Zero-sum stochastic differential game in finite horizon involving impulse controls

TL;DR

This paper studies a two-player zero-sum stochastic differential game with impulse controls over a finite horizon, establishing existence, uniqueness, and the equivalence of value functions via viscosity solutions of the HJBI PDE.

Contribution

It introduces a framework for analyzing such games with weak assumptions on cost functions and proves key properties of the value functions.

Findings

Existence and uniqueness of the solution to the HJBI PDE.

The upper and lower value functions coincide.

Validation of the dynamic programming principle for the game.

Abstract

This paper considers the problem of two-player zero-sum stochastic differential game with both players adopting impulse controls in finite horizon under rather weak assumptions on the cost functions ($c$ and $\chi$ not decreasing in time). We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation (PDE) of the game. We prove that the upper and lower value functions coincide.

Equations

$$\begin{array}[]{ll}X_{s}=x+\displaystyle\int_{t}^{s}b(r,X_{r})dr+\displaystyle\int_{t}^{s}\sigma(r,X_{r})dW_{r}+\sum\limits_{m\geq 1}\xi_{m}1\!\!1_{[\tau_{m},T]}(s)\prod_{l\geq 1}1\!\!1_{\{\tau_{m}\neq\rho_{l}\}}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum\limits_{l\geq 1}\eta_{l}1\!\!1_{[\rho_{l},T]}(s),\qquad\qquad s\geq t,\end{array}$$

$$u(s)=\sum _{m\ge 1}{\xi }_{m}11_{[{\tau }_m,T]}(s)\text{mbox}andv(s)=\sum _{l\ge 1}{\eta }_{l}11_{[{\rho }_l,T]}(s)$$

$$\begin{array}[]{ll}J(t,x;u,v)=\mathbb{E}\bigg{[}\displaystyle\int_{t}^{T}f(s,X_{s}^{t,x,u,v})ds-\sum\limits_{m\geq 1}c(\tau_{m},\xi_{m})1\!\!1_{[\tau_{m}\leq T]}\prod_{l\geq 1}1\!\!1_{\{\tau_{m}\neq\rho_{l}\}}\\ \qquad\qquad\qquad\qquad+\sum\limits_{l\geq 1}\chi(\rho_{l},\eta_{l})1\!\!1_{[\rho_{l}\leq T]}+g(X_{T}^{t,x,u,v})\bigg{]},\end{array}$$

$$c(t,y)\ge c(t^{\prime },y)\text{mbox}and\chi (t,z)\ge \chi (t^{\prime },z),$$

$$\left\{\begin{array}[]{c}max\Big{\{}min\Big{[}-\cfrac{\partial V}{\partial t}-\mathcal{L}V-f,V-H^{c}_{sup}V\Big{]},V-H^{\chi}_{inf}V\Big{\}}=0\qquad[0,T)\times\mathbb{R}^{n}\\ \\ V(T,x)=g(x)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall x\in\mathbb{R}^{n}.\end{array}\right.$$

$$H_{sup}^{c}V(t,x)=\underset{\xi \in \mathcal{U}}{\sup }[V(t,x+\xi )-c(t,\xi )],H_{inf}^{\chi }V(t,x)=\underset{\eta \in \mathcal{V}}{\inf }[V(t,x+\eta )+\chi (t,\eta )],$$

$$|\sigma (t,x)-\sigma (t,x^{\prime })|+|b(t,x)-b(t,x^{\prime })|\le C|x-x^{\prime }|.$$

$$|\sigma (t,x)|+|b(t,x)|\le C(1+|x|).$$

$$\underset{[0,T]\times \mathcal{U}}{\inf }c\ge k,\underset{[0,T]\times \mathcal{V}}{\inf }\chi \ge k,$$

$$c(t,{\xi }_1+{\xi }_2)\le c(t,{\xi }_1)+c(t,{\xi }_2)$$

$$\chi (t,{\eta }_1+{\eta }_2)\le \chi (t,{\eta }_1)+\chi (t,{\eta }_2)$$

$$\underset{\xi \in \mathcal{U}}{\sup }[g(x+\xi )-c(T,\xi )]\le g(x)\le \underset{\eta \in \mathcal{V}}{\inf }[g(x+\eta )+\chi (T,\eta )].$$

$$\left\{\begin{array}[]{c}max\Big{\{}min\Big{[}-\cfrac{\partial V}{\partial t}-\mathcal{L}V-f,V-H^{c}_{sup}V\Big{]},V-H^{\chi}_{inf}V\Big{\}}=0\qquad[0,T)\times\mathbb{R}^{n},\\ \\ V(T,x)=g(x)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall x\in\mathbb{R}^{n}.\end{array}\right.$$

$$\mathcal{L}V=⟨b,{\nabla }_{x}V⟩+\frac{1}{2}tr[\sigma {\sigma }^{\ast }{\nabla }_x^{2}V],$$

$$H_{sup}^{c}V(t,x)=\underset{\xi \in \mathcal{U}}{\sup }[V(t,x+\xi )-c(t,\xi )],H_{inf}^{\chi }V(t,x)=\underset{\eta \in \mathcal{V}}{\inf }[V(t,x+\eta )+\chi (t,\eta )],$$

$$\begin{array}[]{ll}max\Big{\{}min\Big{[}-\cfrac{\partial\phi}{\partial t}(t_{0},x_{0})-\mathcal{L}\phi(t_{0},x_{0})-f(t_{0},x_{0}),V(t_{0},x_{0})-H^{c}_{sup}V(t_{0},x_{0})\Big{]},\\ \qquad\qquad\qquad V(t_{0},x_{0})-H^{\chi}_{inf}V(t_{0},x_{0})\Big{\}}\leq 0.\end{array}$$

$$\begin{array}[]{ll}max\Big{\{}min\Big{[}-\cfrac{\partial\phi}{\partial t}(t_{0},x_{0})-\mathcal{L}\phi(t_{0},x_{0})-f(t_{0},x_{0}),V(t_{0},x_{0})-H^{c}_{sup}V(t_{0},x_{0})\Big{]},\\ \qquad\qquad\qquad V(t_{0},x_{0})-H^{\chi}_{inf}V(t_{0},x_{0})\Big{\}}\geq 0.\end{array}$$

$$\begin{array}[]{lll}V(s,y)\leq V(t,x)+p(s-t)+\langle q,y-x\rangle+\displaystyle\frac{1}{2}\langle X(y-x),y-x\rangle+o(|s-t|+|y-x|^{2})\end{array}$$

$$\begin{array}[]{lll}\big{(}\text{resp.}\\ V(s,y)\geq V(t,x)+p(s-t)+\langle q,y-x\rangle+\displaystyle\frac{1}{2}\langle X(y-x),y-x\rangle+o(|s-t|+|y-x|^{2})\big{)}.\Box\end{array}$$

$$(D_t\varphi (t,x),D_x\varphi (t,x),D_{xx}^2\varphi (t,x))\in J^{2,-}V(t,x)(\text{resp}.J^{2,+}V(t,x)).\square$$

$$\begin{array}[]{ll}max\Big{\{}min\Big{[}-p-\langle b,q\rangle-\displaystyle\frac{1}{2}Tr\big{[}\sigma^{*}X\sigma\big{]}-f(t,x),V(t,x)-H^{c}_{sup}V(t,x)\Big{]},\\ \qquad\qquad\qquad V(t,x)-H^{\chi}_{inf}V(t,x)\Big{\}}\geq 0\qquad(\text{resp.,}\leq 0).\end{array}$$

$$\begin{array}[]{ll}X_{s}=x+\displaystyle\int_{t}^{s}b(r,X_{r})dr+\displaystyle\int_{t}^{s}\sigma(r,X_{r})dW_{r}+\sum\limits_{m\geq 1}\xi_{m}1\!\!1_{[\tau_{m},T]}(s)\prod_{l\geq 1}1\!\!1_{\{\tau_{m}\neq\rho_{l}\}}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum\limits_{l\geq 1}\eta_{l}1\!\!1_{[\rho_{l},T]}(s),\qquad\qquad s\geq t,\end{array}$$

$$\begin{array}[]{ll}J(t,x,u,v):=\mathbb{E}\bigg{[}\displaystyle\int_{t}^{T}f(s,X_{s}^{t,x,u,v})ds-\sum\limits_{m\geq 1}c(\tau_{m},\xi_{m})1\!\!1_{[\tau_{m}\leq T]}\prod_{l\geq 1}1\!\!1_{\{\tau_{m}\neq\rho_{l}\}}\\ \qquad\qquad\qquad\qquad+\sum\limits_{l\geq 1}\chi(\rho_{l},\eta_{l})1\!\!1_{[\rho_{l}\leq T]}+g(X_{T}^{t,x,u,v})\bigg{]},\end{array}$$

$$u_{[\tau ,\sigma ]}(s)=\sum _{m\ge 1}{\xi }_{{\mu }_{t,\tau }(u)+m}11_{[{\tau }_{{\mu }_{t,\tau }(u)+m}\le s\le \sigma ]}(s),\tau \le s\le \sigma ,$$

$${\mu }_{t,\tau }(u):=\sum _{m\ge 1}11_{[{\tau }_m\le \tau ]}.$$

$$\mathbb{E}[{\mu }_{t,T}(u)]<\infty (resp.,\mathbb{E}[{\mu }_{t,T}(v)]<\infty ).$$

$$\mathbb{P}(\{u=u^{\prime }\text{a.e. on}[t,T]\})=1.$$

$$\begin{array}[]{ll}\text{if}\quad v_{1}\equiv v_{2}\quad\text{on}\quad[t,\tau],\quad\text{then}\\ \\ \;\alpha(v_{1})\equiv\alpha(v_{2})\quad\text{on}\quad[t,\tau].\end{array}$$

$$V^{-}(t,x):=\underset{\beta \in {\mathcal{B}}_{t,T}}{\inf }\underset{u\in {\mathcal{U}}_{t,T}}{\sup }J(t,x,u,\beta (u))$$

$$V^{+}(t,x):=\underset{\alpha \in {\mathcal{A}}_{t,T}}{\sup }\underset{v\in {\mathcal{V}}_{t,T}}{\inf }J(t,x,\alpha (v),v)$$

Full text

AMS subject classifications: 93E20, 49L20, 49L25, 49N70

$\bf Keywords$: stochastic differential game, impulse control, quasi-variational inequality, viscosity solution

1 Introduction

In this paper we consider the state process of the stochastic differential game, defined as the solution of the following stochastic equation:

[TABLE]

for all $s\in[t,T]$, P-a.s., with $X_{t^{-}}=x$. Here $W$ is a d-dimensional Wiener process, while

[TABLE]

are the impulse controls of player I and player II, respectively. The random variables $\xi_{m}$ and $\eta_{l}$ take values in two convex cones $\mathcal{U}$ and $\mathcal{V}$ of $\mathbb{R}^{n}$, respectively, called the spaces of control actions. The infinite product $\prod_{l\geq 1}1\!\!1_{\{\tau_{m}\neq\rho_{l}\}}$ has the following meaning: When the two players act together on the system at the same time, we take into account only the action of player II. We denote by $X^{t,x,u,v}=\{X^{t,x,u,v}_{s},t\leq s\leq T\}$ the state trajectory of the game with initial time $t$, initial state $x$, and impulse controls $u$ and $v$. The gain functional for player I (resp., cost functional for player II) is given by

[TABLE]

$f$ is the running gain and $g$ is the payoff. The function $c$ is the cost function for player I and is a gain function for player II, meaning that when player I performs an action he/she has to pay a cost, resulting in a gain for player II. Analogously, $\chi$ is the cost function for player II and is a gain function for player I. The zero-sum stochastic differential games problems we will investigate is to show the upper and lower value functions coincide and the game admits a value.

The stochastic differential games problem have recently attracted a lot of research activities, especially in connection with mathematical finance, commodities, and in particular energy markets, etc (see e.g. [1, 3, 4, 9, 15, 16, 18, 19, 20, 23, 24, 25] and the references therein). In order to tackle those problems, authors use mainly two approaches. Either a probabilistic one [8, 13] or an approach which uses the Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation (PDE)[17].

In the finite horizon framework, Cosso [6] have studied a two-player zero-sum stochastic differential game with both players adopting impulse controls using viscosity solutions to the HJBI equation. They have proved existence and uniqueness for the HJBI equation under stronger constraint (see also Tang and Yong [28]): the cost functions $c$ and $\chi$ are decreasing in time:

[TABLE]

for every $0\leq t\leq t^{\prime}\leq T$, $y\in\mathcal{U}$ and $z\in\mathcal{V}$.

Our aim in this work lies in the fact that we investigate the solution to the zero-sum stochastic differential games under rather weak assumptions on the cost functions($c$ and $\chi$ are not decreasing in time). Therefore the main objective of our work, and this is the novelty of the paper is to characterize the value function as the only solution in viscosity sense of the associated Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation (PDE) for the finite horizon problem.

We prove the lower and the upper value functions of stochastic differential game satisfy the dynamic programming principle. We show that the HJBI equation associated to the stochastic differential game, which turns out to be the same for the two value functions because the two players cannot act simultaneously on the system, is the unique solution of the following system:

[TABLE]

Where $\mathcal{L}$ is the second-order local operator, and the nonlocal operators $H_{\sup}^{c}$ and $H_{\inf}^{\chi}$ are given by

[TABLE]

for every $(t,x)\in[0,T]\times\mathbb{R}^{n}.$

This paper is organized as follows: In Section 2, we formulate the problem and we give the related definitions. In Section 3, we shall introduce the stochastic differential game problem and give some preliminary results of the lower and the upper value functions of stochastic differential game. Further we provide some estimate for the optimal strategy of problem which in combination with the dynamic programming principle plays a crucial role in the proof of existence the value functions. In Section 4, we prove the dynamic programming principle. Further, we show the existence and continuity of the lower and the upper value functions. Section 5, is devoted to the connection between the zero sum stochastic differential game problem and Hamilton-Jacobi-Bellman-Isaacs equation. In Section 6, we show that the solution of HJBI is unique in the subclass of bounded continuous functions. Further, the upper and the lower value functions coincide and the game admits a value.

2 Assumptions and formulation of the problem

Throughout this paper $T$ (resp. $n,\,d$) is a fixed real (resp. integers) positive numbers.

Let us assume the following assumptions:

(H1) $b:[0,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and $\sigma:[0,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times d}$ be two continuous functions for which there exists a constant $C>0$ such that for any $t\in[0,T]$ and $x,x^{\prime}\in\mathbb{R}^{n}$

[TABLE]

Also there exists a constant $C>0$ such that for any $(t,x)\in[0,T]\times\mathbb{R}^{n}$

[TABLE]

(H2) $f:[0,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ is uniformly continuous and bounded on $[0,T]\times\mathbb{R}^{n}$. $g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is uniformly continuous and bounded on $\mathbb{R}^{n}$.

(H3) The cost functions $c:[0,T]\times\mathcal{U}\rightarrow\mathbb{R}$ and $\chi:[0,T]\times\mathcal{V}\rightarrow\mathbb{R}$ are measurable and uniformly continuous. Furthermore

[TABLE]

where $k>0$.

Moreover,

[TABLE]

[TABLE]

for every $t\in[0,T],\xi_{1},\xi_{2}\in\mathcal{U}$ and $\eta_{1},\eta_{2}\in\mathcal{V}$.

(H4) (no terminal impulse). For any $x\in\mathbb{R}^{n},\eta\in\mathcal{V}$ and $\xi\in\mathcal{U}$

[TABLE]

$\mathcal{U}$ and $\mathcal{V}$ are two convex cones of $\mathbb{R}^{n}$ with $\mathcal{U}\subset\mathcal{V}$.

Remark 1

The above assumptions (2.4) and (2.5) ensures that multiple impulses occurring at the same time are suboptimal. $\Box$

We now consider the HJBI equation:

[TABLE]

Where $\mathcal{L}$ is the second-order local operator

[TABLE]

and the nonlocal operators $H_{\sup}^{c}$ and $H_{\inf}^{\chi}$ are given by

[TABLE]

for every $(t,x)\in[0,T]\times\mathbb{R}^{n}.$

The main objective of this paper is to focus on the existence and uniqueness of the solution in viscosity sense of (2.7) whose definition is:

Definition 1

Let $V$ be a continuous function defined on $[0,T]\times\mathbb{R}^{n}$ and such that $V(T,x)=g(x)$ for any $x\in\mathbb{R}^{n}$. The $V$ is called:

(i) A viscosity subsolution of (2.7) if for any $(t_{0},x_{0})\in[0,T)\times\mathbb{R}^{n}$ and any function $\phi\in C^{1,2}([0,T)\times\mathbb{R}^{n})$, such that $(t_{0},x_{0})$ is a local maximum of $V-\phi$, we have:

[TABLE]

(ii) A viscosity supersolution of (2.7) if for any $(t_{0},x_{0})\in[0,T)\times\mathbb{R}^{n}$ and any function $\phi\in C^{1,2}([0,T)\times\mathbb{R}^{n})$, such that is $(t_{0},x_{0})$ a local minimum of $V-\phi$, we have:

[TABLE]

(iii) A viscosity solution if it is both a viscosity supersolution and subsolution. $\Box$

There is an equivalent formulation of this definition (see e.g. [7]) which we give because it will be useful later. So firstly, we define the notions of superjet and subjet of a continuous function $V$.

Definition 2

Let $V\in C((0,T)\times\mathbb{R}^{n}),(t,x)$ an element of $(0,T)\times\mathbb{R}^{n}$ and finally $S_{n}$ the set of $n\times n$ symmetric matrices. We denote by $J^{2,+}V(t,x)$ (resp. $J^{2,-}V(t,x))$, the superjets (resp. the subjets) of $\,V$ at $(t,x)$, the set of triples $(p,q,X)\in\mathbb{R}\times\mathbb{R}^{n}\times S_{n}$ such that:

[TABLE]

[TABLE]

Note that if $\phi-V$ has a local maximum (resp. minimum) at $(t,x)$, then we obviously have:

[TABLE]

We now give an equivalent definition of a viscosity solution of HJBI equation(2.7):

Definition 3

Let $V$ be a continuous function defined on $[0,T]\times\mathbb{R}^{n}$ and such that $V(T,x)=g(x)$ for any $x\in\mathbb{R}^{n}$. Then V is a viscosity supersolution (resp., subsolution) to the HJBI equation (2.7) if and only if for every $(t,x)\in[0,T)\times\mathbb{R}^{n}$ and $(p,q,X)\in J^{2,-}V(t,x)\,($resp. $J^{2,+}V(t,x))$,

[TABLE]

It is called a viscosity solution it is both a viscosity subsolution and supersolution. $\Box$

As pointed out previously we will show that system (2.7) has a unique solution in viscosity sense. This system is the deterministic version of the stochastic differential game problem will describe briefly in the next section.

3 The stochastic differential game problem

3.1 Setting of the problem

Let ($\Omega,\mathcal{F},\mathbb{P})$ is a fixed probability space on which is defined a standard $d$-dimensional Brownian motion $W=(W_{t})_{t\leq T}$, whose natural filtration is $(\mathcal{F}^{0}_{t}:=\sigma\{W_{s};s\leq t\})_{0\leq t\leq T}.$ We denote by $\mathbb{F}=(\mathcal{F}_{t})_{t\leq T}$ the completed filtration of $(\mathcal{F}^{0}_{t})_{t\leq T}$ with the $\mathbb{P}$-null sets of $\mathcal{F}$. We are given two convex cones $\mathcal{U}$ and $\mathcal{V}$ of $\mathbb{R}^{n}$, with $\mathcal{V}\subset\mathcal{U}$. We call $\mathcal{U}$ and $\mathcal{V}$ the spaces of controle actions. We begin by introducing the concept of impulse control.

Definition 4

An impulse control $u=\sum\limits_{m\geq 1}\xi_{m}1\!\!1_{[\tau_{m},T]}$ for player I (resp, $v=\sum\limits_{l\geq 1}\eta_{l}1\!\!1_{[\rho_{l},T]}$ for player II) on $[t,T]\subset\mathbb{R}^{+}=[0,+\infty)$, is such that:

(i) $(\tau_{m})_{m}$ (resp., $(\rho_{l})_{l})$, the action times, is a sequence of $\mathbb{F}$-stopping times, valued in $[t,T]\cup\{+\infty\}$ such that $\mathbb{P}$-a.s. $\tau_{m}\leq\tau_{m+1}$ (resp.,  $\rho_{l}\leq\rho_{l+1})$.

(ii) $(\xi_{m})_{m}$ (resp., $(\eta_{l})_{l})$, the actions, is a sequence of $\mathcal{U}$-valued (resp, $\mathcal{V}$-valued) random variables, where each $\xi_{m}$ (resp., $\eta_{l}$) is $\mathcal{F}_{\tau_{m}}$-measurable (resp., $\mathcal{F}_{\rho_{l}}$-measurable).

Let $t\in[0,T]$ be the initial time of the game and $x\in\mathbb{R}^{n}$ the initial state. Then, given the impulse controls $u$ and $v$ on $[t,T]$, the state process of the stochastic differential game is defined as the solution to the following stochastic equation:

[TABLE]

Then, under (H1), the stochastic differential equation (3.1) admits a unique solution $X^{t,x,u,v}=\{X^{t,x,u,v}_{s},t\leq s\leq T\}$ for every $(t,x)\in$[0,T]$\times\mathbb{R}^{n},u\in\mathcal{U}$ and $v\in\mathcal{V}$. The gain functional for player I (resp., cost functional for player II) is given by

[TABLE]

$f$ is the running gain and $g$ is the payoff. The function $c$ is the cost function for player I and is a gain function for player II, meaning that when player I performs an action he/she has to pay a cost, resulting in a gain for player II. Analogously, $\chi$ is the cost function for player II and is a gain function for player I.

Definition 5

Let $u=\sum\limits_{m\geq 1}\xi_{m}1\!\!1_{[\tau_{m},T]}$ be an impulse control on $[t,T]$, and let $\tau\leq\sigma$ be two $[t,T]$-valued $\mathbb{F}$-stopping times. Then we define the restriction $u_{[\tau,\sigma]}$ of the impulse control $u$ by:

[TABLE]

$\mu_{t,\tau}$* is the number of impulses up to time $\tau$, i.e.,*

[TABLE]

Definition 6

(Admissible impulse control). An admissible impulse control u for player I (resp., $v$ for player II) on $[t,T]\subset\mathbb{R}^{+},$ is an impulse control for player I (resp., II) on $[t,T]$ with a finite average number of impulses, i.e.,

[TABLE]

The set of all admissible impulse controls for player I (resp., II) on $[t,T]$ is denoted by $\mathcal{U}_{t,T}$ (resp., $\mathcal{V}_{t,T})$.

Next, we adopt the notion of Control identification and nonanticipative strategy.

Definition 7

*(Control identification).

Let $u=\sum\limits_{m\geq 1}\xi_{m}1\!\!1_{[\tau_{m},T]}$ and $u^{\prime}=\sum\limits_{m\geq 1}\xi^{\prime}_{m}1\!\!1_{[\tau^{\prime}_{m},T]}$ in $\mathcal{U}_{t,T},$ we write $u\equiv u^{\prime}$ on $[t,T]$ if*

[TABLE]

Similarly, we interpret $v\equiv v^{\prime}$ on $[t,T]$ in $\mathcal{V}_{t,T}.$

Definition 8

(nonanticipative strategy). The nonanticipative strategy set $\mathcal{A}_{t,T}$ for player I is the collection of all nonanticipative maps $\alpha$ from $\mathcal{V}_{t,T}$ to $\mathcal{U}_{t,T}$, i.e for any $[t,T]$-valued $\mathbb{F}$-stopping times $\tau$ and any $v_{1}$,$v_{2}\in\mathcal{V}_{t,T}$,

[TABLE]

Analogously, the nonanticipative strategy set $\mathcal{B}_{t,T}$ for player II is the collection of all nonanticipative maps $\beta$ from $\mathcal{U}_{t,T}$ to $\mathcal{V}_{t,T}$.

We are now ready to introduce the upper and lower values of the game:

For every $(t,x)\in[0,T]\times\mathbb{R}^{n}$ let us set

[TABLE]

and

[TABLE]

The maps $V^{-}$ and $V^{+}$ are called the lower value and the upper value of the game, respectively. The game is said to admit a value if  $V^{-}=V^{+}$.

The HJBI equation associated to the stochastic differential game, which turns out to be the same for the two value functions because the two players cannot act simultaneously on the system, is given by (2.7).

Remark 2

The infinite product $\prod_{l\geq 1}1\!\!1_{\{\tau_{m}\neq\rho_{l}\}}$ in (3.1) means that when the two players act together on the system, we take into account only the action of player II. When take into account only the action of player I instead of player II. Then, using arguments analogous to those presented in the sections below, it can be proved that, with this assumption, the corresponding HJBI equation is given by

[TABLE]

3.2 Preliminary results

In this section we present some properties of the lower and upper value functions of our differential game.

We begin by the following lemma, which is concerned with the continuous dependence of $X^{t,x,u,v}$ with respect to $x$.

Lemma 1

Under assumption (H1) there exists a constant $C>0$ such that, for every $t\in[0,T],\,x,x^{\prime}\in\mathbb{R}^{n},u\in\mathcal{U}_{t,T}$ and $v\in\mathcal{V}_{t,T}$ we have:

[TABLE]

$Proof$. see [[26], Appendix]

Next, in the following proposition, we prove that the two value functions are bounded.

Theorem 1

Under the standing assumptions (H1), (H2) and (H3), the lower and upper value functions are bounded.

$Proof$: We make the proof only for the lower value function $V^{-}$, the other case being analogous.

Using the definition of lower value function, we have, for $(t,x)\in[0,T]\times\mathbb{R}^{n}$

[TABLE]

where $v_{0}$ is the control with no impulses.

Let $\epsilon>0$, then there exists a strategy $u^{\epsilon}\in\mathcal{U}_{t,T}$ such that

[TABLE]

Since the cost $c(\tau^{\epsilon}_{m},\xi^{\epsilon}_{m})$ are non negative functions, then we have

[TABLE]

Therefore we find, using also the boundedness of $f$ and $g$, that there exists a constant $C>0$ such that

[TABLE]

In a similar way we can prove that there exists a constant $C>0$ such that

[TABLE]

from which we deduce the thesis.$\quad\Box$

We are now giving some properties of nearly nearly optimal strategies.

Proposition 1

Let $u\in\mathcal{U}_{t,T}$ and $v\in\mathcal{V}_{t,T}$ be a nearly optimal strategies composed of impulses control $(\tau,\xi)=((\tau_{n})_{n\geq 1},(\xi_{n})_{n\geq 1})$ and $(\rho,\eta)=((\rho_{n})_{n\geq 1},(\eta_{n})_{n\geq 1}).$ Then:

[TABLE]

There exists a positive constant $C>0$ which does not depend on $t$ and $x$ such that:

[TABLE]

We denote by $\widehat{\mathcal{U}}_{t,T}$ and $\widehat{\mathcal{V}}_{t,T}$ the set which satisfies the conditions (3.9) and (3.10), respectively. Similarly, by $\widehat{\mathcal{A}}_{t,T}$ and $\widehat{\mathcal{B}}_{t,T}$ the sets that contain all the nonanticipative strategies with values in $\widehat{\mathcal{U}}_{t,T}$ and $\widehat{\mathcal{V}}_{t,T}$, respectively.

$Proof$. Let us choose a nearly optimal strategy $u\in\mathcal{U}_{t,T}$ composed of impulse control $((\tau_{n})_{n\geq 1},(\xi_{n})_{n\geq 1})$ such that, for $(t,x)\in[0,T]\times\mathbb{R}^{n}$,

[TABLE]

Since $V^{-}$, $f$ and $g$ are bounded, then we have

[TABLE]

Next we show $(\ref{condition2})$. Taking into account that $c(\tau,\xi)\geq k>0$ for any $(\tau,\xi)\in[0,T]\times\mathcal{U}_{t,T}$, we obtain:

[TABLE]

But for any $m\leq n,\,[\tau_{n}\leq T]\subseteq[\tau_{m}\leq T]$, then:

[TABLE]

Finally taking into account $k>0$, we obtain the desired result.

The other case being analogous.                      $\Box$

Corollary 1

Under the standing assumptions (H1), (H2) and (H3), $J(t,x,u,v)$ is bounded for every $(t,x)\in[0,T]\times\mathbb{R}^{n},u\in\widehat{\mathcal{U}}_{t,T}$ and $v\in\widehat{\mathcal{V}}_{t,T}$.

In the following proposition, using Lemma 1, we prove that the lower and upper value functions are continuous in the state variable, together with the gain functional.

Proposition 2

The gain functional, lower and upper value functions are continuous in $x$.

$Proof.$ It is enough to show that the conclusion holds for the gain functional $J$.

For every $t\in[0,T],\,x,x^{\prime}\in\mathbb{R}^{n},u\in\mathcal{U}_{t,T},v\in\mathcal{V}_{t,T}$, we have

[TABLE]

From Lemma 1 and continuity of $f$ and $g$ in $x$ we get the thesis.    $\Box$

4 The value functions

4.1 Dynamic Programming Principle

In this section we present the dynamic programming principle (DPP) for the stochastic differential game. We begin with the following lemma.

Lemma 2

([6], Lemma 3.3). The lower and upper value functions are given by

[TABLE]

and

[TABLE]

for every $(t,x)\in[0,T)\times\mathbb{R}^{n}$, where $\bar{\mathcal{U}}_{t,T}$ and $\bar{\mathcal{V}}_{t,T}$ contain all the impulse controls in $\widehat{\mathcal{U}}_{t,T}$ and $\widehat{\mathcal{V}}_{t,T}$, respectively, which have no impulses at time t. Similarly, $\bar{\mathcal{A}}_{t,T}$ and $\bar{{\mathcal{B}}}_{t,T}$ are subsets of $\widehat{\mathcal{A}}_{t,T}$ and $\widehat{\mathcal{B}}_{t,T}$, respectively. In particular, they contain all the nonanticipative strategies with values in $\bar{\mathcal{U}}_{t,T}$ and $\bar{\mathcal{V}}_{t,T}$, respectively.

Theorem 2

Under assumptions (H1), (H2) and (H3), given $0\leq t\leq s\leq T$, $x\in\mathbb{R}^{n}$, and each family of $\mathbb{Q}_{[t,T]}-$valued $\mathcal{F}_{t,s}$-stopping times $\{\tau^{u,v},(u,v)\in\widehat{\mathcal{U}}_{t,T}\times\widehat{\mathcal{V}}_{t,T}\}$, we have

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where $\mathbb{Q}_{[t,T]}=(\mathbb{Q}\cap[t,T])\cup\{t,T\},$ and $V_{*}$ (resp., $V^{*}$) its lower (resp. upper) semicontinuous envelope defined by:

[TABLE]

$Proof$. We prove the dynamic programming principle only for $V^{-}$, the other case being analogous.

Let $\epsilon>0$ and consider an arbitrary function

[TABLE]

For each $(s,y)\in[0,T]\times\mathbb{R}^{n}$, there exists $\beta^{\epsilon}\in\bar{\mathcal{B}}_{s,T}$ such that

[TABLE]

where $u_{[s,T]}$ is as introduced in Definition 5. Let $\mathbb{Q}_{[t,T]}:=\{t_{i}\}_{i\geq 1}$ and fix one of the points $t_{i}$ in time. For each $y\in\mathbb{R}^{n}$, the continuity of $J$ established in Proposition 2 and that of $\phi$ imply that there exists a neighborhood $B^{i}(y)\subseteq\mathbb{R}^{n}$ (of size depending on $y,i,\epsilon$) such that

[TABLE]

for all $y^{\prime}\in B^{i}(y)$ and $v\in\bar{\mathcal{V}}_{t_{i},T}$.

Therefore, since the family $\{B^{i}(y):y\in\mathbb{R}^{n}\}$ forms an open cover of $\mathbb{R}^{n}$, there exists a sequence $(y_{j})_{j\geq 1}$ in $\mathbb{R}^{n}$ such that $\{B^{i}(y_{j})\}_{j\geq 1}$ is a countable subcover of $\mathbb{R}^{n}$. We set $\beta_{i,j}:=\beta^{t_{i},y_{j}}\in\bar{\mathcal{B}}_{t_{i},T}$ and $B^{i}_{j}:=B^{i}(y_{j})$.

We can now define, for $i$ still being fixed, a measurable partition $(A^{i}_{j})_{j\geq 1}$ by

[TABLE]

Since $A^{i}_{j}\subseteq B^{i}_{j}$, the inequalities $(\ref{progeq1})$ and $(\ref{progeq2})$ yield that

[TABLE]

Now, let $(u,\beta)\in\widehat{\mathcal{U}}_{t,T}\times\widehat{\mathcal{B}}_{t,T}$ be arbitrary and set $\tau=\tau^{\beta(u),u}$. Fix an integer $k\geq 1$, we now focus on $(t_{i})_{1\leq i\leq k}.$ We may assume that $t_{1}<t_{2}<...<t_{k},$ by eliminating and relabeling some of the $t_{i}$. We define the $\mathcal{F}^{t}_{\tau}$-measurable sets

[TABLE]

Since the $t_{i}$ are distinct and $A^{i}_{j}\cap A^{i}_{j^{\prime}}=\emptyset$ for $j\neq j^{\prime}$, we have $\Gamma^{i}_{j}\cap\Gamma^{i^{\prime}}_{j^{\prime}}=\emptyset$ for $(i,j)\neq(i^{\prime},j^{\prime}).$ We construct the strategy $\beta^{k}$ by

[TABLE]

Then

[TABLE]

We deduce via $(\ref{progeq3})$ that

[TABLE]

for every $k\geq 1$. Letting $k\rightarrow\infty$, therefore,

[TABLE]

by dominated convergence and Corollary 1. Moreover, monotone convergence yields

[TABLE]

Therefore, we deduce the existence of an integer $k_{0}\geq 1$ such that

[TABLE]

Let a sequence of continuous functions $(\phi_{n})_{n}$ such that $\phi_{n}\geq(V^{-})^{*}$ for all $n\geq 1$ and such that $\phi_{n}$ converges pointwise to $(V^{-})^{*}$. Set $\varphi_{N}:=max_{n\geq N}\phi_{n}$ for $N\geq 1$ and observe that the sequence $(\varphi_{N})_{N}$ is decreasing and converges pointwise to $(V^{-})^{*}.$ By the monotone convergence theorem, we then obtain:

[TABLE]

Therefore, we have

[TABLE]

As $\beta$ and $\epsilon$ are arbitrary then sending $\epsilon\to 0,$ and take infimums in right-hand side of $(\ref{progeq6})$, to obtain:

[TABLE]

In a similar way we can prove the reverse inequality, hence deducing the thesis.     $\Box$

4.2 Continuity of value functions in time

In this section we prove the continuity of  lower value function and upper value function in t .

First, we know that any stopping time $\tau$ can be approximated by a sequence of rational stopping times $\tau^{(n)}=2^{-n}([2^{n}\tau]-1)$. Then we can pose

(H5) The action times $\rho_{l}$ and $\tau_{m}$ takes values in $\mathbb{Q}_{[t,T]}$

Theorem 3

The lower and upper value functions are continuous in $t$.

$Proof.$ We make the proof only for $V^{-}$, the other case being analogous.

First let us show that $V^{-}$ is lower semi-continuous.

Recall the characterization of dynamical programming principle that reads as

[TABLE]

[TABLE]

Fix an arbitrary $\epsilon>0$, and we assume that $t<t^{\prime}$. Pick $\beta^{\epsilon}\in\widehat{\mathcal{B}}_{t,T}$ such that

[TABLE]

where $u\in\widehat{\mathcal{U}}_{t,T}$ will be chosen later. On the other hand, pick $u^{\epsilon}\in\widehat{\mathcal{U}}_{t^{\prime},T}$ such that

[TABLE]

where

[TABLE]

Therefore

[TABLE]

We note that, there exists a constant $C>0$ such that

[TABLE]

Also, taking the limit as $t\rightarrow t^{\prime}$, and using the uniform continuity of $f$, $g$ and $\chi$ to obtain:

[TABLE]

As n and $\epsilon$ are arbitrary then sending $\epsilon\rightarrow 0$ and $n\rightarrow+\infty,$ to obtain:

[TABLE]

Therefore $V^{-}$ is lower semi-continuous.

Now we show that $V^{-}$ is upper semi-continuous.

Fix an arbitrary $\epsilon>0$, and we assume that $t<t^{\prime}$.

Pick $u^{\epsilon}\in\widehat{\mathcal{U}}_{t,T}$ such that

[TABLE]

where $\beta\in\widehat{\mathcal{B}}_{t,T}$ will be chosen later. On the other hand, pick $\beta^{\epsilon}$ belongs to $\bar{\mathcal{B}}_{t^{\prime},T}$ such that

[TABLE]

where

[TABLE]

Therefore

[TABLE]

[TABLE]

We note that, there exists a constant $C>0$ such that

[TABLE]

Also, taking the limit as $t\rightarrow t^{\prime}$, and using the uniform continuity of $f$, $g$ and $c$ to obtain:

[TABLE]

As n and $\epsilon$ are arbitrary then sending $\epsilon\rightarrow 0$ and $n\rightarrow+\infty,$ to obtain:

[TABLE]

Therefore $V^{-}$ is upper semi-continuous. We then proved that $V^{-}$ is continuous in $t$.  $\Box$

Corollary 2

The lower (resp,. upper) value functions are continuous on $[0,T]\times\mathbb{R}^{n},$ therefore we have $V^{-}=(V^{-})_{*}=(V^{-})^{*}$ (resp,. $V^{+}=(V^{+})_{*}=(V^{+})^{*}$). As a consequence, $V^{-}$ (resp,. $V^{+}$) satisfies the classical dynamic programming principle: given $0\leq t\leq s\leq T$, $x\in\mathbb{R}^{n}$, and each family of $\mathbb{Q}_{[t,T]}-$valued $\mathcal{F}_{t,s}$-stopping times $\{\tau^{u,v},(u,v)\in\widehat{\mathcal{U}}_{t,T}\times\widehat{\mathcal{V}}_{t,T}\}$, we have

[TABLE]

and

[TABLE]

5 Hamilton-Jacobi-Bellman-Isaacs Equation

In this section we prove that the two value functions are viscosity solutions of the Hamilton-Jacobi-Bellman-Isaacs equation (2.7) associated to the stochastic differential game. We begin with the following Proposition.

Proposition 3

*The lower and upper value functions satisfy the following properties:

for all $t\in[0,T)$ and $x\in\mathbb{R}^{n}$,*

[TABLE]

[TABLE]

$Proof$. We make the proof only for $V^{-}$, the other case being analogous.

(i) Let $\eta\in\mathcal{V}$, for $\beta^{\prime}(u)=\eta 1\!\!1_{[t,T]}+\sum\limits_{l\geq 1}\eta^{\prime}_{l}1\!\!1_{[\rho^{\prime}_{l},T]}$ we have

[TABLE]

Choose $\beta(u)=\sum\limits_{l\geq 1}\eta^{\prime}_{l}1\!\!1_{[\rho^{\prime}_{l},T]}$. Then

[TABLE]

from which we deduce that the following inequality holds:

[TABLE]

(ii) the proof proceeds by a case distinction, for $\tau=t$ in dynamic programming principle, and the suboptimality of multiple impulses at the same time (Assumptions (2.4) and (2.5)) we have

[TABLE]

As a consequence we have

[TABLE]

If $H^{\chi}_{inf}V(t,x)>V(t,x),$ then

[TABLE]

Therefore,

[TABLE]

Now we prove that the two value functions satisfy in the viscosity sense.

Theorem 4

The lower and upper value functions are viscosity solutions to the Hamilton-Jacobi-Bellman-Isaacs equation (2.7).

$Proof.$ We give the proof for the lower value function $V^{-}$, the other case being analogous. First, we prove the supersolution property. Suppose $V^{-}-\phi$ achieves a local minimum in $[t_{0},t_{0}+\delta)\times B(x_{0},\delta)$ with $V^{-}(t_{0},x_{0})=\phi(t_{0},x_{0})$. We have by Proposition 3,

[TABLE]

If

[TABLE]

then we are done. Now suppose

[TABLE]

we prove by contradiction that

[TABLE]

Suppose otherwise, i.e.,$\,-\cfrac{\partial\phi}{\partial t}(t_{0},x_{0})-\mathcal{L}\phi(t_{0},x_{0})-f(t_{0},x_{0})<0$. Then without loss of generality we can assume that $-\cfrac{\partial\phi}{\partial t}(t,x)-\mathcal{L}\phi(t,x)-f(t,x)<0$ on $[t_{0},t_{0}+\delta)\times B(x_{0},\delta)$.

Define the stopping time $\tau$ by

[TABLE]

By Ito’s formula

[TABLE]

Let $\epsilon_{1}>0$, using the dynamic programming principle, we deduce the existence of a strategy $\beta^{\epsilon}\in\widehat{\mathcal{B}}_{t_{0},T}$ such that

[TABLE]

Therefore, without loss of generality, we only need to consider $\beta^{\epsilon}\in\widehat{\mathcal{B}}_{t_{0},T}$ such that $\rho_{1}>\tau$. Then

[TABLE]

[TABLE]

Then from (5.4) and sending $\epsilon_{1}\rightarrow 0$ we get

[TABLE]

which is a contradiction. Therefore, we must have -\bigg{(}\cfrac{\partial\phi}{\partial t}+\mathcal{L}\phi+f\bigg{)}(t_{0},x_{0})\geq 0.

Thanks to Proposition 3, we have

[TABLE]

Therefore,

[TABLE]

which is the supersolution property. The subsolution property is proved analogously.  $\Box$

Now we give an equivalent of Hamilton-Jacobi-Bellman-Isaacs equation (2.7). We consider the new function $\Gamma$ given by the classical change of variable $\Gamma(t,x)=\exp(t)V(t,x)$, for any $t\in[0,T]$ and $x\in\mathbb{R}^{n}$.

A second property is given by the:

Proposition 4

$V$* is a viscosity solution of (2.7) if and only if $\Gamma$ is a viscosity solution to the Hamilton-Jacobi-Bellman-Isaacs equation in $[0,T)\times\mathbb{R}^{n}$,*

[TABLE]

where

[TABLE]

[TABLE]

The terminal condition for $\Gamma$ is: $\Gamma(T,x)=\exp(T)g(x)$ in $\mathbb{R}^{n}.$

6 Uniqueness of the solution of Hamilton-Jacobi-

Bellman-Isaacs equation

In this section we deal with the issue of uniqueness of the solution of system (2.7) and to do so. We need to impose additional assumption of cost functions

(H6) There exists a function $h:[0,T]\to(0,\infty)$ such that for all $t\in[0,T]$,

[TABLE]

and

[TABLE]

for every $\xi_{1},\xi_{2}\in\mathcal{U}$, and $\eta,\eta_{1},\eta_{2}\in\mathcal{V}$.

To prove uniqueness viscosity solution to the HJBI equation (2.7), we begin with the following technical lemma which also appears in [[30], Lemma $4.4$], [[6], Lemma $6.1$].

Lemma 3

Suppose $U,V:[0,T]\times\mathbb{R}^{n}\to\mathbb{R}$ are uniformly continuous functions viscosity subsolution and a viscosity supersolution to the HJBI equation (2.7), respectively. Let $(t_{0},x_{0})\in[0,T]\times\mathbb{R}^{n}$, be such that

[TABLE]

or

[TABLE]

Then for every $\epsilon>0$, there exists $\bar{x}\in\mathbb{R}^{n}$ and $\delta>0$ such that

[TABLE]

and

[TABLE]

when $(t,x)\in[(t_{0}-\delta)\vee 0,t_{0}+\delta]\times\bar{B}(\bar{x},\delta),$ with $t_{0}+\delta<T.$

$Proof$: Fix $\epsilon>0,$ We divide the proof into three steps.

**Step 1 ** Let (6.3) hold. Then for $\alpha_{1}\in(0,1),$ there exists $\eta_{0}\in\mathcal{V}$ such that

[TABLE]

Then, we observe that

[TABLE]

Thus, by taking $\alpha_{1}$ sufficiently small and (6.1), we get

[TABLE]

If $U(t_{0},x_{0}+\eta_{0})>H^{c}_{sup}U(t_{0},x_{0}+\eta_{0})$, we take $(t_{0},\bar{x}):=(t_{0},x_{0}+\eta_{0}).$ Otherwise at

$(t_{0},x_{0}+\eta_{0})$ condition (6.4) holds.

On the other hand, it’s easy to check that

[TABLE]

**Step 2 ** Now, suppose that (6.4) holds at $(t_{0},x_{0})$, then for $\alpha_{2}\in(0,1)$, there exists $\xi_{0}\in\mathcal{U}$ such that

[TABLE]

Then, we have

[TABLE]

By taking $\alpha_{2}$ sufficiently small and (6.2), we can show that

[TABLE]

If $V(t_{0},x_{0}+\xi_{0})<H_{inf}^{\chi}V(t_{0},x_{0}+\xi_{0})$, we take $(t_{0},\bar{x}):=(t_{0},x_{0}+\xi_{0})$. Otherwise we can proceed as in Step 1 and we find $\eta_{0}\in\mathcal{V}$ such that

[TABLE]

[TABLE]

and

[TABLE]

On the other hand, we have

[TABLE]

Thus, by (6.2), we get

[TABLE]

Therefore we take $(t_{0},\bar{x}):=(t_{0},x_{0}+\xi_{0}+\eta_{0})$.

**Step 3 ** We can find $\alpha>0$ such that

[TABLE]

Thus, by uniform continuity of $U,V,c,$ and $\chi$ we have

[TABLE]

in which $u,v,h$ and $w$ are the modulus of continuity of $U,V,c$ and $\chi$ respectively. Therefore, there exists $\delta>0$ such that

[TABLE]

when $(t,x)\in[(t_{0}-\delta)\vee 0,t_{0}+\delta]\times\bar{B}(\bar{x},\delta),$ with $t_{0}+\delta<T.$  $\Box$

We are going now to address the question of uniqueness of the viscosity solution of Hamilton-Jacobi-Bellman-Isaacs equation (2.7). We have the following:

Theorem 5

The solution in viscosity sense of Hamilton-Jacobi-Bellman-Isaacs equation (2.7) is unique in the space of bounded uniformly continuous functions on $[0,T]\times\mathbb{R}^{n}$.

Proof. We will show by contradiction that if $U$ and $V$ is a subsolution and a supersolution respectively for $(\ref{I})$, then $U\leq V.$ Therefore if we have two solutions of $(\ref{I})$ then they are obviously equal. Actually for some $R>0$ (large enough) suppose there exists $(\hat{t},\hat{x})\in[0,T]\times B(0,R)$ such that $\sup\limits_{t,x}(U(t,x)-V(t,x))=U(\hat{t},\hat{x})-V(\hat{t},\hat{x})>0$.

**Step 1. ** Using Lemma 3, we can find $(\bar{t},\bar{x})\in[0,T)\times\mathbb{R}^{n}$ and $\delta$ such that

[TABLE]

and

[TABLE]

for all $(t,x)\in I\times\bar{B}(\bar{x},\delta)$, where $I:=[\bar{t}-\delta,\bar{t}+\delta]\subset[0,T).$

Let $(t_{0},x_{0})\in I\times\bar{B}(\bar{x},\delta)$ such that

[TABLE]

For a small $\epsilon,\beta,\theta>0$, let us define:

[TABLE]

By the boundedness of $U$ and $V$, that there exists a $(t_{\epsilon},x_{\epsilon},y_{\epsilon})\in I\times\bar{B}(\bar{x},\delta)\times\bar{B}(\bar{x},\delta)$, attaining the maximum of $\Phi_{\epsilon}$ on $I\times\bar{B}(\bar{x},\delta)\times\bar{B}(\bar{x},\delta)$.

On the other hand, from $2\Phi_{\epsilon}(t_{\epsilon},x_{\epsilon},y_{\epsilon})\geq\Phi_{\epsilon}(t_{\epsilon},x_{\epsilon},x_{\epsilon})+\Phi_{\epsilon}(t_{\epsilon},y_{\epsilon},y_{\epsilon})$, we have

[TABLE]

and consequently $\displaystyle\frac{1}{2\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2}$ is bounded, and as $\epsilon\rightarrow 0$, $|x_{\epsilon}-y_{\epsilon}|\rightarrow 0$. Since $U$ and $V$ are uniformly continuous on $I\times\bar{B}(\bar{x},\delta)$, then $\displaystyle\frac{1}{2\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2}\rightarrow 0$ as $\epsilon\rightarrow 0.$

Since $\Phi_{\epsilon}(t_{\epsilon},x_{\epsilon},y_{\epsilon})\geq\Phi_{\epsilon}(t_{0},x_{0},x_{0})$, we have

[TABLE]

it follow from the continuity of $U$ and $V$ that, up to a subsequence,

[TABLE]

Next, since $x_{0}\in\bar{B}(\bar{x},\delta)$ then, for $\epsilon$ small enough and at least for a subsequence which we still index by $\epsilon$, we obtain

[TABLE]

**Step 2. ** We now show that $t_{\epsilon}<T.$ Actually if $t_{\epsilon}=T$ then,

[TABLE]

and,

[TABLE]

since $U(T,x_{\epsilon})=\exp(T)g(x_{\epsilon})$, $V(T,y_{\epsilon})=\exp(T)g(y_{\epsilon})$ and $g$ is uniformly continuous on $\bar{B}(\bar{x},\delta)$. Then as $\epsilon\rightarrow 0$, we have,

[TABLE]

which yields a contradiction and we have $t_{\epsilon}\in[0,T)$.

**Step 3 ** To complete the proof it remains to show contradiction. Let us denote

[TABLE]

Then we have:

[TABLE]

Then applying the result by Crandall et al. (Theorem 8.3, [7]) to the function

[TABLE]

at the point $(t_{\epsilon},x_{\epsilon},y_{\epsilon})$, for any $\epsilon_{1}>0$, we can find $c,d\in I\!\!R$ and $X,Y\in S_{n},$ such that:

[TABLE]

Then by definition of viscosity solution, we get:

[TABLE]

and

[TABLE]

which implies that:

[TABLE]

But from $(\ref{derive})$ there exist a constant $C>0$ such that:

[TABLE]

As

[TABLE]

Then

[TABLE]

It follows that:

[TABLE]

where $C$ which hereafter may change from line to line. Choosing now $\epsilon_{1}=\epsilon,$ yields the relation

[TABLE]

Now, from (H1), (6.16) and (6.21) we get:

[TABLE]

Next

[TABLE]

And finally,

[TABLE]

So that by plugging into (LABEL:viscder) we obtain:

[TABLE]

By sending $\epsilon\rightarrow 0$, $\beta\rightarrow 0$, $\theta\rightarrow 0$, and taking into account of the continuity of $f$, we obtain $\eta\leq 0$, which is a contradiction. The proof of Theorem 5 is now complete.   $\Box$

Corollary 3

The lower and upper value functions coincide, and the value function of the stochastic differential game is given by $V(t,x):=V^{-}(t,x)=V^{+}(t,x)$ for every $(t,x)\in[0,T)\times\mathbb{R}^{n}.$