#  Stochastic differential switching game in infinite horizon

Brahim EL ASRI and Sehail MAZID Université Ibn Zohr, Equipe. Aide à la
decision, ENSA, B.P. 1136, Agadir, Maroc. e-mail<EMAIL_ADDRESS>Université Ibn Zohr, Equipe. Aide à la decision, ENSA, B.P. 1136, Agadir,
Maroc. e-mail<EMAIL_ADDRESS>
###### Abstract

We study a zero-sum stochastic differential switching game in infinite
horizon. We prove the existence of the value of the game and characterize it
as the unique viscosity solution of the associated system of quasi-variational
inequalities with bilateral obstacles. We also obtain a verification theorem
which provides an optimal strategy of the game. Finally, some numerical
examples with two regimes are given.

AMS subject classifications: 49N70, 49L25, 60H30, 90C39, 93E20

$\bf Keywords$: stochastic differential game, switching strategies, quasi-
variational inequality, value function, viscosity solution

##  1 Introduction

Differential game theory involves multiple persons (also called players or
individuals) decision making in the context of dynamic system. The study of
the two-person zero-sum differential games could be traced back to the
pioneering work by Isaacs [21], which inspired further research in this area.

In 1989, Fleming and Souganidis [15], for the first time, studied two-person
zerosum stochastic differential games. They adopted the definitions of upper
and lower value functions introduced by Elliott and Kalton [13] and proved
that the two value functions satisfy the dynamic programming principle and
they are the unique viscosity solutions to the associated
Hamilton–-Jacobi–-Bellman–-Isaacs partial differential equations (HJBI PDEs).
Their work generalized that of Evans and Souganidis [14] from the
deterministic framework to the stochastic one and is now regarded as one of
the outstanding results in the field of stochastic differential games. In
1995, Hamadène and Lepeltier [17] introduced the backward stochastic
differential equation (BSDE) theory into the study of stochastic differential
games, and there exist also many other works using this BSDEs approach, such
as Hamadène, Lepeltier, and Peng [18]. Subsequently, Buckdahn and Li [2]
extended the findings presented in [17, 18], and generalized the framework
introduced in [15].

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

$\begin{array}[]{ll}X_{s}=x+\displaystyle\int_{t}^{s}b(r,X_{r},a_{t},b_{t})dr+\displaystyle\int_{t}^{s}\sigma(r,X_{r},a_{t},b_{t})dW_{r}\qquad
s\geq t,\end{array}$

with $X_{t^{-}}=x$. Here $W$ is a d-dimensional Wiener process, while

$a_{t}=\sum_{m\geq 1}\xi_{m}1\\!\\!1_{(\tau_{m}\leq
t<\tau_{m+1}]}(t)\quad\quad\text{and}\quad\quad b_{t}=\sum_{n\geq
1}\eta_{n}1\\!\\!1_{(\rho_{n}\leq t<\rho_{n+1}]}(t).$

with the cost functional

$\begin{array}[]{ll}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\infty}e^{-rs}f(X_{s},a_{t},b_{t})ds-\sum\limits_{m\geq
1}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})+\sum\limits_{n\geq
1}e^{-r\rho_{n}}\chi(\eta_{n-1},\eta_{n})\bigg{]}.\end{array}$ (1.1)

The first player chooses the control a from a given finite set $\mathcal{I}$
to maximize the payoff (1.1), and each actions is related with one cost C,
while the second player chooses the control b from $\mathcal{I}$ to minimize
the payoff (1.1), and each of his actions is associated with the other cost
$\chi$. 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 Isaacs’ system of quasi-variational inequalities for
this switching game is the following: for any $i,j\in\mathcal{I}$, and
$x\in\mathbb{R}^{m}$,

$\begin{array}[]{c}max\Big{\\{}min\Big{[}rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f(x,i,j);\\\
\qquad
V^{ij}(x)-M^{ij}[V](x)\Big{]};V^{ij}(x)-N^{ij}[V](x)\Big{\\}}=0,\end{array}$
(1.2)

and

$\begin{array}[]{c}min\Big{\\{}max\Big{[}rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f(x,i,j);\\\
\qquad
V^{ij}(x)-N^{ij}[V](x)\Big{]};V^{ij}(x)-M^{ij}[V](x)\Big{\\}}=0.\end{array}$
(1.3)

where,

$M^{ij}[V](x)=\max_{k\neq i}\\{V^{kj}(x)-C(i,k)\\},\qquad
N^{ij}[V](x)=\min_{l\neq j}\\{V^{il}(x)+\chi(j,l)\\}.$

In the finite horizon framework, the switching game have been studied by
several authors. The most recent work discussing this topic includes the
papers by Djehiche et al. (2017) ([8]), Tang and Hou (2007) ([32]).

The objective of this work is to establish existence and uniqueness of a
continuous viscosity solution of (1.2) and (1.3). The second contribution of
our paper is the Verification Theorem which provides an optimal strategy of
the game, To the best of our knowledge, this issue have not been addressed in
the literature yet.

This paper is organized as follows: in Section 2 we list all the notations,
state the full set of assumptions, and define viscosity sub- and
supersolutions along with equivalent characterizations. 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. In Section 4, by the dynamic programming principle we prove that the
lower and upper value functions of the game satisfy the Isaacs’ system of
quasi-variational inequalities in the viscosity solution sense. In Section 5,
we show that the solution of Isaacs’ system of quasi-variational inequalities
is unique. Further, the upper and the lower value functions coincide and the
game admits a value. In Section 6, we present a verification theorem which
gives an optimal strategy of the switching game, while finally Section 7 will
present some numerical examples.

##  2 Assumptions and problem formulation

Throughout this paper $m$ and $d$ are two integers. Let
$\mathcal{I}=\\{1,...,q\\}$ the finite set of regimes, and assume the
following assumptions:

[H1]
$b:\mathbb{R}^{m}\times\mathcal{I}\times\mathcal{I}\rightarrow\mathbb{R}^{m}$
and
$\sigma:\mathbb{R}^{m}\times\mathcal{I}\times\mathcal{I}\rightarrow\mathbb{R}^{m\times
d}$ be two continuous functions for which there exists a constant $C>0$ such
that for any $x,x^{\prime}\in\mathbb{R}^{m}$ and $i,j\in\mathcal{I}$

$|\sigma(x,i,j)-\sigma(x^{\prime},i,j)|+|b(x,i,j)-b(x^{\prime},i,j)|\leq
C|x-x^{\prime}|.$ (2.1)

Thus they are also of linear growth. i.e., there exists a constant $C$ such
that for any $x\in\mathbb{R}^{m}$ and $i,j\in\mathcal{I}$

$|\sigma(x,i,j)|+|b(x,i,j)|\leq C(1+|x|).$ (2.2)

[H2] $f:\mathbb{R}^{m}\times\mathcal{I}\times\mathcal{I}\rightarrow\mathbb{R}$
is a continuous function for which there exists a constant $C$ such that for
each $i,j\in\mathcal{I}$, $x,x^{\prime}\in\mathbb{R}^{m}$:

$|f(x,i,j)|\leq C(1+|x|)\qquad\text{and}$ (2.3)
$\qquad\qquad|f(x,i,j)-f(x^{\prime},i,j)|\leq
C|x-x^{\prime}|.\qquad\qquad\quad$ (2.4)

[H3] For any $i,j\in\mathcal{I}$ The switching costs $C(i,j)$ and $\chi(i,j)$
are constants, and we assume the triangular condition :

$C(i,k)<C(i,j)+C(j,k),\quad j\neq i,k.$ (2.5)
$\chi(i,k)<\chi(i,j)+\chi(j,k),\quad j\neq i,k,$ (2.6)

which means that it is less expensive to switch directly in one step from
regime $i$ to $k$ than in two steps via an intermediate regime $j$. Notice
that a switching costs $C(i,j)$ and $\chi(i,j)$ may be negative, and
conditions (2.5) and (2.6) for $i=k$ prevents an arbitrage by simply switching
back and forth, i.e.

$0<C(i,j)+C(j,i).$ (2.7) $0<\chi(i,j)+\chi(j,i).$ (2.8)

We set $b^{ij}(.)=b(.,i,j),\sigma^{ij}(.)=\sigma(.,i,j)$ and
$f^{ij}(.)=f(.,i,j)$.

We now consider the following Isaacs’ system of quasi-variational
inequalities: for any $i,j\in\mathcal{I}$, and $x\in\mathbb{R}^{m}$,

$\begin{array}[]{c}max\Big{\\{}min\Big{[}rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f^{ij}(x);\\\
\qquad
V^{ij}(x)-M^{ij}[V](x)\Big{]};V^{ij}(x)-N^{ij}[V](x)\Big{\\}}=0,\end{array}$
(2.9)

and

$\begin{array}[]{c}min\Big{\\{}max\Big{[}rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f^{ij}(x);\\\
\qquad
V^{ij}(x)-N^{ij}[V](x)\Big{]};V^{ij}(x)-M^{ij}[V](x)\Big{\\}}=0.\end{array}$
(2.10)

Where
(i) $r$ is a positive discount factor and $\mathcal{A}$ is the following
infinitesimal generator:

$\mathcal{A}V^{ij}=\langle
b^{ij},\nabla_{x}V^{ij}\rangle+\cfrac{1}{2}tr[\sigma^{ij}(\sigma^{ij})^{*}\nabla^{2}_{x}V^{ij}].$

(ii) For any $x\in\mathbb{R}^{m}$ and $i,j\in\mathcal{I},$

$M^{ij}[V](x)=\max_{k\neq i}\\{V^{kj}(x)-C(i,k)\\},\qquad
N^{ij}[V](x)=\min_{l\neq j}\\{V^{il}(x)+\chi(j,l)\\}.$

We now define the notions of viscosity solution of the system (2.9). We can
similarly define the notions for (2.10).

###### Definition 1

Let
$\vec{V}=(V^{kl}(x))_{(k,l)\in\mathcal{I}\times\mathcal{I}}:I\\!\\!R^{m}\to
I\\!\\!R$ such that for any $(i,j)\in\mathcal{I}\times\mathcal{I}$, $V^{ij}$
is continuous, is called:
(i) A viscosity supersolution to (2.9) if for any $i,j\in\mathcal{I}$, for any
$x_{0}\in I\\!\\!R^{m}$ and any function $\phi^{ij}\in C^{2}(I\\!\\!R^{m})$
such that $\phi^{ij}(x_{0})=V^{ij}(x_{0})$ and $x_{0}$ is a local maximum of
$\phi^{ij}-V^{ij}$, we have:

$\begin{array}[]{lll}max\Big{\\{}min\Big{[}r\phi^{ij}(x_{0})-\mathcal{L}\phi^{ij}(x_{0})-f^{ij}(x_{0}),\\\
\qquad
V^{ij}(x_{0})-M^{ij}[V](x_{0})\Big{]},V^{ij}(x_{0})-N^{ij}[V](x_{0})\Big{\\}}\geq
0.\end{array}$ (2.11)

(ii) A viscosity subsolution to (2.9) if for any $i,j\in\mathcal{I}$, for any
$x_{0}\in I\\!\\!R^{m}$ and any function $\phi^{ij}\in C^{2}(I\\!\\!R^{m})$
such that $\phi^{ij}(x_{0})=V^{ij}(x_{0})$ and $x_{0}$ is a local minimum of
$\phi^{ij}-V^{ij}$, we have:

$\begin{array}[]{lll}max\Big{\\{}min\Big{[}r\phi^{ij}(x_{0})-\mathcal{L}\phi^{ij}(x_{0})-f^{ij}(x_{0}),\\\
\qquad
V^{ij}(x_{0})-M^{ij}[V](x_{0})\Big{]},V^{ij}(x_{0})-N^{ij}[V](x_{0})\Big{\\}}\leq
0.\end{array}$ (2.12)

(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.[5]) which we
give since it will be useful later. So firstly we define the notions of
superjet and subjet of a continuous function $V$.

###### Definition 2

Let $V\in C(I\\!\\!R^{m})$, $x$ an element of $I\\!\\!R^{m}$ and finally
$\mathbb{S}_{m}$ the set of $m\times m$ symmetric matrices. We denote by
$J^{2,+}V(x)$ (resp. $J^{2,-}V(x))$, the superjets (resp. the subjets) of $V$
at x, the set of pairs $(q,X)\in I\\!\\!R^{m}\times\mathbb{S}_{m}$ such that:

$\begin{array}[]{ll}V(y)\leq V(x)+\langle q,y-x\rangle+\frac{1}{2}\langle
X(y-x),y-x\rangle+o(|y-x|^{2})\\\ \\\ (\text{resp.}\;V(y)\geq V(x)+\langle
q,y-x\rangle+\frac{1}{2}\langle
X(y-x),y-x\rangle+o(|y-x|^{2})).\Box\end{array}$

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

$(D_{x}\phi(x),D^{2}_{xx}\phi(x))\in
J^{2,-}V(x)\;(\text{resp}.\;J^{2,+}V(x)).\Box$

We now give an equivalent definition of a viscosity solution of of the system
(2.9).

###### Definition 3

Let
$\vec{V}=(V^{kl}(x))_{(k,l)\in\mathcal{I}\times\mathcal{I}}:I\\!\\!R^{m}\to
I\\!\\!R$ such that for any $(i,j)\in\mathcal{I}\times\mathcal{I}$; $V^{ij}$
continuous , is called a viscosity supersolution (resp. a viscosity
subsolution) to (2.9) if for any $i,j\in\mathcal{I}$, for any $x\in
I\\!\\!R^{m}$ and any $(q,X)\in J^{2,-}V^{ij}(x)$ (resp. $J^{2,+}V^{ij}(x)),$

$\begin{array}[]{lll}max\Big{\\{}min\Big{[}rV^{ij}-\frac{1}{2}Tr[(\sigma^{ij})^{*}X\sigma^{ij}]-\langle
b^{ij},q\rangle-f^{ij}(x);\\\ \qquad\qquad
V^{ij}(x)-M^{ij}[V](x)\Big{]};V^{ij}(x)-N^{ij}[V](x)\Big{\\}}\geq
0\quad(resp.\leq 0).\end{array}$ (2.13)

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

As pointed out previously we will show that system (2.9) and (2.9) has a
unique solution in viscosity sense. This systems are the deterministic version
of the zero-sum switching game which we will describe briefly in the next
section.

##  3 The zero-sum switching game

###  3.1 Setting of the problem

Consider a complete probability space ($\Omega,\mathcal{F},\mathbb{P}$) and a
$d$-dimensional standard Wiener process $W=(W_{t})$ defined on it. Let
$\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$ be the natural filtration generated
by the Wiener process, completed with the $\mathbb{P}$-null sets of
$\mathcal{F}$. We begin by description of the zero-sum switching game.

###### Definition 4

Assume we have two players I and II who intervene on a system (e.g. the
production of energy from several sources such as oil, cole, hydro-electric,
etc.) with the help of switching strategies. An admissible switching process
for I (resp. II) is a sequence $\delta:=(\tau_{m},\xi_{m})_{m\geq 0}$ (resp.
$\nu:=(\rho_{n},\eta_{n})_{n\geq 0})$ where:
(i) $(\tau_{m})_{m}$ (resp. $(\rho_{n})_{n})$, the action times, is a sequence
of $\mathcal{F}$-stopping times such that $\tau_{m}<\tau_{m+1}$ (resp.
$\rho_{n}<\rho_{n+1})$ and $\tau_{n}\to\infty$ (resp. $\rho_{n}\to\infty$).
(ii) $(\xi_{m})_{m}$ (resp. $(\eta_{n})_{n})$, the actions, is a sequence of
$\mathcal{I}$-valued random variables, where each $\xi_{m}$ (resp. $\eta_{n}$)
is $\mathcal{F}_{\tau_{m}}$-measurable (resp.
$\mathcal{F}_{\rho_{n}}$-measurable).

Next let $(i,j)\in\mathcal{I}\times\mathcal{I}$ be fixed. We say that the
admissible switching strategy $\delta:=(\tau_{m},\xi_{m})_{m\geq 0}$ of I
(resp. $\nu:=(\rho_{n},\eta_{n})_{n\geq 0})$ (resp. II) belongs to
$\mathcal{A}^{i}$ (resp. $\mathcal{B}^{j}$) if $\tau_{0}=0,\xi_{0}=i$ (resp.
$\rho_{0}=0,\eta_{0}=j$).

Given an admissible strategy $\delta$ (resp. $\nu$) of I (resp. II), one
associates a stochastic process $(a_{t})_{t\geq 0}$ (resp. $(b_{t})_{t\geq
0})$ which indicates along with time the current mode of I (resp. II) and
which is defined by: $\forall t\geq 0$

$a_{t}=\xi_{0}1\\!\\!1_{\\{\tau_{0}\\}}(t)+\sum_{m\geq
1}\xi_{m-1}1\\!\\!1_{(\tau_{m-1},\tau_{m}]}(t)\;(\text{resp.}\;b_{t}=\eta_{0}1\\!\\!1_{\\{\rho_{0}\\}}(t)+\sum_{n\geq
1}\eta_{n-1}1\\!\\!1_{(\rho_{n-1},\rho_{n}]}(t)).$ (3.1)

The evolution of the state of the game is described by the following
stochastic equation:

$\left\\{\begin{array}[]{lllllll}dX_{t}=b^{a_{t}b_{t}}(X_{t})dt+\sigma^{a_{t}b_{t}}(X_{t})dW_{t},\qquad\;t\geq
0,\\\ X_{0}=x\end{array}\right.$ (3.2)

We assume that for any $x\in\mathbb{R}^{m}$ and $i,j\in\mathcal{I}$, there
exists a unique strong solution to (3.2). Let now
$\delta:=(\tau_{m},\xi_{m})_{m\geq 0}$ (resp. $\nu:=(\rho_{n},\eta_{n})_{n\geq
0})$ be an admissible strategy for I (resp. II) which belongs to
$\mathcal{A}^{i}$ (resp. $\mathcal{B}^{j}$). The interventions of the players
are not free and generate a payoff which is a reward (resp. cost) for I (resp.
II) and whose expression is given by

$\begin{array}[]{ll}J(x,\delta,\nu)=\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\infty}e^{-rs}f^{a_{s}b_{s}}(X^{x}_{s})ds-\sum\limits_{m\geq
1}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})+\sum\limits_{n\geq
1}e^{-r\rho_{n}}\chi(\eta_{n-1},\eta_{n})\bigg{]}.\end{array}$ (3.3)

Following Buckdahn and Li [2], we define nonanticipative strategy as follows.

###### Definition 5

A nonanticipative strategy for player I is a mapping

$\alpha:\cup_{j\in\mathcal{I}}\mathcal{B}^{j}\rightarrow\mathcal{A}^{i}$

such that for any stopping time $\tau$ and any $b$,
$b^{\prime}\in\mathcal{B}^{j}$, with $b\equiv b^{\prime}$ on $\llbracket
0,\tau\rrbracket,$ it holds that $\alpha(b)\equiv\alpha(b^{\prime})$ on
$\llbracket 0,\tau\rrbracket$. (with the notation $\llbracket
0,\tau\rrbracket=\\{(s,\omega)\in[0,T]\times\Omega,\,0\leq
s\leq\tau(\omega)\\}).$
A nonanticipative strategy for player II

$\beta:\cup_{i\in\mathcal{I}}\mathcal{A}^{i}\rightarrow\mathcal{B}^{j}$

is defined similarly. The set of all nonanticipative strategies $\alpha$
(resp. $\beta$) for player I (resp. II) is denoted by $\Gamma^{i}$ (resp,
$\Delta^{j})$.
We define the upper (resp. lower) value of the game by

$\overline{V}^{\;ij}(x)=\inf\limits_{\beta\in{\Delta^{j}}}\sup\limits_{\delta\in\mathcal{A}^{i}}J(x,\delta,\beta(\delta))$
(3.4)

(resp.

$\underline{V}^{ij}(x)=\sup\limits_{\alpha\in\Gamma^{i}}\inf\limits_{\nu\in\mathcal{B}^{j}}J(x,\alpha(\nu),\nu)\big{)}.$
(3.5)

The game has a value if $\underline{V}^{ij}=\overline{V}^{\;ij}.$

###  3.2 Preliminary results

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

###### Lemma 1

(see e.g. [29]) The process $X^{x}$ satisfies the following estimate: There
exists a constant $\rho$ such that

$E[|X^{x}_{t}|]\leq e^{\rho t}(1+|x|)\qquad\forall t\geq 0.$ (3.6)

###### Proposition 1

Under the standing assumptions (H1), (H2) and (H3), there exists some positive
constant $\rho$ such that for $r\geq\rho$, the lower and upper value function
are satisfy a linear growth condition: for all $i,j\in\mathcal{I}$ and $x\in
I\\!\\!R^{m}$ there exists a constant $C$ such that

$|\overline{V}^{\;ij}(x)|,|\underline{V}^{ij}(x)|\leq C(1+|x|).$ (3.7)

$Proof$: We make the proof only for the lower value function
$\overline{V}^{\;ij}$, the other case being analogous. for every
$\delta\in\mathcal{A}^{i}$ we consider the particular strategy:
$\tilde{\beta}(\delta)=(\tilde{\rho}_{n},\tilde{\eta}_{n})$ given by
$\tilde{\rho}_{n}=\infty,\tilde{\eta_{n}}=j$ for all $n\geq 1$ we have

$\overline{V}^{\;ij}(x)\leq\sup\limits_{\delta\in\mathcal{A}^{i}}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\infty}e^{-rs}f^{a_{s}b_{s}}(X_{s}^{x})ds-\sum\limits_{m\geq
1}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})\bigg{]}.$ (3.8)

Then for every $\epsilon>0$, there exists a strategy
$\delta\in\mathcal{A}^{i}$ such that

$\overline{V}^{\;ij}(x)\leq\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\infty}e^{-rs}f^{a_{s}b_{s}}(X_{s}^{x})ds-\sum\limits_{m\geq
1}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})\bigg{]}+\epsilon.$ (3.9)

The switching cost is dominated by

$-\sum_{m=1}^{N}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})\leq\max_{k\in\mathcal{I}}(-C(i,k))$
(3.10)

for any $N\geq 1$, which can be proved by induction as in [26]. Indeed, (3.10)
obviously holds for N= 1. Suppose now that it is also verified for some $N-1$
and let us show that it is valid for $N$. When $C(\xi_{N-1},\xi_{N})\geq 0$
holds as $r\geq 0$. When $C(\xi_{N-1},\xi_{N})<0$, we have

$\begin{array}[]{ll}-\sum_{m=1}^{N}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})\\\ \\\
\qquad\qquad\qquad=-\sum_{m=1}^{N-2}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})-(e^{-r\tau_{N-1}}C(\xi_{N-2},\xi_{N-1})+e^{-r\tau_{N}}C(\xi_{N-1},\xi_{N}))\\\
\\\
\qquad\qquad\qquad\leq-\sum_{m=1}^{N-2}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})-e^{-r\tau_{N-1}}(C(\xi_{N-2},\xi_{N-1})+C(\xi_{N-1},\xi_{N}))\\\
\\\
\qquad\qquad\qquad\leq-\sum_{m=1}^{N-2}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})-e^{-r\tau_{N-1}}C(\xi_{N-2},\xi_{N})\leq\max_{k\in\mathcal{I}}(-C(i,k)).\end{array}$

Combining (3.9) and (3.10) gives us

$\overline{V}^{\;ij}(x)\leq\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\infty}e^{-rs}f^{a_{s}b_{s}}(X_{s}^{x})ds+\max_{k\in\mathcal{I}}(-C(i,k))\bigg{]}+\epsilon.$
(3.11)

Now, from estimate (3.6) and the polynomial growth condition of $f^{ij}$ in
(H2), there exists $C$ and $\rho$ such that

$\overline{V}^{\;ij}(x)\leq\frac{C}{r}+C(1+|x|)\displaystyle\int_{0}^{\infty}e^{(\rho-r)s}ds+\max_{k\in\mathcal{I}}(-C(i,k)).$
(3.12)

Therefore if $r>\rho$ we have

$\overline{V}^{\;ij}(x)\leq C(1+|x|).$

On the other hand, by considering the particular strategy
$\tilde{\delta}=(\tilde{\tau}_{m},\tilde{\xi}_{m})$ given by
$\tilde{\tau}_{m}=\infty,\tilde{\xi}_{m}=i$ for all $m\geq 1$, and for
$\epsilon>0$ there exists a strategy $\nu\in\mathcal{B}^{j}$ such that

$\overline{V}^{\;ij}(x)\geq\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\infty}e^{-rs}f^{a_{s}b_{s}}(X_{s}^{x})ds+\sum\limits_{n\geq
1}e^{-r\rho_{n}}\chi(\eta_{n-1},\eta_{m})\bigg{]}.$ (3.13)

For all $N\geq 1,$ we have

$\sum_{n=1}^{N}e^{-r\rho_{n}}\chi(\eta_{n-1},\eta_{n})\geq\min_{l\in\mathcal{I}}(\chi(j,l)).$

Then, we get

$\overline{V}^{\;ij}(x)\geq\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\infty}e^{-rs}f^{a_{s}b_{s}}(X_{s}^{x})ds+\min_{l\in\mathcal{I}}(\chi(j,l))\bigg{]}.$
(3.14)

As a consequence, there exists $C$ and $\rho$ such that

$\overline{V}^{\;ij}(x)\geq-\frac{C}{r}-C(1+|x|)\displaystyle\int_{0}^{\infty}e^{(\rho-r)s}ds+\min_{l\in\mathcal{I}}(\chi(j,l)).$
(3.15)

Therefore if $r>\rho$, we have

$\overline{V}^{\;ij}(x)\geq-C(1+|x|).$

from which we deduce the thesis. $\Box$

###### Lemma 2

There exists some positive constant $\rho$ such that for $r>\rho$,
$x,x^{\prime}\in\mathbb{R}^{m},\delta\in\mathcal{A}^{i}$ and
$\nu\in\mathcal{B}^{j}$ we have

$|J(x,\delta,\nu)-J(x^{\prime},\delta,\nu)|+|\overline{V}^{\;ij}(x)-\overline{V}^{\;ij}(x^{\prime})|+|\underline{V}^{ij}(x)-\underline{V}^{ij}(x^{\prime})|\leq
C|x-x^{\prime}|,$

for some positive constant C.

$Proof.$ It is enough to show that the conclusion holds for the gain
functional $J$.
For every
$x,x^{\prime}\in\mathbb{R}^{m},u\in\mathcal{A}^{i},\nu\in\mathcal{B}^{j}$, we
have

$\qquad\qquad|J(x,\delta,\nu)-J(x^{\prime},\delta,\nu)|\leq\mathbb{E}\bigg{[}\int_{0}^{\infty}e^{-rs}|f^{a_{s}b_{s}}(X_{s}^{x})-f^{a_{s}b_{s}}(X_{s}^{x^{\prime}})|ds\bigg{]}.\qquad\qquad\qquad\qquad\qquad$

By a standard estimate for the SDE applying Ito’s formula to
$|X^{x}-X^{x^{\prime}}|^{2}$ and using Gronwall’s lemma, we then obtain from
the Lipschitz condition on $b^{ij},\sigma^{ij}$, the following inequality

$E[|X_{t}^{x}-X_{t}^{x^{\prime}}|\,]\leq e^{\rho t}|x-x^{\prime}|\qquad\forall
x,x^{\prime}\in\mathbb{R}^{m},t\geq 0.$

From the Lipschitz condition on $f^{ij}$, we deduce

$\qquad\qquad|J(x,\delta,\nu)-J(x^{\prime},\delta,\nu)|\leq
C\mathbb{E}\bigg{[}\int_{0}^{\infty}e^{-rs}|X_{s}^{x}-X_{s}^{x^{\prime}}|ds\bigg{]}\qquad\qquad\qquad\qquad\qquad\\\
\leq C\mathbb{E}\bigg{[}\int_{0}^{\infty}e^{-rs}e^{\rho
s}|x-x^{\prime}|ds\bigg{]}\leq C|x-x^{\prime}|,\qquad\qquad$

for $r>\rho$. This ends the proof. $\Box$

In the sequel, we shall assume that $r$ is large enough, which ensures that
the value functions are Lipschitz continuous. We now present the dynamic
programming principle of the zero-sum switching game.

###### Theorem 1

For any $x\in\mathbb{R}^{m}$ and $i,j\in\mathcal{I}$, we have

$\begin{array}[]{ll}\overline{V}^{\;ij}(x)=\inf\limits_{\beta\in{\Delta^{j}}}\sup\limits_{\delta\in\mathcal{A}^{i}}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\theta}e^{-rs}f^{a_{s}b_{s}}(X^{x}_{s})ds-\sum\limits_{\tau_{m}\leq\theta}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})\\\
\qquad\qquad\qquad\qquad\qquad\qquad+\sum\limits_{\rho_{n}\leq\theta}e^{-r\rho_{n}}\chi(\eta_{n-1},\eta_{n})+e^{-r\theta}\overline{V}^{\;a_{\theta}b_{\theta}}(X^{x}_{\theta})\bigg{]}.\end{array}$
(3.16)
$\begin{array}[]{ll}\underline{V}^{ij}(x)=\sup\limits_{\alpha\in\Gamma^{i}}\inf\limits_{\nu\in\mathcal{B}^{j}}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\theta}e^{-rs}f^{a_{s}b_{s}}(X^{x}_{s})ds-\sum\limits_{\tau_{m}\leq\theta}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})\\\
\qquad\qquad\qquad\qquad\qquad\qquad+\sum\limits_{\rho_{n}\leq\theta}e^{-r\rho_{n}}\chi(\eta_{n-1},\eta_{n})+e^{-r\theta}\underline{V}^{a_{\theta}b_{\theta}}(X^{x}_{\theta})\bigg{]},\end{array}$
(3.17)

where $\theta$ is any stopping time.

This principle was formally stated in [2] and proved rigorously for the finite
horizon case in [33]. The arguments for the infinite horizon case may be
adapted in a straightforward way.

##  4 Isaacs’ system of quasi-variational inequalities

In the present section we prove that the two value functions are viscosity
solutions to (2.9) and (2.10). But we make the proof only for (2.10), the
other case is analogous. We begin with the following lemma.

###### Lemma 3

For any $i,j\in\mathcal{I}$, the lower and upper value functions satisfy the
following properties: for all $x\in\mathbb{R}^{m}$,

$V^{ij}(x)\leq N^{ij}[V](x)$ (4.1) $M^{ij}[V](x)\leq V^{ij}(x)$ (4.2)

$Proof:$ We make the proof only for the lower value function
$\overline{V}^{\;ij}$, the other case being analogous. For every
$j,\tilde{j}\in\mathcal{I},j\neq\tilde{j}$ and
$\tilde{\nu}\in\mathcal{B}^{\tilde{j}}$ we define $\nu\in\mathcal{B}^{j}$ by

$\tilde{\rho}_{l-1}=\rho_{l},\;\rho_{0}=0,\;\tilde{\eta}_{l-1}=\eta_{l},\;\eta_{0}=j.$

Note that $\rho_{0}=\rho_{1}=0.$ Let
$\tilde{\nu}:=\\{\tilde{\rho}_{l},\tilde{\eta}_{l}\\}$ and
$\nu:=\\{\rho_{l},\eta_{l}\\}$, then

$\begin{array}[]{ll}\overline{V}^{\;ij}(x)\leq\sup\limits_{\delta\in\mathcal{A}^{i}}J(x,\delta,\nu)=\sup\limits_{\delta\in\mathcal{A}^{i}}\big{[}J(x,\delta,\tilde{\nu})+\chi(j,\tilde{j})\big{]},\end{array}$

from which we deduce that the following inequality holds:

$\overline{V}^{\;ij}(x)\leq\overline{V}^{\;i\tilde{j}}(x)+\chi(j,\tilde{j}).$

Therefore

$\overline{V}^{\;ij}(x)\leq\inf_{\tilde{j}\neq
j}\\{\overline{V}^{\;i\tilde{j}}(x)+\chi(j,\tilde{j})\\}.$

In a similar way we can prove $(\ref{Hsup})$, hence deducing the thesis.
$\Box$

###### Theorem 2

For any $i,j\in\mathcal{I}$, the lower value function $\overline{V}^{\;ij}$
and the upper value function $\underline{V}^{ij}$ are viscosity solutions of
(2.9).

$Proof.$ We state the proof only for lower value function, the other case is
analogous. First, we prove the supersolution property. Fix $i,j\in\mathcal{I}$
and let $\bar{x}\in I\\!\\!R^{m},\varphi^{ij}\in C^{2}(I\\!\\!R^{m})$ such
that $\bar{x}$ is a minimum of $\overline{V}^{\;ij}-\varphi^{ij}$ with
$\overline{V}^{\;ij}(\bar{x})=\varphi^{ij}(\bar{x})$. We have, by Lemma 3

$\overline{V}^{\;ij}(\bar{x})-N^{ij}[\overline{V}](\bar{x})\leq 0.$

If

$\overline{V}^{\;ij}(\bar{x})-N^{ij}[\overline{V}](\bar{x})=0,$

then we are done. Now suppose

$\overline{V}^{\;ij}(\bar{x})-N^{ij}[\overline{V}](\bar{x})<-2\epsilon<0,$

we prove by contradiction that

$r\varphi^{ij}(\bar{x})-\mathcal{A}\varphi^{ij}(\bar{x})-f^{ij}(\bar{x})\geq
0.$

Suppose otherwise, i.e.,
$r\varphi^{ij}(\bar{x})-\mathcal{A}\varphi^{ij}(\bar{x})-f^{ij}(\bar{x})<0$.
Then without loss of generality we can assume that
$r\varphi^{ij}(\bar{x})-\mathcal{A}\varphi^{ij}(\bar{x})-f^{ij}(\bar{x})<0$
and $\overline{V}^{\;ij}(x)-N^{ij}[\overline{V}](x)<-\epsilon<0$ on
$B(\bar{x},\delta)$.
Define the stopping time $\tau$ by

$\tau=\inf\\{t\geq 0:X_{t}\notin B(\bar{x},\delta)\\}.$

Let $\epsilon_{1}>0$, using the dynamic programming principle, we deduce the
existence of a strategy $\beta\in\Delta^{j}$ such that

$\begin{array}[]{llllll}\overline{V}^{\;ij}(\bar{x})\geq\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\tau\wedge\rho_{1}}e^{-rs}f^{ij}(X_{s}^{\bar{x}})ds+1\\!\\!1_{[\rho_{1}\leq\tau]}e^{-r\rho_{1}}\big{(}\chi(j,\eta_{1})+\overline{V}^{\;i\eta_{1}}(X_{\rho_{1}}^{\bar{x}})\big{)}\bigg{]}\\\
\\\
\qquad\qquad+\mathbb{E}\bigg{[}1\\!\\!1_{[\rho_{1}>\tau]}e^{-r\tau}\overline{V}^{\;ij}(X_{\tau}^{\bar{x}})\bigg{]}-\epsilon_{1}\\\
\\\
\qquad\quad\;\geq\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\tau\wedge\rho_{1}}e^{-rs}f^{ij}(X_{s}^{\bar{x}})ds+1\\!\\!1_{[\rho_{1}\leq\tau]}e^{-r\rho_{1}}N^{ij}[\overline{V}](X_{\rho_{1}}^{\bar{x}})\bigg{]}+\mathbb{E}\bigg{[}1\\!\\!1_{[\rho_{1}>\tau]}e^{-r\tau}\overline{V}^{\;ij}(X_{\tau}^{\bar{x}})\bigg{]}-\epsilon_{1}\\\
\\\
\qquad\quad\;\geq\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\tau\wedge\rho_{1}}e^{-rs}f^{ij}(X_{s}^{\bar{x}})ds+e^{-r(\rho_{1}\wedge\tau)}\overline{V}^{\;ij}(X_{\rho_{1}\wedge\tau}^{\bar{x}})\bigg{]}-\epsilon_{1}.\end{array}$

Therefore, without loss of generality, we only need to consider
$\beta\in\Delta^{j}$ such that $\rho_{1}>\tau$. Then

$\begin{array}[]{llllll}\varphi^{ij}(\bar{x})=\overline{V}^{\;ij}(\bar{x})\geq\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\tau}e^{-rs}f^{ij}(X_{s}^{\bar{x}})ds+e^{-r\tau}\overline{V}^{\;ij}(X_{\tau}^{\bar{x}})\bigg{]}-\epsilon_{1}\\\
\\\
\qquad\qquad\quad\quad\quad\geq\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\tau}e^{-rs}f^{ij}(X_{s}^{\bar{x}})ds+e^{-r\tau}\varphi^{ij}(X_{\tau}^{\bar{x}})\bigg{]}-\epsilon_{1}.\end{array}$

By applying Itô’s formula to $e^{-rt}\varphi^{ij}(X^{\bar{x}}_{t})$ and
plugging into the last inequality, we obtain

$0\leq\mathbb{E}\bigg{[}\displaystyle\int_{0}^{\tau}e^{-rs}(r\varphi^{ij}-\mathcal{A}\varphi^{ij}-f^{ij})(X_{s}^{\bar{x}})ds\bigg{]}-\epsilon_{1},$

which is a contradiction. Therefore, we must have
$0\leq(r\varphi^{ij}-\mathcal{A}\varphi^{ij}-f^{ij})(\bar{x})$.
Thanks to Lemma 3, it is enough to show that given $\bar{x}\in\mathbb{R}^{m}$
such that

$M^{ij}[\overline{V}](\bar{x})\leq\overline{V}^{\;ij}(\bar{x}).$

Then

$\begin{array}[]{c}max\Big{\\{}min\Big{[}r\varphi^{ij}(\bar{x})-\mathcal{A}\varphi^{ij}(\bar{x})-f^{ij}(\bar{x});\\\
\qquad\qquad\overline{V}^{\;ij}(\bar{x})-M^{ij}[\overline{V}](\bar{x})\Big{]};\overline{V}^{\;ij}(\bar{x})-N^{ij}[\overline{V}](\bar{x})\Big{\\}}\geq
0.\end{array}$ (4.3)

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

##  5 Uniqueness of the solution of Isaacs’ system of quasi-variational
inequalities

We prove that the (2.9) admits a unique viscosity solution. We need to make an
additional assumption on the switching costs.
[H5] (The no free loop property)

For any sequence of pairs
$\\{i_{p},j_{p}\\}_{p=1}^{N+1}\subset\mathcal{I}\times\mathcal{I}$ such that
$(i_{N+1},j_{N+1})=(i_{1},j_{1}),$ and for $1\leq p\leq N,$ either
$i_{p+1}=i_{p}$ or $j_{p+1}=j_{p}$, we have

$\sum_{p=1}^{N}\chi(j_{p},j_{p+1})-\sum_{p=1}^{N}C(i_{p},i_{p+1})\neq 0.$

We begin with the technical lemma.

###### Lemma 4

Let $\vec{U}:=(U^{ij})_{(i,j)\in\mathcal{I}\times\mathcal{I}}$ and
$\vec{V}:=(V^{ij})_{(i,j)\in\mathcal{I}\times\mathcal{I}}$ be a viscosity
subsolution and a viscosity supersolution to the equation (2.9), respectively.
Let $x\in\mathbb{R}^{m}$ be fixed and
$(\bar{i},\bar{j})\in\mathcal{I}\times\mathcal{I}$ such that

$V^{\bar{i}\bar{j}}(x)-U^{\bar{i}\bar{j}}(x)=\max_{i,j\in\mathcal{I}}\\{V^{ij}(x)-U^{ij}(x)\\}$
(5.1)

and

$U^{\bar{i}\bar{j}}(x)\leq M^{\bar{i}\bar{j}}[U](x)\quad\text{or}\quad
V^{\bar{i}\bar{j}}(x)\geq N^{\bar{i}\bar{j}}[V](x).$ (5.2)

Then there exists $(i_{0},j_{0})\in\mathcal{I}\times\mathcal{I}$ such that

$V^{\bar{i}\bar{j}}(x)-U^{\bar{i}\bar{j}}(x)=V^{i_{0}j_{0}}(x)-U^{i_{0}j_{0}}(x)$
(5.3)

and

$U^{i_{0}j_{0}}(x)>M^{i_{0}j_{0}}[U](x)\quad\text{and}\quad
V^{i_{0}j_{0}}(x)<N^{i_{0}j_{0}}[V](x).$ (5.4)

$Proof$. Suppose that $U^{\bar{i}\bar{j}}(x)\leq
M^{\bar{i}\bar{j}}[U](x_{\epsilon})$, then there exists $k\neq\bar{i}$, such
that

$U^{\bar{i}\bar{j}}(x)\leq U^{k\bar{j}}(x)-C(\bar{i},k).$ (5.5)

Since $\vec{V}$ is a subsolution, it satisfies

$V^{\bar{i}\bar{j}}(x)\geq V^{k\bar{j}}(x)-C(\bar{i},k)$ (5.6)

which implies that

$U^{\bar{i}\bar{j}}(x)-V^{\bar{i}\bar{j}}(x)\leq
U^{k\bar{j}}(x)-V^{k\bar{j}}(x).$

Hence

$U^{\bar{i}\bar{j}}(x)-V^{\bar{i}\bar{j}}(x)=U^{k\bar{j}}(x)-V^{k\bar{j}}(x).$
(5.7)

Then (5.5), (5.6) and (5.7) imply

$U^{\bar{i}\bar{j}}(x)-U^{k\bar{j}}(x)=V^{\bar{i}\bar{j}}(x)-V^{k\bar{j}}(x)=-C(\bar{i},k).$

Similarly if $V^{\bar{i}\bar{j}}(x)\geq N^{\bar{i}\bar{j}}[V](x)$ hold, then
there exists $l\neq\bar{j}$ such that

$U^{\bar{i}\bar{j}}(x)-V^{\bar{i}\bar{j}}(x)=U^{\bar{i}l}(x)-V^{\bar{i}l}(x)$

and

$U^{\bar{i}\bar{j}}(x)-U^{\bar{i}l}(x)=V^{\bar{i}\bar{j}}(x)-V^{\bar{i}l}(x)=\chi(\bar{j},l).$

Now if the new index ($k$ or $l$) verify (5.2) , we repeat this reasoning . If
this case continues to occur, finally we find a loop
$(i_{1},j_{1}),...,(i_{N},j_{N}),(i_{N+1},j_{N+1})=(i_{1},j_{1})$ such that

$\sum_{q=1}^{N}\chi(j_{q},j_{q+1})-\sum_{q=1}^{N}C(i_{q},i_{q+1})=0.$

Hence we obtain a contradiction to the assumption (H5). Thus (5.3) and (5.4)
holds for some $(i_{0},j_{0})$ $\Box$

###### Theorem 3

Let $\vec{U}=(U^{ij})_{(i,j)\in\mathcal{I}\times\mathcal{I}}$ (resp.
$\vec{V}=(V^{ij})_{(i,j)\in\mathcal{I}\times\mathcal{I}}),$ a family of
continuous viscosity subsolutions (resp. supersolutions) to (2.9), and
satisfying a linear growth condition. Then, $U^{ij}\leq V^{ij}$ for all
$i,j\in\mathcal{I}.$

$Proof.$ Let us proceed by contradiction. For some $R>0$ suppose there exists
there exists $(\bar{x},\bar{i},\bar{j})\in
B_{R}\times\mathcal{I}\times\mathcal{I}\;(B_{R}:=\\{x\in\mathbb{R}^{m};|x|<R\\})$
such that

$\max\limits_{x\in
B_{R}}\max_{i,j}(U^{ij}-V^{ij})(x)=(U^{\bar{i}\bar{j}}-V^{\bar{i}\bar{j}})(\bar{x})=\eta>0.$
(5.8)

We divide the proof into two steps.

Step 1. Using Lemma 4 we derive the existence of $i_{0}$ and $j_{0}$ such that

$(U^{i_{0}j_{0}}-V^{i_{0}j_{0}})(\bar{x})=\eta>0.$ (5.9)

and

$V^{i_{0}j_{0}}(x)<N^{i_{0}j_{0}}[V](x)\quad\text{and}\quad
U^{i_{0}j_{0}}(x)>M^{i_{0}j_{0}}[U](x)$ (5.10)

For a small $\epsilon>0$, let $\Phi_{\epsilon}$ be the function defined as
follows.

$\begin{array}[]{ll}\Phi_{\epsilon}(x,y)=U^{i_{0}j_{0}}(x)-V^{i_{0}j_{0}}(y)-\displaystyle\frac{1}{2\epsilon}|x-y|^{2}\qquad\qquad(x,y)\in\mathbb{R}^{m}\times\mathbb{R}^{m}.\end{array}$
(5.11)

Let $(x_{\epsilon},y_{\epsilon})$ be such that

$\begin{array}[]{ll}\Phi^{i_{\epsilon}j_{\epsilon}}_{\epsilon}(x_{\epsilon},y_{\epsilon})=\max\limits_{x,y\in
B_{R}}\max\limits_{i,j\in\mathcal{I}}\Phi^{ij}_{\epsilon}(x,y).\end{array}$
(5.12)

From
$2\Phi_{\epsilon}(x_{\epsilon},y_{\epsilon})\geq\Phi_{\epsilon}(x_{\epsilon},x_{\epsilon})+\Phi(y_{\epsilon},y_{\epsilon})$
we have

$\displaystyle\frac{1}{\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2}\leq(U^{i_{0}j_{0}}(x_{\epsilon})-U^{i_{0}j_{0}}(y_{\epsilon}))+(V^{i_{0}j_{0}}(x_{\epsilon})-V^{i_{0}j_{0}}(y_{\epsilon})),$
(5.13)

and consequently
$\displaystyle\frac{1}{\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2}$ is bounded,
and as $\epsilon\rightarrow 0$, $|x_{\epsilon}-y_{\epsilon}|\rightarrow 0$.
Since $U^{i_{\epsilon}j_{\epsilon}}$ and $V^{i_{\epsilon}j_{\epsilon}}$ are
uniformly continuous, then
$\displaystyle\frac{1}{2\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2}\rightarrow 0$
as $\epsilon\rightarrow 0.$
Next, recalling that $U^{i_{0}j_{0}}$ and $V^{i_{0}j_{0}}$ are continuous,
then, for $\epsilon$ small enough and at least for a subsequence which we
still index by $\epsilon$, we obtain

$V^{i_{0}j_{0}}(y_{\epsilon})<N^{i_{0}j_{0}}[V](y_{\epsilon})\quad\text{and}\quad
U^{i_{0}j_{0}}(x_{\epsilon})>M^{i_{0}j_{0}}[U](x_{\epsilon})$ (5.14)

Step 2. Let us denote

$\varphi_{\epsilon}(x,y)=\displaystyle\frac{1}{2\epsilon}|x-y|^{2}.$ (5.15)

Then we have:

$\left\\{\begin{array}[]{lllllll}D_{x}\varphi_{\epsilon}(x,y)=\displaystyle\frac{1}{\epsilon}(x-y),\\\
D_{y}\varphi_{\epsilon}(x,y)=-\displaystyle\frac{1}{\epsilon}(x-y)\\\ \\\
B(x,y)=D^{2}_{x,y}\varphi_{\epsilon}(x,y)=\displaystyle\frac{1}{\epsilon}\begin{pmatrix}I&-I\\\
-I&I\end{pmatrix}.\end{array}\right.$ (5.16)

Then applying the result by Crandall et al. (Theorem 3.2, [5]) to the function

$U^{i_{0}j_{0}}(x)-V^{i_{0}j_{0}}(y)-\varphi_{\epsilon}(x,y)$

at the point $(x_{\epsilon},y_{\epsilon})$, for any $\epsilon_{1}>0$, we can
find $X,Y\in\mathbb{S}_{m},$ such that:

$\left\\{\begin{array}[]{lllllll}\big{(}\displaystyle\frac{1}{\epsilon}(x_{\epsilon}-y_{\epsilon}),X\big{)}\in
J^{2,+}(U^{i_{0}j_{0}}(x_{\epsilon})),\\\ \\\
\big{(}\displaystyle\frac{1}{\epsilon}(x_{\epsilon}-y_{\epsilon}),Y\big{)}\in
J^{2,-}(V^{i_{0}j_{0}}(y_{\epsilon})),\\\ \\\
-\big{(}\displaystyle\frac{1}{\epsilon_{1}}+\|B(x_{\epsilon},y_{\epsilon})\|\big{)}I\leq\begin{pmatrix}X&0\\\
0&-Y\end{pmatrix}\leq
B(x_{\epsilon},y_{\epsilon})+\epsilon_{1}B(x_{\epsilon},y_{\epsilon})^{2}.\end{array}\right.$
(5.17)

Then by definition of viscosity solution, we get:

$\begin{array}[]{lll}rU^{i_{0}j_{0}}(x_{\epsilon})-\langle\displaystyle\frac{1}{\epsilon}(x_{\epsilon}-y_{\epsilon}),\\\
\qquad\qquad
b(x_{\epsilon})\rangle-\displaystyle\frac{1}{2}tr[\sigma^{*}(x_{\epsilon})X\sigma(x_{\epsilon})]-f^{i_{0}j_{0}}(x_{\epsilon})]\leq
0,\end{array}$ (5.18)

and

$\begin{array}[]{l}rV^{i_{0}j_{0}}(y_{\epsilon})-\langle\displaystyle\frac{1}{\epsilon}(x_{\epsilon}-y_{\epsilon}),\\\
\qquad\qquad\qquad\qquad
b(y_{\epsilon})\rangle-\displaystyle\frac{1}{2}tr[\sigma^{*}(y_{\epsilon})Y\sigma(y_{\epsilon})]-f^{i_{0}j_{0}}(y_{\epsilon})\geq
0,\end{array}$ (5.19)

which implies that:

$\begin{array}[]{llllll}&rU^{i_{0}j_{0}}(x_{\epsilon})-rV^{i_{0}j_{0}}(y_{\epsilon})\\\
\\\
&\leq\langle\displaystyle\frac{1}{\epsilon}(x_{\epsilon}-y_{\epsilon}),b(x_{\epsilon})-b(y_{\epsilon})\rangle+\displaystyle\frac{1}{2}tr[\sigma^{*}(x_{\epsilon})X\sigma(x_{\epsilon})-\sigma^{*}(y_{\epsilon})Y\sigma(y_{\epsilon})]\\\
\\\ &+f^{i_{0}j_{0}}(x_{\epsilon})-f^{i_{0}j_{0}}(y_{\epsilon}).\end{array}$
(5.20)

we have :

$B+\epsilon_{1}B^{2}\leq\frac{\epsilon+\epsilon_{1}}{\epsilon^{2}}\begin{pmatrix}I&-I\\\
-I&I\end{pmatrix},$ (5.21)

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

$B+\epsilon_{1}B^{2}\leq\frac{2}{\epsilon}\begin{pmatrix}I&-I\\\
-I&I\end{pmatrix}.$ (5.22)

Now, from (H1), (5.17) and (5.22) we get:

$\displaystyle\frac{1}{2}tr[\sigma^{*}(x_{\epsilon})X\sigma(x_{\epsilon})-\sigma^{*}(y_{\epsilon})Y\sigma(y_{\epsilon})]\leq\frac{C}{\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2}.$

Next

$\langle\frac{1}{\epsilon}(x_{\epsilon}-y_{\epsilon}),b(x_{\epsilon})-b(y_{\epsilon})\rangle\leq\frac{C^{2}}{\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2}.$

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

$\begin{array}[]{llllll}rU^{i_{0}j_{0}}(x_{\epsilon})-rV^{i_{0}j_{0}}(y_{\epsilon})\\\
\\\
\qquad\qquad\leq\displaystyle\frac{C}{\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2}+\displaystyle\frac{C^{2}}{\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2}+f^{i_{0}j_{0}}(x_{\epsilon})-f^{i_{0}j_{0}}(y_{\epsilon}).\end{array}$
(5.23)

By sending $\epsilon\rightarrow 0$, and taking into account of the continuity
of $f^{i_{0}j_{0}}$, we obtain $\eta\leq 0$ which is a contradiction. Thus,
$U^{ij}\leq V^{ij}$, for any $(i,j)\in\mathcal{I}\times\mathcal{I}$, which is
the desired result. $\Box$

###### Corollary 1

The lower and upper value functions coincide, and the value function of the
stochastic differential game is given by
$V^{ij}(x):=\overline{V}^{\;ij}(x)=\underline{V}^{ij}(x)$ for every
$i,j\in\mathcal{I}$ and $x\in\mathbb{R}^{m}.$ As a consequence the two
equations (2.9) and (2.10) coincide.

##  6 A verification theorem

In this section, we present a verification theorem which gives an optimal
strategies of our zero-sum stochastic differential game.

We suppose that a classical solution of (2.9) exists, denoted by
$(V^{ij})_{(i,j)\in\mathcal{I}\times\mathcal{I}}$. Then for each
$i,j\in\mathcal{I},V^{ij}$ separates the space $\mathbb{R}^{m}$ into four
regions:

$\begin{array}[]{ll}\mathcal{C}:=\\{x\in\mathbb{R}^{n}:V^{ij}(x)<N^{ij}[V](x);V^{ij}(x)>M^{ij}[V](x)\;\mbox{and}\;rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f^{ij}(x)=0\\}\\\
\\\
\mathcal{I}_{1}:=\\{x\in\mathbb{R}^{n}:V^{ij}(x)<N^{ij}[V](x);V^{ij}(x)=M^{ij}[V](x)\;\mbox{and}\;rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f^{ij}(x)\geq
0\\}\\\ \\\
\mathcal{I}_{2}:=\\{x\in\mathbb{R}^{n}:V^{ij}(x)=N^{ij}[V](x);V^{ij}(x)>M^{ij}[V](x)\;\mbox{and}\;rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f^{ij}(x)\leq
0\\}\\\ \\\
\mathcal{I}_{3}:=\\{x\in\mathbb{R}^{n}:V^{ij}(x)=N^{ij}[V](x)\quad\mbox{and}\quad
V^{ij}(x)=M^{ij}[V](x)\\}\end{array}$

Let us define the strategies $\delta^{*}:=(\tau^{*}_{m},\xi^{*}_{m})_{m\geq
0}$ (resp. $\nu^{*}:=(\rho^{*}_{n},\eta^{*}_{n})_{n\geq 0})$ as follows:

$\tau^{*}_{0}=0,\xi^{*}_{0}=i\;(\text{resp}.\;\rho^{*}_{0}=0,\eta^{*}_{0}=j)$

and for any $m\geq 1$,

$\left\\{\begin{array}[]{ll}\tau^{*}_{m}=\left\\{\begin{array}[]{ll}\text{inf}\\{s\geq\tau^{*}_{m-1},V^{\xi^{*}_{m-1}b_{s}}(X_{s})=\text{max}_{k\neq\xi^{*}_{m-1}}\\{V^{k\,b_{s}}(X_{s})-C(\xi^{*}_{m-1},k)\\}\\}\\\
\\\ +\infty\quad\text{if the above set is empty}\end{array}\right.\\\ \\\
\text{and}\\\ \\\
\xi^{*}_{m}=\left\\{\begin{array}[]{ll}\text{max}\\{k\neq\xi^{*}_{m-1},V^{k\,a_{\tau^{*}_{m}}}(X_{\tau^{*}_{m}})-C(\xi^{*}_{m-1},k)\\}\quad\mbox{if}\quad\tau^{*}_{m}<+\infty\\\
\\\
\xi^{*}_{m-1}\quad\mbox{if}\quad\tau^{*}_{m}=+\infty\end{array}\right.\end{array}\right.$

(resp.

$\left\\{\begin{array}[]{ll}\rho^{*}_{m}=\left\\{\begin{array}[]{ll}\inf\\{s\geq\rho^{*}_{m-1},V^{a_{s}\eta^{*}_{m-1}}(X_{s})=\text{min}_{l\neq\eta^{*}_{m-1}}\\{V^{a_{s}l}(X_{s})+\chi(\eta^{*}_{m-1},l)\\}\\}\\\
\\\ +\infty\quad\text{if the above set is empty}\end{array}\right.\\\ \\\
\text{and}\\\ \\\
\eta^{*}_{m}=\left\\{\begin{array}[]{ll}\min\\{l\neq\eta^{*}_{m-1},V^{a_{\rho^{*}_{m}}l}(X_{\rho^{*}_{m}})+\chi(\eta^{*}_{m-1},l)\\}\quad\text{if}\quad\rho^{*}_{m}<+\infty\\\
\\\
\eta^{*}_{m-1}\quad\text{if}\quad\rho^{*}_{m}=+\infty\end{array}\right.\bigg{)}\end{array}\right.$

We are now ready to present the verification theorem for our switching game.

###### Theorem 4

For each $i,j\in\mathcal{I}$, $x\in\mathbb{R}^{m}$, and assume that
$(\delta^{*},\nu^{*})\in\mathcal{A}^{i}\times\mathcal{B}^{j}$. Then we have
$V^{ij}(x)=J(x,\delta^{*},\nu^{*}).$

$Proof:$ First for each $\delta:=(\tau_{m},\xi_{m})_{m\geq
0}\in\mathcal{A}^{i}$ and $\nu:=(\rho_{n},\eta_{n})_{n\geq
0}\in\mathcal{B}^{j}$ we define $\theta=(\theta_{k})$ an increasing sequence
of stopping times by

$\begin{array}[]{ll}\qquad\qquad\theta_{0}=\tau_{0}=\rho_{0}=0\\\ \\\
\qquad\qquad\theta_{1}=\min(\tau_{1},\rho_{1})\\\ \\\
\qquad\qquad\theta_{2}=\min(\tau_{1}1\\!\\!1_{[\tau_{1}>\theta_{1}]}+\tau_{2}1\\!\\!1_{[\tau_{1}\leq\theta_{1}]},\rho_{1}1\\!\\!1_{[\rho_{1}>\theta_{1}]}+\rho_{2}1\\!\\!1_{[\rho_{1}\leq\theta_{1}]}),\\\
\\\
\qquad\qquad\theta_{3}=\min(\tau_{1}1\\!\\!1_{[\tau_{1}>\theta_{2}]}+\tau_{3}1\\!\\!1_{[\tau_{1}\leq\theta_{2}]},\rho_{1}1\\!\\!1_{[\rho_{1}>\theta_{2}]}+\rho_{3}1\\!\\!1_{[\rho_{1}\leq\theta_{2}]}\\\
\qquad\qquad\qquad\qquad\quad,\tau_{2}1\\!\\!1_{[\tau_{2}>\theta_{2}]}+\tau_{3}1\\!\\!1_{[\tau_{2}\leq\theta_{2}]},\rho_{2}1\\!\\!1_{[\rho_{2}>\theta_{2}]}+\rho_{3}1\\!\\!1_{[\rho_{2}\leq\theta_{2}]}),\\\
\\\
\qquad\qquad\qquad\qquad\qquad.....................................................\\\
\\\
\qquad\qquad\theta_{k}=\min(\tau_{1}1\\!\\!1_{[\tau_{1}>\theta_{k-1}]}+\tau_{k}1\\!\\!1_{[\tau_{1}\leq\theta_{k-1}]},\rho_{1}1\\!\\!1_{[\rho_{1}>\theta_{k-1}]}+\rho_{k}1\\!\\!1_{[\rho_{1}\leq\theta_{k-1}]},...\par\\\
\qquad\qquad\qquad\qquad,\tau_{k-1}1\\!\\!1_{[\tau_{k-1}>\theta_{k-1}]}+\tau_{k}1\\!\\!1_{[\tau_{k-1}\leq\theta_{k-1}]},\rho_{k-1}1\\!\\!1_{[\rho_{k-1}>\theta_{k-1}]}+\rho_{k}1\\!\\!1_{[\rho_{k-1}\leq\theta_{k-1}]}).\end{array}$

Then the cost functional is rewritten as:

$\begin{array}[]{ll}J(x,\delta,\nu)=\displaystyle\sum\limits_{k\geq
1}\mathbb{E}\bigg{[}\displaystyle\int_{\theta_{k-1}}^{\theta_{k}}e^{-rs}f^{a_{s}b_{s}}(X_{s})ds-\sum\limits_{m\geq
1}e^{-r\tau_{m}}C(\xi_{m-1},\xi_{m})1\\!\\!1_{[\tau_{m}=\theta_{k}]}\\\
\qquad\qquad\qquad\qquad+\displaystyle\sum\limits_{n\geq
1}e^{-r\rho_{n}}\chi(\eta_{n-1},\eta_{n})1\\!\\!1_{[\rho_{n}=\theta_{k}]}\bigg{]}.\end{array}$
(6.1)

Now, let $(\theta^{*})_{k}$ associated with $\delta^{*}$ and $\nu^{*}$, then
when $\theta^{*}_{k-1}<t<\theta^{*}_{k}$ we have

$rV^{a_{s}b_{t}}(X_{t})-\mathcal{A}V^{a_{t}b_{t}}(X_{t})=f^{a_{t}b_{t}}$ (6.2)

Then by Itô’s formula (see, e.g., Sect. IV.45 of [31]), we obtain

$\begin{array}[]{ll}\mathbb{E}\bigg{[}\displaystyle\int_{\theta^{*}_{k-1}}^{\theta^{*}_{k}}e^{-rs}f^{a_{s}b_{s}}(X_{s})ds\bigg{]}=\mathbb{E}\bigg{[}\displaystyle\int_{\theta^{*}_{k-1}}^{\theta^{*}_{k}}rV^{a_{s}b_{s}}(X_{s})-\mathcal{A}V^{a_{s}b_{s}}(X_{s})ds\bigg{]}\\\
\\\
\qquad\qquad\qquad\qquad\qquad\quad\;=\mathbb{E}\big{[}e^{-r\theta^{*}_{k-1}}V^{a_{\theta^{*}_{k-1}}b_{\theta^{*}_{k-1}}}(X_{\theta^{*}_{k-1}})-e^{-r\theta^{*}_{k}}V^{a_{\theta^{*}_{k-1}}b_{\theta^{*}_{k-1}}}(X_{\theta^{*}_{k}})\big{]}\end{array}$
(6.3)

Substituting this into (6.1), we obtain

$\begin{array}[]{ll}J(x,\delta^{*},\nu^{*})=\mathbb{E}\bigg{[}V^{ij}(x)+\displaystyle\sum_{k\geq
1}\big{[}\\{-V^{a_{\theta^{*}_{k-1}}b_{\theta^{*}_{k-1}}}(X_{\theta^{*}_{k}})+V^{a_{\theta^{*}_{k}}b_{\theta^{*}_{k}}}(X_{\theta^{*}_{k}})\\\
\qquad\qquad-\displaystyle\sum\limits_{m\geq
1}C(\xi^{*}_{m-1},\xi^{*}_{m})1\\!\\!1_{[\tau^{*}_{m}=\theta^{*}_{k}]}+\displaystyle\sum\limits_{n\geq
1}\chi(\eta^{*}_{n-1},\eta^{*}_{n})1\\!\\!1_{[\rho^{*}_{n}=\theta^{*}_{k}]}\\}e^{-r\theta^{*}_{k}}\big{]}\bigg{]}.\end{array}$
(6.4)

We now estimate the term in the right-hand side of (6.4) for each $k\geq 1$.

  * •

If $\theta^{*}_{k}=\tau^{*}_{m}$ for some $m\geq 1$ we have

$V^{\xi^{*}_{m}b_{\theta^{*}_{k-1}}}(X_{\theta^{*}_{k}})-V^{\xi^{*}_{m-1}b_{\theta^{*}_{k-1}}}(X_{\theta^{*}_{k}})-C(\xi^{*}_{m-1},\xi^{*}_{m})=0$
(6.5)

  * •

If $\theta^{*}_{k}=\rho^{*}_{n}$ for some $n\geq 1$ we have

$V^{a_{\theta^{*}_{k-1}}\eta^{*}_{n}}(X_{\theta^{*}_{k}})-V^{a_{\theta^{*}_{k-1}}\eta^{*}_{n-1}}(X_{\theta^{*}_{k}})+\chi(\eta^{*}_{n-1},\eta^{*}_{n})=0$
(6.6)

  * •

If $\theta^{*}_{k}=\tau^{*}_{m}=\rho^{*}_{n}$ for some $m,n\geq 1$ we have

$V^{\xi^{*}_{m}\eta^{*}_{n}}(X_{\theta^{*}_{k}})-V^{\xi^{*}_{m-1}\eta^{*}_{n-1}}(X_{\theta^{*}_{k}})+\chi(\eta^{*}_{n-1},\eta^{*}_{n})-C(\xi^{*}_{m-1},\xi^{*}_{m})=0.$
(6.7)

By (6.4) and the estimates in (6.5)-(6.7) it follows that

$J(x,\delta^{*},\nu^{*})=V^{ij}(x)\qquad\qquad\Box$

##  7 Numerical results

In this section, we present the results of some numerical simulations on the
game by using MATLAB, here we apply the policy iteration algorithm for solving
numerically(2.9).
In particular, for the numerical example we consider a two-regime switching
problem where the diffusion is independent of the regime and follows a
geometric Brownian motion, i.e $b^{ij}(x)=bx$ and $\sigma^{ij}(x)=\sigma x$
for some $b\in\mathbb{R}$ and $\sigma>0$ .
We consider the following game problem:

$\begin{array}[]{ll}\qquad\quad\mathcal{I}=\\{1,2\\}\\\ \\\ f^{11}=5x,\qquad
f^{12}=x,\\\ \\\ f^{21}=-x,\qquad f^{22}=-4x,\\\ \\\
r=0.15,\quad\sigma=0.2,\quad b=0.01,\end{array}$ (7.1)

In the flowing figures, we plot the value functions for different switching
costs.

(a) $C(1,2)=2,C(2,1)=5,\chi(1,2)=2,\chi(2,1)=5$

(b) $C(1,2)=5,C(2,1)=-1,\chi(1,2)=2,\chi(2,1)=2$

(c) $C(1,2)=2,C(2,1)=2,\chi(1,2)=-1,\chi(2,1)=5$

(d) $C(1,2)=4,C(2,1)=-2,\chi(1,2)=-1,\chi(2,1)=3$

Figure 1: Value
functions:$V^{11}$(black),$V^{12}$(red),$V^{21}$(green),$V^{22}$(blue)

.

Next, we plot the optimal strategies of our zero-sum stochastic differential
game. We focus on exemple $(a)$ in figure $1$ . First let $\mathcal{I}_{ij\to
kl}=\\{x\in\mathbb{R}:V^{ij}(x)=V^{kl}(x)-C(i,k)+\chi(j,l)\\}$. Then by figure
$1$ (exemple $(a))$ we get

$\begin{array}[]{ll}\mathcal{I}_{11\to
21}=(-\infty,-0.25]\qquad\quad\mathcal{I}_{11\to 12}=[1.46,+\infty)\\\
\mathcal{I}_{12\to 21}=(-\infty,-0.62]\qquad\quad\mathcal{I}_{12\to
22}=[-0.62,-0.25]\\\ \mathcal{I}_{21\to
11}=[0.33,1.46]\qquad\quad\;\;\;\mathcal{I}_{21\to 12}=[1.46,+\infty)\\\
\mathcal{I}_{22\to 21}=(-\infty,-0.62]\qquad\quad\mathcal{I}_{22\to
12}=[0.33,+\infty)\end{array}$ (7.2)

Figure 2: optimal strategies and state simulation

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