^T Z^{v,m}_sdW_s+ K^{v,m}_T-K^{v,m}_t,\quad \forall t\in [0,T], \\ Y^{v,m}_t\ge \sup _{b\in U}\{Y^{v\circ (t,b),m}_t-\ell (t,X^{v}_{t},b))\}, \quad \forall t\in [0,T],\\ \int _0^T(Y^{v,m}_t-\sup _{b\in U}\{Y^{v\circ (t,b),m}_t -\ell (t,X^{v}_{t},b))\})dK^{v,m}_t=0, \end{array}\right. } \end{aligned}$$

(4.16)

where $H^{*,u,m}(t,\omega ,z):=\inf _{\alpha \in A} H^{u,m}(t,\omega ,z,\alpha )$, with

$$\begin{aligned} H^{u,m}(t,\omega ,z,\alpha ):=z\breve{a}(t,(X^{u}_s)_{s\le t}, \alpha )+\phi ^{+,m}(t,X^{u}_t,\alpha )-\phi ^{-}(t,X^{u}_t,\alpha ), \end{aligned}$$

and $\psi ^{m}(x):=\psi ^{+,m}(x)-\psi ^{-}(x)$.

Proof

Let $\eta _k:=\inf \{s\ge 0:|X^{v}_s|\ge k\}\wedge T$. Then, for all $n\ge k$ it follows that $(Y^{v,m,n},Z^{v,m,n},K^{v,m,n})$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} Y^{v,m,n}_t=Y^{v,m,n}_{\eta _k}+\int _t^{\eta _k} H^{*,v,m} (s,Z^{v,m,n}_s)ds-\int _t^{\eta _k} Z^{v,m,n}_sdW_s+ K^{v,m,n}_{\eta _k}\\ \qquad \qquad \qquad -K^{v,m,n}_t,\quad \forall t\in [0,{\eta _k}], \\ Y^{v,m,n}_t\ge \sup _{b\in U}\{Y^{v\circ (t,b),m,n}_t -\ell (t,X^{v}_{t},b))\},\quad \forall t\in [0,{\eta _k}],\\ \int _0^{\eta _k}(Y^{v,m,n}_t-\sup _{b\in U}\{Y^{v\circ (t,b),m,n}_t -\ell (t,X^{v}_{t},b))\})dK^{v,m,n}_t=0. \end{array}\right. } \end{aligned}$$

Moreover, by Proposition 3.19 and Theorem 3.20 there is a unique family of triples $(\hat{Y}^{v,m},\hat{Z}^{v,m},\hat{K}^{v,m})$ that solves

$$\begin{aligned} {\left\{ \begin{array}{ll} \hat{Y}^{v,m}_t=Y^{v,m}_{\eta _k}+\int _t^{\eta _k} H^{*,v,m} (s,\hat{Z}^{v,m}_s)ds-\int _t^{\eta _k} \hat{Z}^{v,m}_sdW_s +\hat{K}^{v,m}_{\eta _k}-\hat{K}^{v,m}_t,\quad \forall t\in [0,{\eta _k}], \\ \hat{Y}^{v,m}_t\ge \sup _{b\in U}\{\hat{Y}^{v\circ (t,b),m}_t -\ell (t,X^{v}_{t},b))\},\quad \forall t\in [0,{\eta _k}],\\ \int _0^{\eta _k}(\hat{Y}^{v,m}_t-\sup _{b\in U}\{\hat{Y}^{v\circ (t,b),m}_t-\ell (t,X^{v}_{t},b))\})d\hat{K}^{v,m}_t=0. \end{array}\right. } \end{aligned}$$

Letting, $n\rightarrow \infty$ we have by Lemma 4.9 and Proposition 2.4 that

$$\begin{aligned}&\Vert (\hat{Y}^{v,m} - Y^{v,m,n}){\mathbbm {1}}_{[0,\eta _k]} \Vert _{{\mathcal {S}}^2} \\&\quad + \Vert (\hat{Z}^{v,m} - Z^{v,m,n}) {\mathbbm {1}}_{[0,\eta _k]}\Vert _{{\mathcal {H}}^2} + \Vert (\hat{K}^{v,m} -K^{v,m,n}){\mathbbm {1}}_{[0,\eta _k]}\Vert _{{\mathcal {S}}^2} \rightarrow 0 \end{aligned}$$

and we find that there is a family of pairs $(Z^{v,m},K^{v,m}:v\in {\mathcal {U}}^f)$ such that for each $k\ge 0$,

$$\begin{aligned} {\left\{ \begin{array}{ll} Y^{v,m}_t=Y^{v,m}_{\eta _k}+\int _t^{\eta _k} H^{*,v,m}(s,Z^{v,m}_s)ds -\int _t^{\eta _k} Z^{v,m}_sdW_s+ K^{v,m}_{\eta _k}-K^{v,m}_t, \quad \forall t\in [0,{\eta _k}], \\ Y^{v,m}_t\ge \sup _{b\in U}\{Y^{v\circ (t,b),m}_t-\ell (t,X^{v}_{t},b))\},\quad \forall t\in [0,{\eta _k}],\\ \int _0^{\eta _k}(Y^{v,m}_t-\sup _{b\in U}\{Y^{v\circ (t,b),m}_t -\ell (t,X^{v}_{t},b))\})dK^{v,m}_t=0. \end{array}\right. } \end{aligned}$$

Now, since there is a ${\mathbb {P}}$-a.s. finite $k_0(\omega )$ such that $\eta _k=T$ for all $k\ge k_0$, existence of a solution to (4.16) follows.

Uniqueness is established by repeating steps in the proof of Theorem 3.20. $\square$

By an identical argument to that used in Lemma 4.9 we find that

$$\begin{aligned} \lim _{m,m'\rightarrow \infty }\Vert \sup _{b\in U}|Y^{v\circ (\cdot ,b),m'}_\cdot -Y^{v\circ (\cdot ,b),m}_\cdot |\Vert _{{\mathcal {S}}^p}= 0 \end{aligned}$$

for all $p\ge 1$ and we conclude that:

Proposition 4.11

The sequential system of reflected BSDEs (4.11) has a unique solution.

Proof

The result follows by letting $m\rightarrow \infty$ and using an identical argument to that of Proposition 4.10. $\square$

Remark 4.12

We note that the verification result of Theorem 3.20 holds in the present setting as well.

4.3 A Solution to the Stochastic Differential Game

We are now ready to solve the SDG by relating optimal controls to solutions of the sequential system of reflected BSDEs (4.11). However, before proceeding we need to narrow down the set of admissible impulse controls that we search over in order to guarantee that (4.13) admits a unique solution.

Definition 4.13

We let ${\mathcal {U}}^m$ be the set subset of ${\mathcal {U}}^f$ with all $u=(\tau _1,\ldots ,\tau _N;\beta _1,\ldots ,\beta _N)$ such that N has moments of all orders, i.e. for each $k\ge 0$ we have ${\mathbb {E}}\big [(N)^k\big ]<\infty$.

Lemma 4.14

For each $u\in {\mathcal {U}}^f$, there is a $\hat{u}\in {\mathcal {U}}^m$ such that

$$\begin{aligned} J(u,\alpha )\le J(\hat{u},\alpha ) \end{aligned}$$

for all $\alpha \in {\mathcal {A}}$.

Proof

For $(u,\alpha )\in {\mathcal {U}}^f\times {\mathcal {A}}$ we let $(B^{u,\alpha },E^{u,\alpha })$ solve

$$\begin{aligned} B^{u,\alpha }_t&=\psi (X^{u,\alpha }_T) +\int _t^T\phi (s,X^{u,\alpha }_s,\alpha _s)ds-\int _t^TE^{u,\alpha }_s dW_s \nonumber \\&- \sum _{j=1}^n{\mathbbm {1}}_{[t\le \tau _j]}\ell (\tau _j, X^{[u]_{j-1},\alpha }_{\tau _j},\beta _j), \end{aligned}$$

(4.17)

whenever a solution exists in ${\mathcal {S}}^2_l\times {\mathcal {H}}^2$. We define the set of sensible impulse controls, ${\mathcal {U}}^s$, as the subset of $u\in {\mathcal {U}}^f$ such that for each $t\in [0,T]$ and $\alpha \in {\mathcal {A}}$,

$$\begin{aligned}&{\mathbb {E}}\Big [\psi (X^{u,\alpha }_T)+\int _t^T\phi (s,X^{u,\alpha }_s,\alpha _s)ds - \sum _{t\le \tau _j}\ell (\tau _j,X^{[u]_{j-1},\alpha }_{\tau _j},\beta _j) \Big |{\mathcal {F}}_t\Big ]\nonumber \\&\quad \ge -C^g(1+T)(1+\mathop {\mathrm{ess}\,\sup }_{u'\in {\mathcal {U}}^f_t,\alpha '\in {\mathcal {A}}_t}{\mathbb {E}}\Big [\sup _{s\in [t,T]}|X^{u(t-)\circ u',\alpha (t-)\circ \alpha '}_s|^\rho \Big |{\mathcal {F}}_t\Big ]), \end{aligned}$$

(4.18)

where $u(t-)=(\tau _1,\ldots ,\tau _{N(t-)};\beta _1,\ldots , \beta _{N(t-)})$ with $N(t-):=\max \{j\ge 0:\tau _j<t\}$ and $\alpha (t-)\circ \alpha ':={\mathbbm {1}}_{[0,t)}\alpha +{\mathbbm {1}}_{[t,T]}\alpha '$. Then for each $u\in {\mathcal {U}}^f$ we obtain a $u'\in {\mathcal {U}}^s$ by removing all future interventions whenever (4.18) does not hold. Moreover, $u'$ dominates u in the sense that $J(u',\alpha )>J(u,\alpha )$ for all $\alpha \in {\mathcal {A}}$.

Now, whenever $u\in {\mathcal {U}}^s$ there is a $(B^{u,\alpha },E^{u,\alpha })\in {\mathcal {S}}^2_l\times {\mathcal {H}}^2$ that solves (4.17). We will build on the argument in Lemma 3.14 to show that ${\mathcal {U}}^s\subset {\mathcal {U}}^m$. We thus assume that $u\in {\mathcal {U}}^s$. Rearranging the terms in (4.17) gives

$$\begin{aligned}&\sum _{j=1}^{N} \ell (\tau _j,X^{[u]_{j-1},\alpha }_{\tau _j},\beta _j) =\psi (X^{u,\alpha }_T)-B^{u,\alpha }_0 +\int _0^{T}\phi (s,X^{u,\alpha }_s, \alpha _s)ds \nonumber \\&\quad -\int _0^{T}E^{u,\alpha }_sdW_s, \end{aligned}$$

(4.19)

where we know that all terms on the right hand side, except for the last (martingale) term, have moments of all orders. By Ito’s formula we have

$$\begin{aligned}&|B^{u,\alpha }_0|^2+\int _0^{T}|E^{u,\alpha }_s|^2ds\\&\quad =|\psi (X^{u,\alpha }_T)|^2+2\int _0^{T}B^{u,\alpha }_s \phi (s,X^{u,\alpha }_s,\alpha _s)ds -2\int _0^T B^{u,\alpha }_sE^{u,\alpha }_sdW_s\\&\qquad +\sum _{j=1}^{N} (-2B^{u,\alpha }_{\tau _j} \ell (\tau _j,X^{[u]_{j-1},\alpha }_{\tau _j},\beta _j) +|\ell (\tau _j,X^{[u]_{j-1},\alpha }_{\tau _j},\beta _j)|^2)\\&\quad \le |\psi (X^{u,\alpha }_T)|^2+(1+2\kappa )\sup _{t\in [0,T]}| B^{u,\alpha }_t|^2+\int _0^{T}|\phi (s,X^{u,\alpha }_s,\alpha _s)|^2ds\\&\qquad -2\int _0^TB^{u,\alpha }_sE^{u,\alpha }_sdW_s+\frac{2}{\kappa } \big (\sum _{j=1}^{N} \ell (\tau _j,X^{[u]_{j-1},\alpha }_{\tau _j}, \beta _j)\big )^2, \end{aligned}$$

for $\kappa >0$ (where we have used that $\ell (\tau _j,X^{[u]_{j-1},\alpha }_{\tau _j},\beta _j)\le 2\sup _{t\in [0,T]}|B^{u,\alpha }_t|$). Using (4.19), the growth conditions on $\phi$ and $\psi$ and the fact that $u\in {\mathcal {U}}^s$ together with (4.8) gives that

$$\begin{aligned} \int _0^{T}|E^{u,\alpha }_s|^2ds&\le C\left( 1+\kappa +\frac{1}{\kappa }\right) \left( 1+\sup _{t\in [0,T]}|X^{u,\alpha }|^{2\rho }\right) \\&\qquad -2\int _0^TB^{u,\alpha }_sE^{u,\alpha }_sdW_s +\frac{4}{\kappa }\Big |\int _0^{T}E^{u,\alpha }_sdW_s\Big |^2. \end{aligned}$$

Raising both sides to $p\ge 1$ followed by taking the expectation and applying the Burkholder-Davis-Gundy inequality gives

$$\begin{aligned} {\mathbb {E}}\Big [\big (\int _0^{T}|E^{u,\alpha }_s|^2ds\big )^p\Big ]&\le C\left( 1+\kappa +\frac{1}{\kappa }\right) \\&\quad +C{\mathbb {E}}\Big [\big (\int _0^T|B^{u,\alpha }_s E^{u,\alpha }_s|^2ds\big )^{p/2}+\frac{1}{\kappa } \big (\int _0^{T}|E^{u,\alpha }_s|^2\big )^p\Big ]\\&\le C\left( 1+\kappa +\frac{1}{\kappa }\right) +\frac{C}{\kappa } {\mathbb {E}}\Big [\big (\int _0^{T}|E^{u,\alpha }_s|^2\big )^p\Big ] \end{aligned}$$

where we have used the relation $2ab\le \kappa a^2+\frac{1}{\kappa }b^2$ together with the bound on ${\mathbb {E}}\big [\sup _{t\in [0,T]}|B^{u,\alpha }_t|^{2p}\big ]$ resulting from the fact that $u\in {\mathcal {U}}^s$ to reach the last inequality. Now, choosing $\kappa$ sufficiently large it follows that the left hand side is finite. Finally, as the left hand side of (4.19) is greater that $\delta N$ we conclude that $u\in {\mathcal {U}}^m$. $\square$

By Benes’ selection Theorem ([2], Lemma 5, pp. 460), there exists, for each $v\in {\mathcal {U}}^f$, a ${\mathcal {P}}\otimes {\mathcal {B}}({\mathbb {R}}^d)/{\mathcal {B}}(A)$-measurable function $\alpha ^v(t,\omega ,z)$ such that for any given $(t,\omega , z) \in [0,T]\times \Omega \times {\mathbb {R}}^{d}$, we have

$$\begin{aligned} H^v(t,\omega ,z,\alpha ^v(t,\omega ,z))=\inf _{\alpha \in A} H^v(t,\omega ,z,\alpha ), \end{aligned}$$

${\mathbb {P}}$-a.s.

The following theorem shows that we can extract the optimal pair $(u^*,\alpha ^*)$ from the family of maps $(\alpha ^v(t,\omega ,z):v\in {\mathcal {U}}^f)$ and the solution to (4.11).

Theorem 4.15

Let the family $(Y^v,Z^v,K^v:v\in {\mathcal {U}}^f)$ be a solution to (4.11). Then the pair $(u^*,\alpha ^*)\in {\mathcal {U}}^f\times {\mathcal {A}}$, with $u^*=(\tau ^*_1,\ldots ,\tau ^*_{N^*};\beta ^*_1, \ldots ,\beta ^*_{N^*})\in {\mathcal {U}}^f_0$ defined as:

• $\tau ^*_{j}:=\inf \Big \{s \ge \tau ^*_{j-1}: \,Y_s^{v\circ [u^*]_{j-1}}=\sup _{b\in U} \{Y^{v\circ [u^*]_{j-1} \circ (s,b)}_s-\ell (s,X^{v\circ [u^*]_{j-1}}_s,b)\}\Big \}\wedge T$

• $\beta ^*_j\in \mathop {\arg \max }_{b\in U}\{Y^{v\circ [u^*]_{j-1} \circ (\tau ^*_j,b)}_{\tau ^*_j}-\ell (\tau ^*_j, X^{v\circ [u^*]_{j-1}}_{\tau ^*_j},b)\}$

and $N^*=\sup \{j:\tau ^*_j<T\}$, with $\tau _0^*:=0$ and

$$\begin{aligned} \alpha ^*_t:=\sum _{j=0}^{N^*}{\mathbbm {1}}_{[\tau ^*_j,\tau ^*_{j+1})} (t)\alpha ^{[u^*]_j}(t,Z^{[u^*]_j}_t), \end{aligned}$$

with $\tau ^*_{N+1}:=\infty$ is optimal in the sense of (4.1) and $Y^\emptyset _0=J(u^*,\alpha ^*)$.

Proof

For $u\in {\mathcal {U}}^m$ we let $(U^u,V^u)\in {\mathcal {S}}^2_l\times {\mathcal {H}}^2$ be the unique solution to

$$\begin{aligned} U_t^{u}=\psi (X^u_T)+\int _t^T H^{*,u}(s,V^u_s)ds -\int _t^{T} V^{u}_sdW_s-\sum _{j=1}^N {\mathbbm {1}}_{[\tau _j\ge t]} \ell (\tau _j,X^{[u]_{j-1}}_{\tau _j},\beta _j). \end{aligned}$$

(4.20)

Then $\Vert V^u\Vert _{{\mathcal {H}}^p}<\infty$ for all $p\ge 1$ implying that $V^u\in {\mathcal {H}}^2_{{\mathbb {Q}}}$ for all ${\mathbb {Q}}\in {\mathfrak {P}}_0$ and, since $U^u_0$ is ${\mathcal {F}}_0$-measurable and ${\mathcal {F}}_0$ is trivial, we have

$$\begin{aligned} U^u_0&={\mathbb {E}}^{{\mathbb {Q}}^{u,\alpha ^*}}\Big [\psi (X^u_T)+\int _0^T H^{*,u}(s,V^u_s)ds -\int _0^{T} V^{u}_sdW_s-\sum _{j=1}^N \ell (\tau _j,X^{[u]_{j-1}}_{\tau _j}, \beta _j)\Big ]\\&={\mathbb {E}}^{{\mathbb {Q}}^{u,\alpha ^*}}\Big [\psi (X^u_T)+\int _0^T \phi (s,X^u_s,\alpha ^*_s)ds -\int _0^{T} V^{u}_sdW^{u,\alpha ^*}_s -\sum _{j=1}^N \ell (\tau _j,X^{[u]_{j-1}}_{\tau _j},\beta _j)\Big ]\\&= J(u,\alpha ^*) \end{aligned}$$

where now ${\mathbb {Q}}^{u,\alpha }$ is the measure, equivalent to ${\mathbb {P}}$, under which $W^{u,\alpha }:=W-\int _0^\cdot \breve{a}(s,(X^v_r)_{r\le s},\alpha _s)ds$ is a martingale. Moreover, a straightforward extension of Lemma 3.16 to impulse controls with an unbounded number of interventions shows that $u^*\in {\mathcal {U}}^m$ and we conclude by Theorem 3.20 (see Remark 4.12) that

$$\begin{aligned} Y^\emptyset _0=U^{u^*}_0=\sup _{u\in {\mathcal {U}}^f}U^u_0=\sup _{u\in {\mathcal {U}}^m}U^u_0. \end{aligned}$$

Combined with Lemma 4.14, the above gives

$$\begin{aligned} Y^\emptyset _0=J(u^*,\alpha ^*)=\sup _{u\in {\mathcal {U}}^m}U^u_0 =\sup _{u\in {\mathcal {U}}^m}J(u,\alpha ^*)=\sup _{u\in {\mathcal {U}}^f}J(u,\alpha ^*). \end{aligned}$$

To show that $\alpha ^*$ is an optimal response it is enough to show that $\alpha ^*$ is a minimizer of $\alpha \mapsto J(u^*,\alpha )$. However, for any $(u,\alpha )\in {\mathcal {U}}^m\times {\mathcal {A}}$, we have

$$\begin{aligned} U^u_0&={\mathbb {E}}^{{\mathbb {Q}}^{u,\alpha }}\Big [\psi (X^u_T)+\int _0^T H^{*,u}(s,V^u_s)ds -\int _0^{T} V^{u}_sdW_s-\sum _{j=1}^N \ell (\tau _j,X^{[u]_{j-1}}_{\tau _j}, \beta _j)\Big ]\\&={\mathbb {E}}^{{\mathbb {Q}}^{u,\alpha }}\Big [\psi (X^u_T)+\int _0^T \phi (s,X^u_s,\alpha _s)ds -\int _0^{T} V^{u}_sdW^{u,\alpha }_s-\sum _{j=1}^N \ell (\tau _j,X^{[u]_{j-1}}_{\tau _j},\beta _j)\Big ]\\&\quad +{\mathbb {E}}^{{\mathbb {Q}}^{u,\alpha }}\Big [\int _0^T (H^{*,u}(s,V^{u}_s)-H^{u}(s,V^{u}_s,\alpha _s))ds\Big ]\\&\le J(u,\alpha ) \end{aligned}$$

and we conclude that $(u^*,\alpha ^*)$ is a saddle-point for the game. $\square$

5 Conclusions

The present work considers a sequentially arranged system of reflected BSDEs parameterized by impulse controls. Using only probabilistic arguments, we show existence and uniqueness of solutions under a stochastic Lipschitz condition on the driver. Moreover, we relate the solution to our system of BSDEs to a weakly formulated stochastic differential game where one player implements an impulse control while the adversary player plays a continuous control that does not enter the diffusion coefficient. Since all arguments are probabilistic, we do not need to rely on any Markov property of solutions to the underlying controlled SDE, enabling us to handle path-dependence in the drift and diffusion coefficient as well as in the impulse-to-jump map, $\Gamma$, of the impulse controller.

A practical application of the result lies within robust control where the impulse controller is ambiguous about the model for the drift term in the controlled SDE. A cautions operator could incorporate this ambiguity into the model by considering the worst case, thus rendering a zero-sum stochastic differential game where the adversary (nature) controls the drift term. In this regard, a number of questions are left open. An issue of clear interest is when the adversary player is allowed to control the diffusion coefficient as well. In the robust control framework this setting corresponds to ambiguity about both the drift and the diffusion coefficient. A recent development in this direction is [27] where path-dependent PDEs [8, 9] and second order BSDEs [29] (abbreviated 2BSDEs) are used to represent solutions to path-dependent zero-sum stochastic differential games of continuous versus continuous control in a weak formulation. A natural next step would thus be to develop the present work to incorporate 2BSDEs.

Our game is one of impulse versus continuous control. Alternatively, one could consider the case where both players play impulse controls. This problem was approached in the Markovian framework in [6] but the extension to the path-dependent setting remains an open problem.

Finally, we leave the important issue of tractable numerical approximation algorithms as a topic of future research.

A Proof of Proposition 2.4

In this section we give a proof of Proposition 2.4. Since the result can be useful in other contexts as well, it is written as a stand-alone piece.

1.1 A.1 Classical Result Under Deterministic Lipschitz Condition

Our approach will rely heavily on the available theory of reflected backward SDEs and we, therefore, recall the following important result:

Theorem A.1

(El Karoui et. al. [12]) Assume that

• (a) $\xi \in L^2(\Omega ,{\mathcal {F}}_T,{\mathbb {P}})$.

• (b) The barrier S is real-valued, ${\mathcal {P}}_{\mathbb {F}}$-measurable and continuous with $S^+\in {\mathcal {S}}^2$ and $S_T\le \xi$.

• (c) $f:[0,T]\times \Omega \times {\mathbb {R}}\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}$ is such that $f(\cdot ,y,z)\in {\mathcal {H}}^2$ for all $(y,z)\in {\mathbb {R}}\times {\mathbb {R}}^{d}$ and for some $k_f>0$ and all $(y,y',z,z')\in {\mathbb {R}}^{2(1+d)}$ we have

  $$\begin{aligned} |f(t,y',z')-f(t,y,z)|\le k_f(|y'-y|+|z'-z|). \end{aligned}$$

Then there exists a unique triple $(Y,Z,K):=(Y_t,Z_t,K_t)_{0\le t\le T}$ with $Y,K\in {\mathcal {S}}^2$ and $Z\in {\mathcal {H}}^2$, where K is non-decreasing with $K_0=0$, such that

$$\begin{aligned} {\left\{ \begin{array}{ll} Y_t=\xi +\int _t^T f(s,Y_s,Z_s)ds-\int _t^T Z_s dW_s+K_T-K_t,\\ Y_t\ge S_t,\, \forall t\in [0,T] \text{ and } \int _0^T \left( Y_t-S_t\right) dK_t=0. \end{array}\right. } \end{aligned}$$

(A.1)

Furthermore,

$$\begin{aligned} \Vert Y\Vert _{{\mathcal {S}}^2}^2+\Vert Z\Vert _{{\mathcal {H}}^2}^2+\Vert K\Vert _{{\mathcal {S}}^2}^2 \le C{\mathbb {E}}\Big [|\xi |^{2}+\int _0^T|f(s,0,0)|^2ds +\sup _{t\in [0,T]}|(S_t)^+|^2\Big ]. \end{aligned}$$

(A.2)

In addition, Y can be interpreted as the Snell envelope in the following way

$$\begin{aligned} Y_t=\mathop {\mathrm{ess}\,\sup }_{\tau \in {\mathcal {T}}_t}{\mathbb {E}}\bigg [\int _t^\tau f(s,Y_s,Z_s)ds +S_\tau {\mathbbm {1}}_{[\tau <T]}+\xi {\mathbbm {1}}_{[\tau =T]}\Big |{\mathcal {F}}_t\bigg ] \end{aligned}$$

and with $D_t:=\inf \{r\ge t: Y_r=S_r\}\wedge T$ we have the representation

$$\begin{aligned} Y_t={\mathbb {E}}\bigg [\int _t^{D_t} f(s,Y_s,Z_s)ds+S_{D_t}{\mathbbm {1}}_{[D_t<T]} +\xi {\mathbbm {1}}_{[D_t=T]}\Big |{\mathcal {F}}_t\bigg ] \end{aligned}$$

and $K_{D_t}-K_t=0$, ${\mathbb {P}}$-a.s.

Moreover, if $(\tilde{Y},\tilde{Z},\tilde{K})$ is the solution to the reflected BSDE with parameters $(\tilde{\xi },\tilde{f},\tilde{S})$, then

$$\begin{aligned}&\Vert \tilde{Y}-Y\Vert _{{\mathcal {S}}^2}^2+\Vert \tilde{Z}-Z\Vert _{{\mathcal {H}}^2}^2 +\Vert \tilde{K}-K\Vert _{{\mathcal {S}}^2}^2 \nonumber \\&\quad \le C\left( \Vert \tilde{S} -S\Vert _{{\mathcal {S}}^{2}}\Psi _T^{1/2}+{\mathbb {E}}\Big [|\tilde{\xi }-\xi |^{2} +\int _0^T |\tilde{f}(s,Y_s,Z_s)-f(s,Y_s,Z_s)|^2ds\Big ]\right) , \end{aligned}$$

(A.3)

where

$$\begin{aligned} \Psi _T:={\mathbb {E}}\Big [|\tilde{\xi }|^{2}+|\xi |^{2} +\int _0^T (|\tilde{f}(s,0,0)|^2+|f(s,0,0)|^2)ds +\sup _{t\in [0,T]}|(\tilde{S}_{t})^+ + (S_{t})^+|^{2}\Big ]. \end{aligned}$$

Existence and uniqueness of solutions to (A.1) under the assumption that the Lipschitz coefficient on the z-variable is a stochastic process was recently shown in [10]. In the next section we show that the moment and stability estimates in Theorem A.1 translates to the case with stochastic Lipschitz coefficients and exponents $p\ne 2$ as well.

1.2 A.2 Extension to Stochastic Lipschitz Coefficient

We consider the non-reflected BSDE with parameters $(f,\xi )$,

$$\begin{aligned} Y_t=\xi +\int _t^T f(s,Y_s,Z_s)ds-\int _t^T Z_sdW_s, \quad \forall \,t\in [0,T] \end{aligned}$$

(A.4)

and the reflected BSDE with parameters $(f,\xi ,S)$ given in (2.3) that we recall is

$$\begin{aligned} {\left\{ \begin{array}{ll} Y_t=\xi +\int _t^T f(s,Y_s,Z_s)ds-\int _t^T Z_s dW_s+K_T-K_t,\\ Y_t\ge S_t,\, \forall t\in [0,T] \text{ and } \int _0^T \left( Y_t-S_t\right) dK_t=0. \end{array}\right. } \end{aligned}$$

(A.5)

Throughout this section we will assume that for some $q'> 1$ and all $p\ge 1$ the following holds:

Assumption A.2

• (i) There is a ${\mathbb {P}}$-a.s. non-negative, ${\mathcal {P}}_{\mathbb {F}}$-measurable, continuous process $(L_t:t\in [0,T])$ (our stochastic Lipschitz coefficient) such that for all ${\mathcal {P}}_{\mathbb {F}}$-measurable processes $(\zeta _t:t\in [0,T])$ with $|\zeta _t|\le L_t$ for all $t\in [0,T]$ (outside of a ${\mathbb {P}}$-null set) we have ${\mathbb {E}}[|{\mathcal {E}}(\zeta *W)_T|^{q'}]<\infty$ and ${\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}[|{\mathcal {E}}(-\zeta *W^{\zeta })_T|^{q'}]<\infty$.

• (ii) The terminal value $\xi \in L^{p}(\Omega ,{\mathcal {F}}_T,{\mathbb {P}})$.

• (iii) The driver $(t,\omega ,y,z)\mapsto f(t,\omega ,y,z): [0,T]\times \Omega \times {\mathbb {R}}\times {\mathbb {R}}^{d}\rightarrow {\mathbb {R}}$ is ${\mathcal {P}}_{{\mathbb {F}}}\otimes {\mathcal {B}}({\mathbb {R}})\otimes {\mathcal {B}}({\mathbb {R}}^{d})$-measurable. Furthermore, we have

  • (a) The bound

    $$\begin{aligned} {\mathbb {E}}\Big [\int _0^T|f(s,0,0)|^{p}ds\Big ]<\infty . \end{aligned}$$

    (A.6)

  • (b) There is a constant $k_f>0$ such that

    $$\begin{aligned} |f(t,y',z')-f(t,y,z)|\le k_f|y'-y|+L_t|z'-z| \end{aligned}$$

    (A.7)

    for all $(t,y,y',z,z')\in [0,T]\times {\mathbb {R}}^{2(1+d)}$, ${\mathbb {P}}$-a.s.

  • (c) The barrier S is real-valued, ${\mathcal {P}}_{\mathbb {F}}$-measurable and continuous with $S^+\in {\mathcal {S}}^p$ and $S_T\le \xi$, ${\mathbb {P}}$-a.s.

Under this set of assumptions, letting $q>1$ be such that $\frac{1}{q}+\frac{1}{q'}=1$, we have the following:

Theorem A.3

The BSDE (A.4) has a unique solution $(Y,Z)\in {\mathcal {S}}^p\times {\mathcal {H}}^p$, with

$$\begin{aligned} \Vert Y\Vert _{{\mathcal {S}}^p}^p+\Vert Z\Vert _{{\mathcal {H}}^p}^p\le C{\mathbb {E}}\Big [|\xi |^{q^2 p} +\big (\int _t^T |f(s,0,0)|ds\big )^{q^2 p}\Big ]^{1/q^2}. \end{aligned}$$

Furthermore, if $(\tilde{Y},\tilde{Z})$ is a solution to (A.4) with parameters $(\tilde{f},\tilde{\xi })$, then

$$\begin{aligned}&\Vert \tilde{Y}-Y\Vert _{{\mathcal {S}}^p}^p+\Vert \tilde{Z}-Z\Vert _{{\mathcal {H}}^p}^p\\&\quad \le C{\mathbb {E}}\Big [|\tilde{\xi }-\xi |^{q^2 p}+\big (\int _t^T | \tilde{f}(s,Y_s,Z_s)-f(s,Y_s,Z_s)|ds\big )^{q^2 p}\Big ]^{1/q^2}. \end{aligned}$$

Proof

This follows by repeating the arguments in the proofs of Lemma 3.1 and Lemma 3.2 in [21]. $\square$

As a step towards obtaining moment estimates for solutions to (A.5) we follow the convention in [10] and let $(Y^{m,n},Z^{m,n},K^{m,n})\in {\mathcal {S}}^2\times {\mathcal {H}}^2\times {\mathcal {S}}^2$ be the unique solution to (A.5) with parameters $({\mathbbm {1}}_{[L_\cdot \le m]}f^+-{\mathbbm {1}}_{[L_\cdot \le n]}f^-,\xi ,S)$, i.e.

$$\begin{aligned} {\left\{ \begin{array}{ll} Y^{m,n}_t=\xi +\int _t^Tf^{m,n}(s,Y^{m,n}_s,Z^{m,n}_s)ds+\int _t^TZ^{m,n}_sdW_s+K^{m,n}_T-K^{m,n}_t\\ Y^{m,n}_t\ge S_t,\, \forall t\in [0,T] \text{ and } \int _0^T \left( Y^{m,n}_t-S_t\right) dK^{m,n}_t=0. \end{array}\right. } \end{aligned}$$

where $f^{m,n}(t,y,z):={\mathbbm {1}}_{[L_t\le m]}f^+(t,y,z)-{\mathbbm {1}}_{[L_t\le n]}f^-(t,y,z)$ is a standard Lipschitz driver.

Lemma A.4

For all $m,n\ge 0$ we have $Z^{m,n}\in {\mathcal {H}}^2_{\mathbb {Q}}$ for all ${\mathbb {Q}}\in {\mathfrak {P}}^{L\wedge (m\vee n)}$.

Proof

For each ${\mathbb {Q}}\in {\mathfrak {P}}^{L\wedge (m\vee n)}$, there is a ${\mathcal {P}}_{\mathbb {F}}$-measurable process, $\zeta$, such that $|\zeta _t|\le L_t\wedge (m\vee n)$, for all $t\in [0,T]$ (outside of a ${\mathbb {P}}$-null set) and $d{\mathbb {Q}}:={\mathcal {E}}(\zeta *W)_Td{\mathbb {P}}$. By the Girsanov theorem (see e.g. Chapter 15 in [5]), it follows that under the probability measure ${\mathbb {Q}}$, the process $W^\zeta :=W-\int _0^\cdot \zeta _sds$ is a Brownian motion. Moreover, the triple $(Y^{m,n},Z^{m,n},K^{m,n})$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} Y^{m,n}_t=\xi +\int _t^T (f^{m,n}(s,Y^{m,n}_s,Z^{m,n}_s)-\zeta _sZ^{m,n}_s)ds-\int _t^T Z^{m,n}_s dW^\zeta _s+K^{m,n}_T-K^{m,n}_t,\\ Y^{m,n}_t\ge S_t,\, \forall t\in [0,T] \text{ and } \int _0^T \left( Y^{m,n}_t-S_t\right) dK^{m,n}_t=0. \end{array}\right. } \end{aligned}$$

Since this is a reflected BSDE with standard Lipschitz driver we can apply (A.2) to get that

$$\begin{aligned}&{\mathbb {E}}^{\mathbb {Q}}\Big [\int _0^T|Z^{m,n}_s|^2ds\Big ]\\&\quad \le C{\mathbb {E}}^{\mathbb {Q}}\Big [|\xi |^{2} +\int _0^T|f^{m,n}(s,0,0)|^2ds+\sup _{t\in [0,T]}|(S_t)^+|^2\Big ]\\&\quad \le C{\mathbb {E}}\Big [{\mathcal {E}}(\zeta *W)_T\Big (|\xi |^{2}+\int _0^T|f^{m,n}(s,0,0)|^2ds +\sup _{t\in [0,T]}|(S_t)^+|^2\Big )\Big ]\\&\quad \le C{\mathbb {E}}\big [|{\mathcal {E}}(\zeta *W)_T|^{q'}]^{1/q'} {\mathbb {E}}\Big [|\xi |^{2q}+\int _0^T|f^{m,n}(s,0,0)|^{2q}ds +\sup _{t\in [0,T]}|(S_t)^+|^{2q}\Big ]^{1/q} \end{aligned}$$

and the result follows by Assumption A.2. $\square$

Lemma A.5

There is a $C>0$ such that

$$\begin{aligned}&\Vert Y^{m,n}\Vert _{{\mathcal {S}}^p}^p + \Vert Z^{m,n}\Vert _{{\mathcal {H}}^p}^p +\Vert K^{m,n}\Vert _{{\mathcal {S}}^p}^p \nonumber \\&\quad \le C{\mathbb {E}}\Big [|\xi |^{q^2p} +\big (\int _0^T |f^{m,n}(s,0,0)|ds\big )^{q^2p} +\sup _{t\in [0,T]}|(S_{t})^+|^{q^2p}\Big ]^{1/q^2}. \end{aligned}$$

(A.8)

for all $m,n\ge 0$.

Proof

We define

$$\begin{aligned} \gamma _s:=\frac{f^{m,n}(s,Y^{m,n}_s,Z^{m,n}_s) -f^{m,n}(s,0,Z^{m,n}_s)}{Y^{m,n}_s}{\mathbbm {1}}_{[Y^{m,n}_s\ne 0]} \end{aligned}$$

and set $e_t:=e^{\int _0^t\gamma _sds}$. By Assumption A.2 we have that $e^{-k_fT}\le e_t\le e^{k_fT}$. Applying Ito’s formula to $e_tY^{m,n}_t$ gives that for any $\tau \in {\mathcal {T}}_t$ we have

$$\begin{aligned} e_tY^{m,n}_t&=e_\tau Y^{m,n}_\tau +\int _t^\tau e_s(f^{m,n} (s,Y^{m,n}_s,Z^{m,n}_s)-\gamma _sY^{m,n}_s)ds\\&\quad -\int _t^\tau e_sZ^{m,n}_sdW_s+\int _t^\tau e_sdK^{m,n}_s\\&=e_\tau Y^{m,n}_\tau +\int _t^\tau e_s(f^{m,n}(s,0,0) +\zeta _sZ^{m,n}_s)ds-\int _t^\tau e_sZ^{m,n}_sdW_s\\&\quad +\int _t^\tau e_sdK^{m,n}_s \end{aligned}$$

where

$$\begin{aligned} \zeta _s:=\frac{f^{m,n}(s,0,Z^{m,n}_s)-f^{m,n} (s,0,0)}{|Z^{m,n}_s|^2}(Z^{m,n}_s)^\top {\mathbbm {1}}_{[Z^{m,n}_s\ne 0]}. \end{aligned}$$

By the Girsanov theorem, it follows that under the probability measure ${\mathbb {Q}}^{\zeta }$ defined as $d{\mathbb {Q}}^{\zeta }:={\mathcal {E}}(\zeta *W)_Td{\mathbb {P}}$ the process $W^{\zeta }_t:=W_t-\int _0^t\zeta _sds$ is a Brownian motion. Furthermore, we have

$$\begin{aligned} e_t Y^{m,n}_t=e_\tau Y^{m,n}_\tau +\int _t^\tau e_sf^{m,n} (s,0,0)ds-\int _t^\tau e_sZ^{m,n}_sdW^{\zeta }_s+\int _t^\tau e_sdK^{m,n}_s. \end{aligned}$$

Taking the conditional expectation while picking $\tau =D_t(:=\inf \{t\ge 0:Y^{m,n}_t=S_t\}\wedge T)$ and appealing to Lemma A.4 which implies that the stochastic integral is a ${\mathbb {Q}}^{\zeta }$-martingale gives

$$\begin{aligned} e_tY^{m,n}_t={\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [{\mathbbm {1}}_{[D_t<T]} e_{D_t}S_{D_t}+{\mathbbm {1}}_{[D_t=T]}e_{T}\xi +\int _t^{D_t} e_sf^{m,n}(s,0,0)ds\big |{\mathcal {F}}_t\Big ]. \end{aligned}$$

(A.9)

Doob’s maximal inequality together with the bounds on $e_t$ gives that

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [\sup _{t\in [0,T]}|Y^{m,n}_t|^p\Big ] \le C{\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [\left( \int _0^{T}|f^{m,n} (s,0,0)| ds\right) ^p+\sup _{t\in [0,T]}|(S_{t})^+|^p+|\xi |^p\Big ]. \end{aligned}$$

Changing back to our original probability measure ${\mathbb {P}}$ on the right hand side gives

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\left[ \sup _{t\in [0,T]}|Y^{m,n}_t|^p\right] \\&\quad \le C{\mathbb {E}}\left[ {\mathcal {E}}(\zeta *W)_T\left( \left( \int _0^{T}|f^{m,n} (s,0,0)| ds\right) ^p+\sup _{t\in [0,T]}|(S_{t})^+|^p+|\xi |^p\right) \right] \\&\quad \le C{\mathbb {E}}\big [{\mathcal {E}}(\zeta *W)_T^{q'}]^{1/q'} {\mathbb {E}}\Big [\left( \int _0^{T}|f^{m,n}(s,0,0)|ds\right) ^{qp} +\sup _{t\in [0,T]}|(S_{t})^+|^{qp}+|\xi |^{qp}\Big ]^{1/q}. \end{aligned}$$

Moreover, as

$$\begin{aligned} {\mathbb {E}}\Big [\sup _{t\in [0,T]}|Y^{m,n}_t|^p\Big ]&={\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [{\mathcal {E}}(-\zeta *W^{\zeta })_T \sup _{t\in [0,T]}|Y^{m,n}_t|^p\Big ]\\&\le C{\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [\sup _{t\in [0,T]}| Y^{m,n}_t|^{qp}\Big ]^{1/q}, \end{aligned}$$

the estimate for $Y^{m,n}$ follows.

To arrive at the estimate for $Z^{m,n}$ we apply Ito’s formula to $(Y^{m,n})^2$ and get that

$$\begin{aligned} |Y^{m,n}_t|^2+\int _t^T|Z^{m,n}_s|^2ds&= |\xi |^2 +2\int _t^TY^{m,n}_sf^{m,n}(s,Y^{m,n}_s,Z^{m,n}_s)ds\nonumber \\&\quad -2\int _t^TY^{m,n}_sZ^{m,n}_sdW_s +2\int _t^TY^{m,n}_sdK^{m,n}_s\nonumber \\&= |\xi |^2+2\int _t^T(\gamma _s|Y^{m,n}_s|^2 +Y^{m,n}_sf^{m,n}(s,0,0))ds\nonumber \\&\quad -2\int _t^TY^{m,n}_sZ^{m,n}_sdW^{\zeta }_s +2\int _t^TY^{m,n}_sdK^{m,n}_s. \end{aligned}$$

(A.10)

Furthermore, applying the relation $ab\le \tfrac{1}{2}\kappa a^2+\tfrac{1}{2\kappa }b^2$ with $\kappa >0$ and the Skorokhod condition $\int _0^T(Y^{m,n}_s-S_s)dK^{m,n}_s=0$ yields

$$\begin{aligned} \int _t^TY^{m,n}_sdK^{m,n}_s\le \int _t^T(S_s)^+dK^{m,n}_s \le \frac{\kappa }{2}\sup _{r\in [t,T]}|(S_r)^+|^2 +\frac{1}{2\kappa }|K^{m,n}_T|^2. \end{aligned}$$

Now, Proposition 2.2 in [12] gives that

$$\begin{aligned} K^{m,n}_T-K^{m,n}_t&=\sup _{r\in [t,T]}\left( \xi +\int _r^Tf^{m,n} (s,Y^{m,n}_s,Z^{m,n}_s)ds-\int _r^TZ^{m,n}_sdW_s-S_r\right) ^{-}\\&=\sup _{r\in [t,T]}\left( \xi +\int _r^T(\gamma _sY^{m,n}_s+f^{m,n} (s,0,0))ds-\int _r^TZ^{m,n}_sdW^{\zeta }_s-S_r\right) ^{-} \end{aligned}$$

and, in particular, we have that

$$\begin{aligned} K^{m,n}_T&\le |\xi |+\int _0^T\big (k_f|Y^{m,n}_s| +|f^{m,n}(s,0,0)|\big )ds\nonumber \\&\quad +2\sup _{r\in [0,T]}\Big |\int _0^r Z^{m,n}_sdW^{\zeta }_s\Big | +\sup _{r\in [0,T]}|(S_r)^+| \end{aligned}$$

(A.11)

Inserted into (A.10) this gives (with $\kappa \ge 1$)

$$\begin{aligned} \int _0^T|Z^{m,n}_s|^2ds&\le \frac{C}{\kappa }\sup _{r\in [0,T]} \Big |\int _0^r Z^{m,n}_sdW^{\zeta }_s\Big |^2 -2\int _0^TY^{m,n}_s Z^{m,n}_sdW^{\zeta }_s\\&\quad +C(1+\kappa )\big (|\xi |^2+\sup _{r\in [0,T]}|Y^{m,n}_r|^2 +\sup _{r\in [0,T]}|(S_r)^+|^2\\&\quad +\left( \int _0^T|f^{m,n}(s,0,0)|ds\big )^2\right) \end{aligned}$$

or

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [\left( \int _0^T|Z^{m,n}_s|^2ds\right) ^{p/2} \\&-\frac{C}{\kappa ^{p/2}}\sup _{r\in [0,T]}\Big |\int _0^r Z^{m,n}_sdW^{\zeta }_s\Big |^p -C\Big |\int _0^TY^{m,n}_s Z^{m,n}_sdW^{\zeta }_s\Big |^{p/2}\Big ]\\&\quad \le C(1+\kappa ^{p/2}){\mathbb {E}}^{{\mathbb {Q}}^{\zeta }} \Big [|\xi |^p+\sup _{r\in [0,T]}|Y^{m,n}_r|^p +\sup _{r\in [0,T]}|(S_r)^+|^p+\big (\int _0^T| f^{m,n}(s,0,0)|ds\big )^p\Big ]. \end{aligned}$$

On the other hand, applying the Burkholder-Davis-Gundy inequality gives that

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [\left( \int _0^T|Z^{m,n}_s|^2ds\right) ^{p/2} -\frac{C}{\kappa ^{p/2}}\sup _{r\in [0,T]}\Big |\int _0^TZ^{m,n}_s dW^{\zeta }_s\Big |^p -C\Big |\int _0^TY^{m,n}_sZ^{m,n}_s dW^{\zeta }_s\Big |^{p/2}\Big ]\\&\ge \left( 1-\frac{C}{\kappa ^{p/2}}\right) {\mathbb {E}}^{{\mathbb {Q}}^{\zeta }} \Big [\big (\int _0^T|Z^{m,n}_s|^2ds\big )^{p/2}\Big ] -C{\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [\big (\int _0^T|Y^{m,n}_s Z^{m,n}_s|^2ds\big )^{p/4}\Big ] \end{aligned}$$

By again using that $ab\le \tfrac{1}{2}\kappa a^2+\tfrac{1}{2\kappa }b^2$ we get

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [\big (\int _0^T|Y^{m,n}_s Z^{m,n}_s|^2ds\big )^{p/4}\Big ]\le C {\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [\kappa \sup _{s\in [0,T]}| Y^{m,n}_s|^p+\frac{1}{\kappa } \big (\int _0^T|Z^{m,n}_s|^2ds\big )^{p/2}\Big ] \end{aligned}$$

and the estimate follows by choosing $\kappa >0$ sufficiently large and repeating the steps above to change back to the original measure ${\mathbb {P}}$. Finally, the bound for $K^{m,n}$ is immediate from (A.11) the above. $\square$

Lemma A.6

There is a constant $C>0$ such that, whenever $(Y^{m,n},Z^{m,n},K^{m,n})$ and $(\tilde{Y}^{\tilde{m},\tilde{n}},\tilde{Z}^{\tilde{m}, \tilde{n}},\tilde{K}^{\tilde{m},\tilde{n}})$ are solutions to (A.5) with parameters $(f^{m,n},\xi ,S)$ and $(\tilde{f}^{\tilde{m},\tilde{n}},\tilde{\xi },\tilde{S})$, respectively, then

$$\begin{aligned}&\Vert \tilde{Y}^{\tilde{m},\tilde{n}}-Y^{m,n}\Vert _{{\mathcal {S}}^p}^p +\Vert \tilde{Z}^{\tilde{m},\tilde{n}}-Z^{m,n}\Vert _{{\mathcal {H}}^p}^p + \Vert \tilde{K}^{\tilde{m},\tilde{n}}-K^{m,n}\Vert _{{\mathcal {S}}^p}^p\\&\quad \le C\left( \Vert \tilde{S}-S\Vert _{{\mathcal {S}}^{q^2p}}^{p/2}\Psi _T^{1/2}\right. \\&\qquad \left. +{\mathbb {E}}\Big [|\tilde{\xi }-\xi |^{q^2p}+\big (\int _0^T | \tilde{f}^{\tilde{m},\tilde{n}}(s,Y^{m,n}_s,Z^{m,n}_s) -f^{m,n}(s,Y^{m,n}_s,Z^{m,n}_s)|ds\big )^{q^2p}\Big ]^{1/q^2}\right) , \end{aligned}$$

where

$$\begin{aligned} \Psi _T&:={\mathbb {E}}\Big [|\tilde{\xi }|^{q^4p}+|\xi |^{q^4p} +\big (\int _0^T (|\tilde{f}^{\tilde{m},\tilde{n}} (s,0,0)|+|f^{m,n}(s,0,0)|)ds\big )^{q^4p}\\&\qquad \qquad +\sup _{t\in [0,T]}|(\tilde{S}_{t})^+ +(S_{t})^+|^{q^4p}\Big ]^{1/q^4}. \end{aligned}$$

Proof

For ease of notation we omit the superscripts m, n and $\tilde{m},\tilde{n}$ and have

$$\begin{aligned} \tilde{Y}_{t}-Y_{t}&= \tilde{\xi }-\xi +\int _t^T (\tilde{f}(s,\tilde{Y}_s,\tilde{Z}_s)-f(s,Y_s,Z_s))ds\\&\quad -\int _t^T (\tilde{Z}_s-Z_s)dW_s+\tilde{K}_T-\tilde{K}_t-(K_T-K_t) \end{aligned}$$

Now, let

$$\begin{aligned} \gamma _s:=\frac{\tilde{f}(s,\tilde{Y}_s,\tilde{Z}_s) -\tilde{f}(s,Y_s,\tilde{Z}_s)}{\tilde{Y}_s-Y_s} {\mathbbm {1}}_{[\tilde{Y}_s\ne Y_s]} \end{aligned}$$

and

$$\begin{aligned} \zeta _s:=\frac{\tilde{f}(s,Y_s,\tilde{Z}_s) -\tilde{f}(s,Y_s,Z_s)}{{|}\tilde{Z}_s-Z_s{|}^2} (\tilde{Z}_s-Z_s)^\top {\mathbbm {1}}_{[\tilde{Z}_s\ne Z_s]} \end{aligned}$$

and set $e_t:=e^{\int _0^t\gamma _sds}$. Letting $\delta Y:=\tilde{Y}-Y$, $\delta \xi :=\tilde{\xi } -\xi$, $\delta Z:=\tilde{Z} - Z$ and $\delta f:=\tilde{f}-f$ we get that for any $\tau \in {\mathcal {T}}_t$ we have

$$\begin{aligned} e_t\delta Y_t&= e_\tau \delta Y_\tau +\int _t^\tau e_s (\delta f(s,Y_s,Z_s)+\zeta _s\delta Z_s)ds-\int _t^\tau e_s\delta Z_sdW_s+\int _t^\tau e_sd(\delta K)_s\\&= e_\tau \delta Y_\tau +\int _t^\tau e_s\delta f(s,Y_s,Z_s)ds -\int _t^\tau e_s\delta Z_sdW^{\zeta }_s+\int _t^\tau e_sd(\delta K)_s \end{aligned}$$

where $W^{\zeta }_t=W_t-\int _0^t\zeta _sds$. Now set $\tau =D_t\,(=\inf \{r\ge t:Y_r=S_r\}\wedge T)$ and we find that

$$\begin{aligned} e_t\delta Y_t&= e_\tau \delta Y_{D_t}+\int _t^{D_t} e_s\delta f(s,Y_s,Z_s)ds-\int _t^{D_t} e_s\delta Z_sdW^{\zeta }_s +\int _t^{D_t}e_s(d\tilde{K}_s-dK_s)\\&\ge {\mathbbm {1}}_{[D_t<T]}e_{D_t}\delta S_{D_t} +{\mathbbm {1}}_{[D_t=T]}e_T\delta \xi +\int _t^{D_t} e_s \delta f(s,Y_s,Z_s)ds-\int _t^{D_t} e_s\delta Z_sdW^{\zeta }_s. \end{aligned}$$

On the other hand, by picking $\tau =\tilde{D}_t:=\inf \{r\ge t:\tilde{Y}_r=\tilde{S}_r\}\wedge T$ we get

$$\begin{aligned} e_t\delta Y_t&\le {\mathbbm {1}}_{[\tilde{D}_t<T]} e_{\tilde{D}_t}\delta S_{\tilde{D}_t}+e_T {\mathbbm {1}}_{[\tilde{D}_t=T]}\delta \xi +\int _t^{\tilde{D}_t} e_s\delta f(s,Y_s,Z_s)ds-\int _t^{\tilde{D}_t} e_s\delta Z_sdW^{\zeta }_s. \end{aligned}$$

Taking the conditional expectation with respect to the measure ${\mathbb {Q}}^{\zeta }$, with $d{\mathbb {Q}}^{\zeta }={\mathcal {E}}(\zeta *W)_Td{\mathbb {P}}$, we find that

$$\begin{aligned} e_t|\delta Y_t| \le {\mathbb {E}}^{{\mathbb {Q}}^{\zeta }} \Big [\sup _{r\in [0,T]}e_r|\delta S_{r}| +e_T|\delta \xi |+\int _0^{T} e_s|\delta f(s,Y_s,Z_s)|ds\big |{\mathcal {F}}_t\Big ] \end{aligned}$$

and the bound for $\Vert \tilde{Y}^{\tilde{m}, \tilde{n}}-Y^{m,n}\Vert _{{\mathcal {S}}^p}$ follows by appealing to the argument of Lemma A.5 while noting that $|\gamma _s|\le k_f$ and $|\zeta _s|\le L_s$.

Applying Ito’s formula to $\delta Y^2$ gives (similarly to (A.10) that)

$$\begin{aligned} |\delta Y_t|^2+\int _t^T|\delta Z_s|^2ds&= |\xi |^2+2\int _t^T(\gamma _s|\delta Y_s|^2 +\delta Y_s\delta f(s,Y_s,Z_s))ds\nonumber \\&\quad -2\int _t^T\delta Y_s\delta Z_sdW^{\zeta }_s + 2\int _t^T\delta Y_sd(\delta K)_s\nonumber \\&\le |\xi |^2+2\int _t^T(k_f|\delta Y_s|^2 +\delta Y_s\delta f(s,Y_s,Z_s))ds\nonumber \\&\quad -2\int _t^T\delta Y_s\delta Z_sdW^{\zeta }_s +2\int _t^T\delta S_sd(\delta K)_s, \end{aligned}$$

(A.12)

where the last inequality follows from the fact that $\int _0^T\delta Y_sd(\delta K_s)\le \int _0^T\delta S_sd(\delta K_s)$. Now,

$$\begin{aligned} \int _0^T\delta S_s d(\delta K)_s\le \sup _{r\in [0,T]}| \delta S_r|(\tilde{K}_T+ K_T) \end{aligned}$$

which gives that

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [\big (\int _0^T|\delta Z_s|^2ds\big )^{p/2}\Big ]\\&\quad \le C{\mathbb {E}}^{{\mathbb {Q}}^{\zeta }}\Big [ |\xi |^p+\sup _{r\in [0,T]}|\delta Y_s|^p +\big (\int _0^T|\delta f(s,Y_s,Z_s)| ds\big )^p\\&\qquad +\Big |\int _0^T\delta Y_s\delta Z^{m,n}_sdW^{\zeta }_s\Big |^{p/2} + \sup _{r\in [0,T]}|\delta S_r|^{p/2}((\tilde{K}_T)^{p/2} +(K_T)^{p/2})\Big ] \end{aligned}$$

and repeating the last steps in the proof of Lemma A.5 we have that

$$\begin{aligned} {\mathbb {E}}\Big [\big (\int _0^T|\delta Z_s|^2ds\big )^{p/2}\Big ]&\le C{\mathbb {E}}\Big [ |\xi |^{q^2p}+\sup _{r\in [0,T]}| \delta Y_s|^{q^2p}+\big (\int _0^T|\delta f(s,Y_s,Z_s)|ds\big )^{q^2p}\\&\quad \sup _{r\in [0,T]}|\delta S_r|^{q^2p/2}((\tilde{K}_T)^{q^2p/2} +(K_T)^{q^2p/2})\Big ]^{1/q^2}. \end{aligned}$$

Finally,

$$\begin{aligned}&{\mathbb {E}}\Big [\sup _{r\in [0,T]}|\delta S_r|^{q^2p/2}((\tilde{K}_T)^{q^2p/2} +(K_T)^{q^2p/2})\Big ]\\&\quad \le \sqrt{2}{\mathbb {E}}\Big [\sup _{r\in [0,T]}| \delta S_r|^{q^2p}\Big ]^{1/2}{\mathbb {E}}\Big [(\tilde{K}_T)^{q^2p} +(K_T)^{q^2p}\Big ]^{1/2} \end{aligned}$$

and the result follows by applying Lemma A.5 to the last term. $\square$

Proposition A.7

The reflected BSDE (A.5) admits a unique solution $(Y,Z,K)\in {\mathcal {S}}^2\times {\mathcal {H}}^2\times {\mathcal {S}}^2$, with K non-decreasing and $K_0=0$. Furthermore, we have

$$\begin{aligned} \Vert Y\Vert _{{\mathcal {S}}^p}^p+\Vert Z\Vert _{{\mathcal {H}}^p}^p+\Vert K\Vert _{{\mathcal {S}}^p}^p \le C\left( \Vert S^+\Vert _{{\mathcal {S}}^{q^2p}}^p+{\mathbb {E}}\Big [|\xi |^{q^2p} +\big (\int _0^T |f(s,0,0)|ds\big )^{q^2p}\Big ]^{1/q^2}\right) \end{aligned}$$

(A.13)

and if $(\tilde{Y},\tilde{Z},\tilde{K})$ is a solution to (A.5) with parameters $(\tilde{f},\tilde{\xi },\tilde{S})$ then

$$\begin{aligned}&\Vert \tilde{Y}-Y\Vert _{{\mathcal {S}}^p}^p+\Vert \tilde{Z}-Z\Vert _{{\mathcal {H}}^p}^p +\Vert \tilde{K}-K\Vert _{{\mathcal {S}}^p}^p \nonumber \\&\le C\left( \Vert \tilde{S}-S\Vert _{{\mathcal {S}}^{q^2p}}^{p/2} \Psi _T^{1/2}+{\mathbb {E}}\Big [|\tilde{\xi }-\xi |^{q^2p} +\big (\int _0^T |\tilde{f}(s,Y_s,Z_s)-f(s,Y_s,Z_s)| ds\big )^{q^2p}\Big ]^{1/q^2}\right) , \end{aligned}$$

(A.14)

where

$$\begin{aligned} \Psi _T&:={\mathbb {E}}\Big [|\tilde{\xi }|^{q^4p}+|\xi |^{q^4p} +\big (\int _0^T (|\tilde{f}(s,0,0)|+|f(s,0,0)|)ds\big )^{q^4p}\\&\qquad \qquad +\sup _{t\in [0,T]}|(\tilde{S}_{t})^+ + (S_{t})^+|^{q^4p}\Big ]^{1/q^4}. \end{aligned}$$

Proof

The first part of the proof largely follows that of Lemma 2.3 in [10] and is included for the sake of completeness. Taking the limit on the right-hand side of (A.8) and appealing to dominated convergence we find that

$$\begin{aligned}&\Vert Y^{m,n}\Vert _{{\mathcal {S}}^p}^p+\Vert Z^{m,n}\Vert _{{\mathcal {H}}^p}^p+\Vert K^{m,n} \Vert _{{\mathcal {S}}^p}^p\\&\quad \le C\left( \Vert S^+\Vert _{{\mathcal {S}}^{q^2p}}^p +{\mathbb {E}}\Big [|\xi |^{q^2p}+\big (\int _0^T |f(s,0,0)| ds\big )^{q^2p}\Big ]^{1/q^2}\right) \\&\quad \le C, \end{aligned}$$

by Assumption A.2 for all $m,n\ge 0$ and $p\ge 1$. In particular, since for fixed m comparison implies that the sequence $(Y^{m,n})_{n\ge 0}$ of continuous processes is non-increasing, the above bound implies that $Y^{m}:=\lim _{n\rightarrow \infty }Y^{m,n}$ exists, ${\mathbb {P}}$-a.s., and by Fatou’s lemma it satisfies $\Vert Y^{m}\Vert _{{\mathcal {S}}^p}^p\le C$. Furthermore, we have by Lemma A.6 that

$$\begin{aligned} \Vert Y^{m,n}-Y^{m,n'}\Vert _{{\mathcal {S}}^2}\rightarrow 0, \end{aligned}$$

as $n,n'\rightarrow \infty$ implying that $Y^{m}$ is a continuous process. Moreover, $(Z^{m,n})_{m,n\ge 0}$ is a uniformly bounded double-sequence in ${\mathcal {H}}^2$ and by again appealing to Lemma A.6 we have that

$$\begin{aligned} \Vert Z^{m,n}-Z^{m,n'}\Vert _{{\mathcal {H}}^2}\rightarrow 0, \end{aligned}$$

as $n,n'\rightarrow \infty$. We conclude that there is a $Z^m$ and consequently also a $K^m$ such that $Z^{m,n}\rightarrow Z^m$ in ${\mathcal {H}}^2$ and $K^{m,n}\rightarrow K^m$, in ${\mathcal {S}}^2$ as $n\rightarrow \infty$. Now, let $\eta _l:=\inf \{s>0:L_s\ge l\}\wedge T$ and note that by Theorem A.1 there is a unique triple $(\hat{Y}^m_t,\hat{Z}^m_t,\hat{K}^m_t)_{0\le t\le \eta _l}$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \hat{Y}^m_t=Y^m_{\eta _l}+\int _t^{\eta _l} f^m(s,\hat{Y}^m_s,\hat{Z}^m_s)ds-\int _t^{\eta _l} \hat{Z}^m_sdW_s+\hat{K}^m_{\eta _l}-\hat{K}^m_t,\quad \forall t\in [0, {\eta _l}],\\ \hat{Y}^m_t\ge S^m_t,\, \forall t\in [0, {\eta _l}] \text{ and } \int _0^{\eta _l}(\hat{Y}^m_t-S^m_t)d\hat{K}^m_t=0, \end{array}\right. } \end{aligned}$$

with $f^{m}:={\mathbbm {1}}_{[L_s\le m]}f^+-f^-$. A trivial modification of the stability result in (A.3) to random terminal times gives that

$$\begin{aligned} \Vert (Y^{m,n}-\hat{Y}^{m}){\mathbbm {1}}_{[0,\eta _l]}\Vert _{{\mathcal {S}}^2}^2+\Vert (Z^{m,n} -\hat{Z}^{m}){\mathbbm {1}}_{[0,\eta _l]}\Vert _{{\mathcal {H}}^2}^2 + \Vert (K^{m,n} -\hat{K}^{m}){\mathbbm {1}}_{[0,\eta _l]}\Vert _{{\mathcal {S}}^2}^2 \rightarrow 0, \end{aligned}$$

as $n\rightarrow \infty$ and by uniqueness of limits it follows that $(Y^m,Z^m,K^m){\mathbbm {1}}_{[0,\eta _l]}=(\hat{Y}^m,\hat{Z}^m, \hat{K}^m){\mathbbm {1}}_{[0,\eta _l]}$ in ${\mathcal {S}}^2\times {\mathcal {H}}^2\times {\mathcal {S}}^2$. Since $\eta _l=T$ outside of a ${\mathbb {P}}$-null set for $l\,(=l(\omega ))$ sufficiently large, we conclude that $(Y^m,Z^m,K^m)$ is the unique solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} Y^m_t=\xi +\int _t^{T} f^m(s,Y^m_s,Z^m_s)ds-\int _t^{T} Z^m_sdW_s +K^m_{T}-K^m_t,\quad \forall t\in [0, T],\\ Y^m_t\ge S^m_t,\, \forall t\in [0, T] \text{ and } \int _0^{T}(Y^m_t-S^m_t)dK^m_t=0. \end{array}\right. } \end{aligned}$$

From the above, the bounds (A.13) and (A.14) with $f=f^m$ are obtained through Lemma A.5 and Lemma A.6, respectively, by Fatou’s lemma and dominated convergence.

Letting $m\rightarrow \infty$ and repeating the above argument the result follows. $\square$

Corollary A.8

If (Y, Z, K) solves (A.5), then Y can be interpreted as the Snell envelope in the following way

$$\begin{aligned} Y_t=\mathop {\mathrm{ess}\,\sup }_{\tau \in {\mathcal {T}}_t}{\mathbb {E}}\bigg [\int _t^\tau f(s,Y_s,Z_s)ds+S_\tau {\mathbbm {1}}_{[\tau <T]} +\xi {\mathbbm {1}}_{[\tau =T]}\Big |{\mathcal {F}}_t\bigg ]. \end{aligned}$$

(A.15)

In particular, with $D_t:=\inf \{r\ge t: Y_r=S_r\}\wedge T$ we have the representation

$$\begin{aligned} Y_t={\mathbb {E}}\bigg [\int _t^{D_t} f(s,Y_s,Z_s)ds+S_{D_t} {\mathbbm {1}}_{[D_t<T]}+\xi {\mathbbm {1}}_{[D_t=T]}\Big |{\mathcal {F}}_t\bigg ] \end{aligned}$$

(A.16)

and $K_{D_t}-K_t=0$, ${\mathbb {P}}$-a.s.

Proof

Let $\eta _l:=\inf \{s>0:L_s\ge l\}\wedge T$, then Theorem A.1 gives that for all $l\ge 0$ we have

$$\begin{aligned} Y_t={\mathbb {E}}\bigg [\int _t^{D_t\wedge \eta _l} f(s,Y_s,Z_s)ds+S_\tau {\mathbbm {1}}_{[D_t<\eta _l]}+{\mathbbm {1}}_{[D_t\ge \eta _l]}({\mathbbm {1}}_{[\eta _l<T]} Y_{\eta _l}+{\mathbbm {1}}_{[\eta _l=T]}\xi )\Big |{\mathcal {F}}_t\bigg ] \end{aligned}$$

and (A.16) follows by letting $l\rightarrow \infty$ and using dominated convergence. Optimality of $D_t$ follows similarly, implying that (A.15) holds. $\square$