# Nonzero-sum stochastic impulse games with an application in competitive
retail energy markets

René Aïd111LEDa, Université Paris Dauphine, PSL research university,
<EMAIL_ADDRESS>The author thanks for their support the Finance for
Energy Markets, the Chaire EDF-CACIB Finance & Sustainable Development:
Quantitative Approach and the ANR project EcoREES ANR-19-CE05-0042. Lamia Ben
Ajmia222ENIT-LAMSIN, University of Tunis El Manar, Tunis, Tunisia,
<EMAIL_ADDRESS>M’hamed Gaïgi333ENIT-LAMSIN, University of Tunis El
Manar, Tunis, Tunisia<EMAIL_ADDRESS>Mohamed Mnif444ENIT-LAMSIN,
University of Tunis El Manar, Tunis, Tunisia<EMAIL_ADDRESS>
###### Abstract

We study a nonzero-sum stochastic differential game with both players adopting
impulse controls, on a finite time horizon. The objective of each player is to
maximize her total expected discounted profits. The resolution methodology
relies on the connection between Nash equilibrium and the corresponding system
of quasi-variational inequalities (QVIs in short). We prove, by means of the
weak dynamic programming principle for the stochastic differential game, that
the value function of each player is a constrained viscosity solution to the
associated QVIs system in the class of linear growth functions. We also
introduce a family of value functions converging to our value function of each
player, and which is characterized as the unique constrained viscosity
solutions of an approximation of our QVIs system. This convergence result is
useful for numerical purpose. We apply a probabilistic numerical scheme which
approximates the solution of the QVIs system to the case of the competition
between two electricity retailers. We show how our model reproduces the
qualitative behaviour of electricity retail competition.

Key words : stochastic impulse games, Nash equilibrium, viscosity solution.

MSC Classification (2000) : 93E20, 49L20,49L25,49N70

###### Contents

  1. 1 Introduction
  2. 2 The model
    1. 2.1 The Dynamics of the state process
    2. 2.2 Value Function
    3. 2.3 HJB QVIs
  3. 3 Properties of the value function
    1. 3.1 Bound on the value functions and Boundary properties
    2. 3.2 Terminal condition
  4. 4 Verification theorem
  5. 5 Viscosity characterization
  6. 6 Comparison principle of viscosity solutions
  7. 7 Numerical illustrations

##  1 Introduction

In this paper, we study a general two-players nonzero-sum stochastic
differential game with impulse controls, on a finite horizon by characterising
the associated Nash Equilibrium. Stochastic differential games have been
widely studied. The theory of two-player zero-sum differential games was
pioneered by Isaacs [12]. Evans and Souganidis [10] studied differential games
by means of the viscosity theory, characterizing the upper and the lower value
functions as the unique viscosity solutions to the corresponding Hamilton-
Jacobi-Bellman-Isaacs partial differential equations. In the literature, many
authors focused on stochastic differential game with continuous controls. The
case of impulse games has less been considered. Stochastic impulse games
(SIGs) are at the intersection between differential game theory and stochastic
impulse control. They are more realistic in modeling finance problems for
example. Aid et al. [3] studied a problem of Nash equilibrium in a general
nonzero-sum impulse game for two players. They derived the corresponding
system of quasi-variational inequalities (QVIs) and provides an application in
the case of a one-dimensional state problem of forex trading competition
between two players. Thanks to a verification theorem, they identified
reasonable candidates for the intervention and continuation regions of both
players and their strategies. Cosso [9] studied a two-players zero-sum
stochastic differential game with both players adopting impulse controls, on a
finite time horizon. The HJBI partial differential equation of the game turns
out to be a double-obstacle quasi-variational inequality. The two obstacles
are implicitly given. He proved that the upper and lower value functions
coincide. He also showed by means of the dynamic programming principle for the
stochastic differential game, that the upper and the lower value functions are
the unique viscosity solution to the HJBI equation, and so the game admits a
value.

Our model is inspired by Aid et al. [3] together with the electricity retail
pricing model of Basei [5]. In our model, we assume that retailers buy the
energy (electricity or gas) in the wholesale market at the same price. The
price of the energy is modeled by the Black-Scholes process. Finally,
retailers sell the energy to their final consumers at a fixed price which
could be different for each retailer. Both retailers’ objective is to maximize
their total expected discounted profits. Their instantaneous profits are
composed of two parts: sale revenue (market share times retail price) and
sourcing cost (market share times wholesale market price). The interaction
between the opposing retailers is implicitly considered through the market
share. We formulate the problem as a nonzero-sum stochastic impulse control
game. The impulse control problem of each player is formulated with three
state variables (besides time variable) related to the wholesale price, the
price she offers and the proposed price of the other player which evolve in
$\mathbb{R}_{+}^{3}$. The strategy of the retailers consist on changing the
energy price in discrete time by using impulse control. Such model reproduces
the fierce competition between the two players which characterizes modern
deregulated energy markets.

To tackle our problem, we use the dynamic programming approach. We prove that
the two players can act simultaneously and the order of intervention of the
two players is not important since we can permute between the two impulse
operators of players, which is not the case in the model considered by Aid et
al. [3]. We prove a verification theorem in a general setting which shows that
under some regularity assumptions, if we solve the system of QVIs, then we
provide the Nash equilibrium of the impulse game which characterizes
completely the value function of each player. Then we prove that the value
functions of the players are viscosity solutions of the associated system of
QVIs.

In our set-up, it is not obvious to prove the continuity of the value
functions of the two players, and so it is then natural to consider the
concept of discontinuous viscosity solutions, which provides by now a well
established method for dealing with stochastic control problems. More
precisely, we need to consider constrained viscosity solutions since the state
process is allowed to be everywhere in $\mathbb{R}_{+}^{3}$ and so the system
of QVIs is satisfied even in the boundary of the solvency region. For the
comparison theorem, we consider a small variation of our original model by
adding a fixed cost paid by player $i$ when player $j$ makes an intervention.
We prove that the value functions of the perturbed problem converge pointwise
to the value functions of the original problem. We provide a suitable strict
supersolution of the system of QVIs solution in the spirit of Crandal et al.
[8]. In the comparison theorem, we construct a test function by introducing a
penalization function related to the distance from an element of the solvency
region to the boundary, so that the optimum associated with the strict
supersolution is not attained on the boundary. We adapt the arguments of
Barles [4] which needs a smooth boundary. Similar difficulties have been
studied in Akian et al. [1] and Ly Vath et al. [14]. It is known that the
function distance is continuously differentiable with bounded derivatives in
the smooth part of the boundary (see Gilbarg [11]). In our context, the
boundary is smooth except the corner lines. We prove that one can compare a
subsolution with a supersolution to the system of QVIs provided that one can
compare them at the terminal date but also on the corner lines of the solvency
boundary. It is also known that the comparison theorem is crucial to
approximate numerically the system of QVIs solution. In this work, we limit
our numerical study to present the numerical scheme and some numerical results
for the optimal intervention strategy. We claim that the value functions could
be obtained as the limit of an iterative procedure where each step is an
optimal stopping problem and the reward function is related to the impulse
operator. We use a numerical approximation algorithm based on quantization
procedure instead of finite difference methods to approximate the value
functions, the intervention and the no-intervention regions of each player.
The convergence of our numerical scheme, in particular, the monotonicity, the
stability and the consistency properties will be postponed in a future
research. In Aïd et al. [2], the authors tackled a nonzero-sum stochastic
impulse games by implementing a policy iteration algorithm. To the best of our
knowledge, it is the only paper which is interested to approximate numerically
the solution of a system of QVIs, and it opens up the possibility of finding
other numerical schemes to approximate the value functions and the associated
optimal strategies.

The paper is organized as follows. Section 2 is devoted to present the model,
to introduce rigorously the stochastic differential game, Nash equilibrium and
the associated system of QVIs. Then, we study the properties of the value
functions. In section 3 , we give some properties of the value function. In
section , we prove a verification theorem which gives a characterization of
the Nash equilibrium. In section 5, we prove that the two value functions are
viscosity solutions to the system of QVIs followed in section 6 by a
comparison theorem for an auxiliary family of value functions. Finally, in
section 7, we explain the numerical scheme and we give some numerical results.

##  2 The model

This section presents the details of the model.

###  2.1 The Dynamics of the state process

Let $(\Omega,{\cal F},\mathbb{P})$ be a probability space equipped with a
filtration $\mathbb{F}:=({\cal F}_{t})_{0\leq t\leq T}$ supporting a one
dimensional Brownian motion $W$ on a finite horizon $[0,T]$, $T<\infty$. We
assume that the wholesale energy price is modeled by a Black-Scholes process.
For a fixed $t\in[0,T]$ and $x\in\mathbb{R}_{+}$, the process $X^{t,x}$
satisfies

$\displaystyle dX_{s}^{t,x}=X_{s}^{t,x}(\mu ds+\sigma
dW_{s}),\,s\in[t,T],\,X_{s}^{t,x}=x,\,s\in[0,t],$ (2.1)

where $\mu\in\mathbb{R}$ and $\sigma>0$.
We consider a nonzero-sum impulse game where two players, which are retailers,
can intervene on a continuous time. After buying the energy, they sell it to
consumers. The retailers price is modeled by a stochastic process
$Y:=(Y^{1},Y^{2})\in(0,\infty)\times(0,\infty)$ which is a piecewise-constant
process. More precisely, the control strategy of each player, representing her
intervention, is represented by an impulse control strategy $\alpha^{i}$ $:=$
$(\tau_{n}^{i},\zeta_{n}^{i})_{n\geq 1}$ for $i\in\\{1,2\\}$, where for
$t\in[0,T]$, $t\leq\tau_{1}^{i}$ $\leq$ $\ldots$ $\tau_{n}^{i}$ $\leq$
$\ldots<T$ are $\mathbb{F}$ stopping times representing the intervention times
of the player $i$ and $\zeta_{n}^{i}$, $n$ $\geq$ $1$, are ${\cal
F}_{\tau_{n}^{i}}$-measurable random variables. The sequence
$(\tau_{n}^{i},\zeta_{n}^{i})_{n\geq 1}$ may be a priori finite or infinite.
The process $Y^{i}$ starts from $Y^{i}_{t^{-}}=y^{i}$ and evolves according to
:

$\displaystyle dY^{i}_{s}$ $\displaystyle=$ $\displaystyle
0,\;\;\;\;\;\tau_{n}^{i}\leq s<\tau_{n+1}^{i}$ (2.2) $\displaystyle
Y^{i}_{\tau_{n+1}^{i}}$ $\displaystyle=$
$\displaystyle\Gamma^{i}(Y^{i}_{\tau_{n+1}^{i-}},\zeta_{n+1}^{i}),$ (2.3)

where $\Gamma^{i}(y^{i},\zeta^{i})=y^{i}e^{\lambda\zeta^{i}}$ and $\lambda>0$.
We further assume that, for $i\in\\{1,2\\}$,
$\zeta^{i}\in[\zeta_{min},\zeta_{max}]$ where $\zeta_{min}<0<\zeta_{max}$ are
fixed constants. This assumption could be seen as a restriction imposed on the
players by the regulator to avoid over-shifting prices. Moreover, with our
specific choice of the function $\Gamma^{i}$, on the one hand we ensure the
positivity of the new prices, on the other hand, the players could increase
the price with a positive impulse (i.e. $\zeta>0$) and decrease it with a
negative impulse (i.e. $\zeta<0$).
We then naturally introduce solvency region: ${\cal S}=(0,\infty)^{3}$, and we
denote its closure and its boundary by

$\displaystyle\bar{\cal S}=\mathbb{R}_{+}^{3},\,\partial{\cal S}=\bar{\cal
S}\setminus{\cal S}.$

In the sequel, we explicit the corner lines of the boundary $\partial{\cal S}$
as follows:

$\displaystyle\partial^{x}{\cal
S}=\mathbb{R}^{+}\times\\{(0,0)\\},\,\partial^{y_{1}}{\cal
S}=\\{0\\}\times(0,\infty)\times\\{0\\},\,\partial^{y_{2}}{\cal
S}=\\{(0,0)\\}\times(0,\infty).$

We denote by

$\displaystyle D_{0}=\partial^{x}{\cal S}\cup\partial^{y_{1}}{\cal
S}\cup\partial^{y_{2}}{\cal S}.$

Let $Z=(X,Y^{1},Y^{2})$ the process living in the space ${\cal S}$. We denote
by $Z^{t,z,\alpha_{1},\alpha_{2}}$ the process $Z$ that starts at
$z=(x,y^{1},y^{2})$ at time $t$ and is controlled by $\alpha^{1}$ and
$\alpha^{2}$.
For the rest of the paper, to simplify the notations and when there is no
ambiguity, we will use $Z^{t,z}$ instead of $Z^{t,z,\alpha^{1},\alpha^{2}}$.
Hence when we use $Z^{t,z}$, one has to check under which control the process
$Z$ evolves. In most cases it can be easily guessed.

###  2.2 Value Function

The objective of the player is to maximize the gain function $J^{i}$ defined
as follows:

$\displaystyle J^{i}(t,z,\alpha^{i},\alpha^{j})$ $\displaystyle=$
$\displaystyle\mathbb{E}[\int_{t}^{T}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{s})ds-\sum_{t\leq\tau_{k}^{i}<T}e^{-\rho^{i}(\tau_{k}^{i}-t)}\phi^{i}(Y^{i,t,y^{i},\alpha^{i}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})$
(2.4) $\displaystyle+$ $\displaystyle
e^{-\rho^{i}(T-t)}g^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{T})],$

where, for $i\in\\{1,2\\}$, $j\neq i$, $\rho^{i}$ is a discount factor;
$f^{i}:\mathbb{R}_{+}^{3}\longrightarrow\mathbb{R}$ is the running payoff,
$g^{i}:\mathbb{R}_{+}^{3}\longrightarrow\mathbb{R}$ is the terminal payoff and
$\phi^{i}:\mathbb{R}_{+}\times[\zeta_{min},\zeta_{max}]\longrightarrow\mathbb{R}$
is the intervention cost function. In order to have a realistic model, the
running payoff and the terminal payoff of player $i$ depend on the price she
asks, the opponent pricing choice and the wholesale price proposed in the
market. In our model, the retailer margin of player $i$, represented by
$y^{i}-x$, is truncated by a linear function of the retail price. This
regulatory constraint forces retailers to drop down their prices when the
wholesale price falls. In our model $f^{i}$ and $g^{i}$ take the form:

$\displaystyle
f^{i}(x,y^{i},y^{j})=g^{i}(x,y^{i},y^{j})=((y^{i}-x)\pi^{i}(y^{i},y^{j}))\wedge
K^{i}x,$ (2.5)

here the function $\pi^{i}$ represents the market share and is defined by:

$\displaystyle\pi^{i}(y^{i},y^{j})=\begin{cases}1&\quad\text{if
}y^{i}-y^{j}\leq-\Delta,\\\
-\frac{1}{2\Delta}(y^{i}-y^{j}-\Delta)&\quad\text{if
}-\Delta<y^{i}-y^{j}<\Delta,\\\ 0&\quad\text{if
}y^{i}-y^{j}\geq\Delta,\end{cases}$ (2.6)

$K^{i}$ and $\Delta$ are positive constants. For $y^{i}=y^{j}$, the market is
split equally between the two retailers. The following assumption will be
needed in our work:
$({\cal H})$ (i) There exist positive constants $C_{1}^{\phi}$ and
$C_{2}^{\phi}$ s.t.

$\displaystyle C_{1}^{\phi}\leq\phi^{i}(y^{i},\zeta)\leq
C_{2}^{\phi}\,\,\forall(y^{i},\zeta)\in\mathbb{R}_{+}^{2}\times[\zeta_{min},\zeta_{max}].$

(ii)The function $\phi^{i}$ is continuous.

###### Remark 2.1

Thanks to the choice of $f^{i}$ and $g^{i}$, we deduce that for all $z\in{\cal
S}$, $f^{i}(z)\leq K^{i}x$ and $g^{i}(z)\leq K^{i}x$. Moreover
$f^{i}(z)\geq-(C^{i}+x)$ and $g^{i}(z)\geq-(C^{i}+x)$, where $C^{i}$ is a
positive constant.

###### Definition 2.1

We say that the couple $(\alpha^{i,*},\alpha^{j,*})$ is a Nash equilibrium,
if, for every other couple of strategies $(\alpha^{i},\alpha^{j})$, we have :

$J^{i}(t,z,\alpha^{i,*},\alpha^{j,*})\geq
J^{i}(t,z,\alpha^{i},\alpha^{j,*})\quad\hbox{and}\quad
J^{j}(t,z,\alpha^{i,*},\alpha^{j,*})\geq J^{j}(t,z,\alpha^{i,*},\alpha^{j}).$

We define the number of intervention times between $t$ and $T$ by:

$\displaystyle{\cal N}_{t}(\alpha^{i}):=Card\\{n\;:t\leq\tau_{n}^{i}<T\\}.$
(2.7)

We need to impose a uniform integrability condition on ${\cal
N}_{t}(\alpha^{i})$.

###### Definition 2.2

We define ${\cal D}_{t}^{i}$ as follows

$\displaystyle{\cal
D}_{t}^{i}:=\\{\alpha^{i}=(\tau_{n}^{i},\zeta_{n}^{i})_{n\geq
1},\mbox{s.t.}\,{\cal N}_{t}(\alpha^{i})\mbox{ is uniformly integrable}\\}.$
(2.8)

For technical reason related to the dynamic programming principle, see Remark
5.2 in Bouchard and Touzi [7], we shall restrict the admissible strategies to
the following set. For $i\in\\{1,2\\}$

$\displaystyle{\cal
A}^{i}_{t}:=\\{\alpha^{i}=(\tau^{i}_{n},\zeta^{i}_{n})_{n\geq
1};\;s.t.\;\tau^{i}_{1}\geq t,\,\alpha^{i}\in{\cal D}_{t}^{i}\mbox{ and
}\alpha^{i}\mbox{ is independent of }{\cal F}_{t}\\},$ (2.9)

and therefore, we define the value functions as $(v_{i})_{i\in\\{1,2\\}}$ on
$[0,T]\times{\cal S}$ by

$\displaystyle v^{i}(t,z):=\displaystyle\sup_{\alpha^{i}\in{\cal
A}^{i}_{t}}J^{i}(t,z,\alpha^{i},\alpha^{j})\quad i\neq j,$ (2.10)

where $\alpha^{j}$ is the best response against $\alpha^{i}$.

###### Remark 2.2

It is easy to see that if $\alpha^{i*}=(\tau_{n}^{i*},\zeta_{n}^{i*})$ exists,
then, we have $\mathbb{E}[{\cal N}_{t}(\alpha^{i*})]<\infty$. In our case, to
obtain the growth property of the value function, we need a uniform
integrability condition of the number of intervention times which is stronger
itegrability condition. In fact, for $i\in\\{1,2\\}$, $\alpha^{i}$ an
admissible strategy and $\alpha^{j}$ the best response against $\alpha^{i}$,
from Assumption $({\cal H})$(i), we have:

$\displaystyle J^{i}(t,z,\alpha^{i},\alpha^{j})$ $\displaystyle\leq$
$\displaystyle\mathbb{E}[\int_{t}^{T}f^{i}(Z^{t,z}_{s})ds-
C_{1}^{\phi}e^{-\rho^{i}T}{\cal N}_{t}(\alpha^{i})+g^{i}(Z^{t,z}_{T})]$
$\displaystyle\leq$ $\displaystyle\mathbb{E}[\int_{t}^{T}K^{i}X_{s}^{t,x}ds-
C_{1}^{\phi}e^{-\rho^{i}T}{\cal N}_{t}(\alpha^{i})+K^{i}X_{T}^{t,x}].$

where the second inequality is obtained from Remark 2.1. If $\mathbb{E}[{\cal
N}_{t}(\alpha^{i})]=\infty$, then $J^{i}(t,z,\alpha^{i},\alpha^{j})=\infty$.
Then, from (2.2), we deduce that:

$\displaystyle J^{i}(t,z,\alpha^{i},\alpha^{j})=-\infty$

On the other hand, by choosing the null strategy by player $i$,
$J^{i}(t,z,0,\alpha^{j})>-\infty$, where here $\alpha^{j}$ denotes the best
response against the null strategy. Hence, an optimal strategy can not have
infinite expected number of interventions.

###  2.3 HJB QVIs

In this subsection, we provide an heuristic derivation of the system of QVIs.
We define the non-local operator called intervention operator ${\cal M}^{i}$
for $i\in\\{1,2\\}$ by:

${\cal
M}^{i}h(t,z)=\sup_{\zeta^{i}\in[\zeta_{min},\zeta_{max}]}\Big{\\{}h(t,x,\Gamma^{i}(y^{i},\zeta^{i}),y^{j})-\phi^{i}(z,\zeta^{i})\Big{\\}},$

for all locally bounded $h:[0,T]\times S\longrightarrow{\mathbb{R}}$. It is
known that if the function $h$ is upper semicontinuous, by selection
measurable theorem, there exists a Borel-measurable function
$\hat{\zeta}^{i*}:[0,T]\times S\longrightarrow{\mathbb{R}}$ such that

${\cal
M}^{i}h(t,z)=h(t,x,\Gamma^{i}(y^{i},\hat{\zeta}^{i*}(t,y^{i})),y^{j})-\phi^{i}(z,\hat{\zeta}^{i*}(t,y^{i})),$

(See for example Proposition 7.33, chapter 7, in [6]). Similarly, we define
the second intervention operator ${\cal H}^{i}$ for $i\in\\{1,2\\}$ by

${\cal H}^{i}h(t,z)=h(t,x,y^{i},\Gamma^{j}(y^{j},\zeta^{j})),$

where
$\zeta^{j}=\displaystyle\operatorname*{argmax}_{\zeta\in[\zeta_{min},\zeta_{max}]}\Big{\\{}h(t,y^{i},\Gamma^{j}(y^{j},\zeta))-\phi^{j}(z,\zeta)\Big{\\}}$.
Thanks to the following Proposition, the two players could act together. In
fact, we can permute between ${\cal M}^{i}$ and ${\cal H}^{i}$ for
$i\in\\{1,2\\}$ and so we get rid of an assumption in Cosso [9] and Aid et al.
[3]. Such result is obtained as a consequence from the structure of gain of
the two players. In our model, when player $i$ acts an intervention, she pays
a cost $\phi^{i}(y^{i},\zeta^{i})$. The permutation between the two
intervention operators holds.

###### Proposition 2.1

For all $(t,z)\in[0,T]\times{\cal S}$, we have

${\cal M}^{i}{\cal H}^{i}v^{i}(t,z)={\cal H}^{i}{\cal M}^{i}v^{i}(t,z),\forall
i,j\in\\{1,2\\},$

_Proof:_ From the definition of ${\cal M}^{i}$ and ${\cal H}^{i}$ we have

$\displaystyle{\cal M}^{i}{\cal H}^{i}v^{i}(t,z)=$
$\displaystyle\sup_{\zeta^{i}\in[\zeta_{min},\zeta_{max}]}\Big{\\{}{\cal
H}^{i}v^{i}(t,x,\Gamma^{i}(y^{i},\zeta^{i}),y^{j})-\phi^{i}(y^{i},\zeta^{i})\Big{\\}},$
$\displaystyle=$
$\displaystyle\sup_{\zeta^{i}\in[\zeta_{min},\zeta_{max}]}\Big{\\{}v^{i}(t,x,\Gamma^{i}(y^{i},\zeta^{i}),\Gamma^{j}(y^{j},\zeta^{j}))-\phi^{i}(y^{i},\zeta^{i})\Big{\\}},$
$\displaystyle=$ $\displaystyle{\cal
M}^{i}v^{i}(t,x,y^{i},\Gamma^{j}(y^{j},\zeta^{j}))$ $\displaystyle=$
$\displaystyle{\cal H}^{i}{\cal M}^{i}v^{i}(t,x,y^{i},y^{j})$ $\displaystyle=$
$\displaystyle{\cal H}^{i}{\cal M}^{i}v^{i}(t,z)$

$\Box$
We define $\overline{{\cal I}}^{i}$ as the intervention region of player $j$
and ${\cal I}^{i}=\bar{\cal S}\setminus\overline{{\cal I}}^{i}$. The
associated QVIs of our control problem are as follows:

$\displaystyle\min\\{v^{i}-{\cal M}^{i}{\cal H}^{i}v^{i},v^{i}-{\cal
H}^{i}v^{i}\\}$ $\displaystyle=$ $\displaystyle
0\quad\hbox{in}\quad\overline{{\cal I}}^{i}$ (2.11)
$\displaystyle\min\\{-\displaystyle\frac{\partial v^{i}}{\partial t}-{\cal
L}v^{i}-f^{i}+\rho^{i}v^{i},v^{i}-{\cal M}^{i}v^{i}\\}$ $\displaystyle=$
$\displaystyle 0\quad\hbox{in}\quad{\cal I}^{i}$ (2.12) $\displaystyle v^{i}$
$\displaystyle=$ $\displaystyle g^{i}\quad\hbox{in
}\quad\\{T\\}\times\bar{\cal S},$ (2.13)

where

$\displaystyle{\cal L}v^{i}(t,z)$ $\displaystyle=$ $\displaystyle\mu
x\frac{\partial v^{i}}{\partial
x}(t,z)+\frac{\sigma^{2}x^{2}}{2}\frac{\partial^{2}v^{i}}{\partial
x^{2}}(t,z),$ (2.14) $\displaystyle{\cal M}^{i}v^{i}(t,z)$ $\displaystyle=$
$\displaystyle\sup_{\zeta^{i}\in[\zeta_{min},\zeta_{max}]}\Big{\\{}v^{i}(t,x,\Gamma^{i}(y^{i},\zeta^{i}),y^{j})-\phi^{i}(z,\zeta^{i})\Big{\\}},$
(2.15) $\displaystyle{\cal H}^{i}v^{i}(t,z)$ $\displaystyle=$ $\displaystyle
v^{i}(t,x,y^{i},\Gamma^{j}(y^{j},\zeta^{j})),$ (2.16)

and
$\zeta^{j}=\displaystyle\operatorname*{argmax}_{\zeta\in[\zeta_{min},\zeta_{max}]}\Big{\\{}v^{j}(t,y^{i},\Gamma^{j}(y^{j},\zeta))-\phi^{j}(z,\zeta)\Big{\\}}$.

###### Remark 2.3

${\cal M}^{i}v^{i}(t,z)$ represents the value function for player $i$ when she
makes the optimal impulse in order to increase her value function. ${\cal
H}^{i}v^{i}(t,z)$ represents the value function for player $i$, when player
$j$ takes the best intervention.

###### Remark 2.4

The system of variational inequalities (2.11)-(2.12) is satisfied up to the
boundary of $\bar{\cal S}$. Since the dynamics of the process $Z$ is allowed
to go everywhere in $\bar{\cal S}$. We recall that the wholesale price $X$ is
modeled by a geometric Brownian motion and the process $(Y^{i},Y^{j})$ are
non-negative piecewise constant and at the intervention times, the price
offered by each player is modeled by the exponential function.

##  3 Properties of the value function

###  3.1 Bound on the value functions and Boundary properties

We start this section with the following lemma which gives a sharp upper bound
on the value function $v^{i}$ for $i\in\\{1,2\\}$.

###### Proposition 3.1

We fix $i\in\\{1,2\\}$. Under Assumption $({\cal H})$, for all
$(t,z)\in[0,T]\times\bar{\cal S}$, we have

$\displaystyle|v^{i}(t,z)|\leq C(1+x),$ (3.1)

where $C$ is a positive constant independent of t and z.

_Proof:_ We consider the no impulse strategy for player i denoted by $0$ and
$\alpha^{j}$ the best response against the null strategy, from the definition
of $f^{i}$ and $g^{i}$, Remark 2.1 and the positivity of the process $Y^{i}$,
we obtain

$\displaystyle v^{i}(t,z)$ $\displaystyle\geq$ $\displaystyle
J^{i}(t,z,0,\alpha^{j})$ (3.2) $\displaystyle=$
$\displaystyle\mathbb{E}[\int_{t}^{T}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z,0,\alpha^{j}}_{s})ds+e^{-\rho^{i}(T-t)}g^{i}(Z^{t,z,0,\alpha^{j}}_{T})]$
$\displaystyle\geq$
$\displaystyle-\mathbb{E}[\int_{t}^{T}e^{-\rho^{i}(s-t)}X_{s}^{t,x}ds+e^{-\rho^{i}(T-t)}X_{T}^{t,x}]$
$\displaystyle\geq$ $\displaystyle-C(1+x),$

where the last inequality is obtained from (2.1), and $C$ is a generic
positive constant which could change from line to line.
Let $(\alpha^{i},\alpha^{j})$ be an admissible strategy. Using Assumption
$({\cal H})(i)$, we have

$\displaystyle J^{i}(t,z,\alpha^{i},\alpha^{j})$ $\displaystyle\leq$
$\displaystyle\mathbb{E}[\int_{t}^{T}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z}_{s})ds+e^{-\rho^{i}(T-t)}g^{i}(Z^{t,z}_{T})]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}[\int_{t}^{T}K^{i}X^{t,x}_{s}ds+K^{i}X^{t,x}_{T}]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}[\int_{t}^{T}K^{i}xe^{\mu(s-t)}ds+K^{i}xe^{\mu(T-t)}].$

Since $(\alpha^{i},\alpha^{j})$ is arbitrary, we obtain

$\displaystyle v^{i}(t,z)\leq Cx.$ (3.3)

From inequalities (3.2) and (3.3), we deduce that (3.1) is proved.
$\Box$
We turn to the behaviour of the values functions on the corner lines of the
boundary $\partial{\cal S}$. The following proposition determines the value
function on the corner lines of the boundary $\partial{\cal S}$.

###### Proposition 3.2

For all $t\in[0,T)$, We have

$\displaystyle v^{i}(t,z)$ $\displaystyle=$ $\displaystyle 0\,\,\mbox{ for all
}z\in\partial^{y^{j}}{\cal S}$ (3.4) $\displaystyle v^{i}(t,z)$
$\displaystyle=$ $\displaystyle 0\,\,\mbox{ for all }z\in\partial^{y^{i}}{\cal
S}$ (3.5) $\displaystyle v^{i}(t,z)$ $\displaystyle=$
$\displaystyle-\frac{x}{2}(\frac{e^{(\mu-\rho^{i})(T-t)}-1}{\mu-\rho^{i}}+e^{(\mu-\rho^{i})(T-t)})\,\,\mbox{
for all }z\in\partial^{x}{\cal S}$ (3.6)

_Proof:_ $\star$ From the definition of (2.5), we have
$f^{j}(0,0,y^{j})=g^{j}(0,0,y^{j})=0$ for all $y^{j}>0$. If player $j$
intervenes, her payoff remains unchanged. On the other hand, as
$\Gamma^{i}(0,\zeta^{i})=0$, any intervention of player $i$ will not change
the proposed retailer price, and so it not optimal for the two players to
intervene. For the price in wholesale market, it remains equal to zero on
$[t,T)$. Then, we deduce (3.4).
$\star$ Since $f^{i}(0,y^{i},0)=g^{i}(0,y^{i},0)=0$ for all $y^{i}>0$, the
payoff of player $i$ remains equal to 0. As above and since $y^{j}=0$, any
intervention of player $j$ will not change her proposed retailer price. This
shows that (3.5) holds.
$\star$ Since $\pi^{i}(0,0)=\frac{1}{2}$, and for any intervention from one of
the two players, their retailer prices remain equal to 0, the market share of
each player remains unchanged. From the definition of the value function
$v^{i}$, we deduce that

$\displaystyle
v^{i}(t,x,0,0)=-\mathbb{E}[\int_{t}^{T}e^{-\rho^{i}(s-t)}\frac{X_{s}^{t,x}}{2}ds+e^{-\rho^{i}(T-t)}\frac{X_{T}^{t,x}}{2}].$

From the definition of the wholesale price, we have
$\mathbb{E}[X_{s}^{t,x}]=xe^{\mu(s-t)}$. This shows that

$\displaystyle
v^{i}(t,x,0,0)=-\frac{x}{2}\big{(}\int_{t}^{T}e^{(\mu-\rho^{i})(s-t)}ds+e^{(\mu-\rho^{i})(T-t)}\big{)},$

and so (3.6) is proved.  $\Box$

###  3.2 Terminal condition

We determine the right terminal condition of the value functions. We recall
that in our model, at the terminal date, each player can not do an
intervention. We obtain such characterization by considering the lower (resp.
upper) semicontinuous envelope of $v^{i}$, for $i\in\\{1,2\\}$. For all
$z\in{\cal S}$, we set

$\displaystyle\bar{v}^{i}(T,z):=\limsup_{\tiny\begin{array}[]{c}(t,z^{\prime})\rightarrow(T,z)\\\
t<T,z^{\prime}\in{\cal S}\end{array}}v(t,z^{\prime}),$
$\displaystyle\underline{v}^{i}(T,z):=\liminf_{\tiny\begin{array}[]{c}(t,z^{\prime})\rightarrow(T,z)\\\
t<T,z^{\prime}\in{\cal S}\end{array}}v(t,z^{\prime})$

###### Proposition 3.3

We have:

$\displaystyle{\underline{v}}^{i}(T,z)\;=\;{\overline{v}}^{i}(T,z)\;=\;g^{i}(z),\;\;\;\forall
z\in\bar{\cal S}.$

_Proof:_ 1) Fix some $z$ $\in$ $\bar{\cal S}$ and consider some sequence
$(t_{m},z_{m})_{m}$ $\in$ $[0,T)\times{\cal S}$ converging to $(T,z)$ and s.t.
$\displaystyle\lim_{m\rightarrow\infty}v^{i}(t_{m},z_{m})$ $=$
$\underline{v}^{i}(T,z)$. By taking the no impulse control strategy for player
$i$ noted by 0, and $\alpha^{j}$ the best response against this strategy by
player $j$ on $[t_{m},T]$, we have

$\displaystyle v^{i}(t_{m},z_{m})$ $\displaystyle\geq$
$\displaystyle\mathbb{E}[\int_{t_{m}}^{T}e^{-\rho^{i}(s-t_{m})}f^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{s})ds+e^{-\rho^{i}(T-t_{m})}g^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{T})].$
(3.8)

where

$\displaystyle
Z^{t_{m},z_{m},0,\alpha^{j}}=(X^{t_{m},x_{m}},y^{i}_{m},Y^{j,t_{m},y^{j}_{m},\alpha^{j}}).$

We fix $\epsilon>0$. As $(t_{m},z_{m})_{m}$ converges to $(T,z)$ when $m$ goes
to infinity, there exists $m_{0}$, such that for $m\geq m_{0}$, we have
$|z_{m}-z|\leq\epsilon$. We fix $m\geq m_{0}$. From (2.1), we have for $s\leq
t_{m}$, $X^{t_{m},x_{m}}_{s}=x$ and for $s\geq t_{m}$,
$X^{t_{m},x_{m}}_{s}=x_{m}e^{(\mu-\frac{\sigma^{2}}{2})(s-t_{m})+\sigma(W_{s}-W_{t_{m}})}$,
and so $\displaystyle\lim_{m\rightarrow\infty}X^{t_{m},x_{m}}_{s}=x$
$dt\otimes d\mathbb{P}$ a.e. For the jump term
$(Y^{j,t_{m},y^{j}_{m},\alpha^{j}}_{s})_{s}$, we have
$Y^{j,t_{m},y^{j}_{m},\alpha^{j}}_{s}=y_{j}^{m}$ for $s\leq t_{m}$ and for
$s\geq t_{m}$, we have

$\displaystyle Y^{j,t_{m},y^{j}_{m}}_{s}=y_{j}^{m}\prod_{n=0}^{{\cal
N}_{t_{m}}^{j}(\hat{\alpha})}e^{\lambda\zeta_{n}^{j}},$

where, we recall that ${\cal N}_{t_{m}}^{j}(\alpha^{j})$ is the number of
interventions of player $j$ between $t_{m}$ and $T$. As it is not allowed to
make an intervention at the terminal date, then
$\displaystyle\lim_{m\rightarrow\infty}{\cal N}_{t_{m}}^{j}(\alpha^{j})=0$
a.s. which shows that
$\displaystyle\lim_{m\rightarrow\infty}Y^{j,t_{m},y^{j}_{m},\alpha^{j}}_{s}=y^{j}$
$dt\otimes d\mathbb{P}$ a.e. We fix $p>1$, from Remark 2.1, for
$h^{i}=f^{i},g^{i}$, we have $|h^{i}(z)|\leq C^{i}(1+x)$, which implies

$\displaystyle\mathbb{E}[\left(\int_{t_{m}}^{T}e^{-\rho^{i}(s-t_{m})}f^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{s})ds+e^{-\rho^{i}(T-t_{m})}g^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{T})\right)^{p}]$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbb{E}[\left(\int_{t_{m}}^{T}f^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{s})ds\right)^{p}]+\mathbb{E}[g^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{T})^{p}]\right)$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}[\int_{t_{m}}^{T}f^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{s})^{p}ds](T-t_{m})^{p-1}+\mathbb{E}[g^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{T})^{p}])$
$\displaystyle\leq$ $\displaystyle
C(1+T^{p})(1+\mathbb{E}[\displaystyle\sup_{s\in[t_{m},T]}|X_{s}^{t_{m},x_{m}}|^{p}]),$

where the second inequality is obtained by using Hölder inequality and $C$ is
a generic constant independent of $m$. From the definition of the wholesale
price and for $m\geq m_{0}$, we have

$\displaystyle\mathbb{E}[\displaystyle\sup_{s\in[t_{m},T]}|X_{s}^{t_{m},x_{m}}|^{p}]\leq
C|x_{m}|^{p}\leq C(1+|x|^{p}).$ (3.10)

From (3.2)-(3.10), we deduce

$\displaystyle\mathbb{E}[\left(\int_{t_{m}}^{T}e^{-\rho^{i}(s-t_{m})}f^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{s})ds+e^{-\rho^{i}(T-t_{m})}g^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{T})\right)^{p}]\leq
C(1+|x|^{p}),$

where $C$ is a positive constant independent of $m$. This shows the
boundedness of
$\int_{t_{m}}^{T}e^{-\rho^{i}(s-t_{m})}f^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{s})ds+e^{-\rho^{i}(T-t_{m})}g^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{T})$
in $L^{p}(\mathbb{P})$ for $p>1$, which implies the uniform integrability of
$\left(\int_{t_{m}}^{T}e^{-\rho^{i}(s-t_{m})}f^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{s})ds+e^{-\rho^{i}(T-t_{m})}g^{i}(Z^{t_{m},z_{m},0,\alpha^{j}}_{T})\right)_{m}$.
It yields that

$\displaystyle{\underline{v}}^{i}(T,z)=\displaystyle\lim_{m\longrightarrow\infty}v^{i}(t_{m},z_{m})\geq
g^{i}(z).$ (3.11)

2) Fix some $z$ $\in$ $\bar{\cal S}$ and consider some sequence
$(t_{m},z_{m})_{m}$ $\in$ $[0,T)\times{\cal S}$ converging to $(T,z)$ and s.t.
$\lim_{m\rightarrow\infty}v^{i}(t_{m},z_{m})$ $=$ $\bar{v}^{i}(T,z)$. For any
$m$, one can find $\hat{\alpha}^{m,i}$ $=$
$(\hat{\tau}_{k}^{m,i},\hat{\zeta}_{k}^{m,i})_{k}$ $\in$ ${\cal
A}^{i}_{t_{m}}$ and $\hat{\alpha}^{m,j}$ $=$
$(\hat{\tau}_{k}^{m,j},\hat{\zeta}_{k}^{m,j})_{k}$ $\in$ ${\cal
A}^{j}_{t_{m}}$ the best response against the strategy $\hat{\alpha}^{m,i}$
s.t.

$\displaystyle
v(t_{m},z_{m})\leq\mathbb{E}\left[\int_{t_{m}}^{T}f^{i}(\hat{Z}^{m}_{s})ds-\sum_{t_{m}\leq\tau_{k}^{i}<T}e^{-\rho^{i}(\tau_{k}^{i}-t_{m})}\phi^{i}(\hat{Y}^{i,m}_{(\tau_{k}^{m,i})^{-}},\zeta_{k}^{i,m})+g^{i}(\hat{Z}_{T}^{m})\right]+\frac{1}{m},$

where $\hat{Z}^{m}$ $=$ $(X^{t_{m},x_{m}},\hat{Y}^{i,m},\hat{Y}^{j,m})$
denotes the state process controlled by $\hat{\alpha}^{m,i}$ and
$\hat{\alpha}^{m,j}$ . Arguing as above, we have
$\displaystyle\lim_{m\rightarrow\infty}{\cal
N}_{t_{m}}^{i}(\hat{\alpha}^{m,i})=0$ a.s. which shows that
$\displaystyle\lim_{m\rightarrow\infty}Y^{i,t_{m},y^{i,m},\hat{\alpha}^{m,i}}_{s}=y^{i}$
$dt\otimes d\mathbb{P}$ a.e. For $p>1$ and $m\geq m_{0}$, we have :

$\displaystyle\mathbb{E}[\left(\int_{t_{m}}^{T}f^{i}(\hat{Z}^{m}_{s})ds+g^{i}(\hat{Z}^{m}_{T})\right)^{p}]$
$\displaystyle\leq$ $\displaystyle
C(1+(T-t_{m})^{p})(1+\mathbb{E}[\displaystyle\sup_{s\in[t_{m},T]}|X_{s}^{t_{m},x_{m}}|^{p}]),$
$\displaystyle\leq$ $\displaystyle C(1+|x|^{2p}),$

where $C$ is a positive constant independent of $m$. This shows the
boundedness of
$\int_{t_{m}}^{T}f^{i}(\hat{Z}^{m}_{s})ds+g^{i}(\hat{Z}^{m}_{T})$ in
$L^{p}(\mathbb{P})$ for $p>1$, which implies the uniform integrability of
$\left(\int_{t_{m}}^{T}f^{i}(\hat{Z}^{m}_{s})ds+g^{i}(\hat{Z}^{m}_{T})\right)_{m}$.
It yields that

$\displaystyle{\underline{v}}^{i}(T,z)=\displaystyle\lim_{m\longrightarrow\infty}v^{i}(t_{m},z_{m})\leq
g^{i}(z).$ (3.12)

The result follows from (3.11) and (3.12).  $\Box$

##  4 Verification theorem

We provide in this section a verification theorem for the problem formalized
previously. It says that if a smooth function $v^{i}$ is solution to the
system of QVIs, then it is associated to a Nash equilibrium. We construct the
optimal strategy of each player in an inductive way.

###### Proposition 4.1

Let $v^{i}$ be a function from $[0,T]\times\bar{\cal S}$ to $\mathbb{R}$, with
$i\in\\{1,2\\}.$ We suppose that Assumption ${\cal H}$ holds and, for
$i\in\\{1,2\\}$ we have:

  * •

$v^{i}\hbox{ is a solution to }{\rm(\ref{HIj})}-\eqref{ct};$

  * •

$v^{i}\in C^{1,2}([0,T)\times\bar{\cal S})$ satisfying the growth condition
(3.1).

Let $(t,z)\in[0,T]\times\bar{\cal S}$ and define the continuation region:

$\displaystyle D^{i}:=\\{(s,\vartheta)\in[t,T]\times{\cal
S};\,v^{i}(s,\vartheta)>{\cal M}^{i}v^{i}(s,\vartheta)\mbox{ or
}v^{i}(s,\vartheta)>{\cal M}^{i}{\cal H}^{i}v^{i}(s,\vartheta)\\}.$

Define the impulse control $\alpha^{i*}=(\tau_{n}^{i*},\zeta_{n}^{i*})_{n\geq
1}$, as follows: $\tau_{0}^{i*}=t^{-}$ and inductively

$\displaystyle\tau_{n+1}^{i*}$ $\displaystyle:=$
$\displaystyle\inf\\{s>\tau_{n}^{i*}\,,(s,Z^{t,z,\alpha^{i*},\alpha^{j*}}_{s})\notin
D^{i}\\}$ (4.1) $\displaystyle\zeta_{n+1}^{i*}$ $\displaystyle:=$
$\displaystyle\hat{\zeta}^{i*}(\tau_{n+1}^{i*},Z^{t,z,\alpha^{i*},\alpha^{j*}}_{\tau_{n+1}^{i*-}})$
(4.2)

Assume that
$(\alpha^{i,*},\alpha^{j,*})\in\mathcal{A}_{t}^{i}\times\mathcal{A}_{t}^{j}$
(i.e. an admissible couple of strategies).
Then, $(\alpha^{i,*},\alpha^{j,*})$ is a Nash equilibrium and
$v^{i}(t,z)=J^{i}(t,z,\alpha^{i,*},\alpha^{j,*})$ for $i\in\\{1,2\\}$, $j\neq
i$.

_Proof:_ We fix $(t,z)\in[0,T]\times\bar{\cal S}$. We have to prove that

$v^{i}(t,z)=J^{i}(t,z;\alpha^{i*},\alpha^{j*}),\,\mbox{ for
}i\in\\{1,2\\},\,j\neq i~{}~{}~{}~{}~{}~{}~{}~{}v^{i}(t,z)\geq
J^{i}(t,z;\alpha^{i},\alpha^{j}).$

for every $\alpha^{i}\in{\cal A}^{i}_{t}$ and $\alpha^{j}\in{\cal A}^{j}_{t}$
the best response against $\alpha^{i}$.
Step 1: We prove $v^{i}(t,z)\geq J^{i}(t,z;\alpha^{i},\alpha^{j}).$
We will use the following shortened notation:

$Z=Z^{t,z;\alpha^{i},\alpha^{j}},~{}~{}~{}~{}~{}~{}\alpha^{i}=(\tau^{i}_{n},\zeta^{i}_{n})_{n\geq
1}.$

For each $r>0$ and $n\in\mathbb{N},$ we set
$\tau_{r}=\displaystyle\inf\\{s>t:X_{s}\in B(x;r)\\}$ is the exit time from
the ball with radius $r$ and center x.
We denote by $IT^{c}$ (resp. $IT^{d}$) the set of intervention times of the
two players which coincide (resp. differ). We apply Itô’s formula to the
process $e^{-\rho^{i}s}v^{i}(s,Z_{s})$ over the interval $[t,\tau_{r}\wedge
T]$. We take the conditional expectations, the stochastic integral vanishes.
Then by taking the expectation, we get:

$\displaystyle e^{-\rho^{i}t}v^{i}(t,z)$ $\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\int^{\tau_{r}\wedge
T}_{t}e^{-\rho^{i}u}\left(-\frac{\partial{v^{i}}}{\partial{t}}(u,Z_{u})-\mathcal{L}v^{i}(u,Z_{u})+\rho^{i}v^{i}(u,Z_{u})\right)du$
$\displaystyle-$ $\displaystyle\sum_{t\leq{\tau_{n}^{i}}<{\tau_{r}\wedge
T}}e^{-\rho^{i}\tau_{n}^{i}}(v^{i}({\tau_{n}^{i}},Z_{\tau_{n}^{i}})-{v^{i}}(\tau_{n}^{i},Z_{\tau_{n}^{i-}}))1_{IT^{d}}(\tau_{n}^{i})$
$\displaystyle-$ $\displaystyle\sum_{t\leq{\tau_{n}^{j}}<{\tau_{r}\wedge
T}}e^{-\rho^{i}\tau_{n}^{j}}(v^{i}({\tau_{n}^{j}},Z_{\tau_{n}^{j}})-{v^{i}}(\tau_{n}^{j},Z_{\tau_{n}^{j-}}))1_{IT^{d}}(\tau_{n}^{j})$
$\displaystyle-$ $\displaystyle\sum_{t\leq{\tau_{n}^{i}}<{\tau_{r}\wedge
T}}e^{-\rho^{i}\tau_{n}^{i}}(v^{i}({\tau_{n}^{i}},Z_{\tau_{n}^{i}})-{v^{i}}(\tau_{n}^{i},Z_{\tau_{n}^{i-}}))1_{IT^{c}}(\tau_{n}^{i})+e^{-\rho^{i}(\tau_{r}\wedge
T)}v^{i}(\tau_{r}\wedge T,Z_{\tau_{r}\wedge T})\Big{]}.$

We now estimate each term in the right-hand side of (4). As for the first
term, two cases are possible: $(s,Z_{s})\in{\cal I}^{i}$ i.e. the QVI (2.12)
holds or $(s,Z_{s})\in\bar{\cal I}^{i}$ i.e. the second player makes impulses
which could be simultaneously with the first player. As $\alpha^{i}\in{\cal
D}^{i}_{t}$, we are interested only in the strategies whose number of
interventions is finite almost surely, it follows that:

$\displaystyle-\frac{\partial{v^{i}}}{\partial{t}}(s,Z_{s})-\mathcal{L}v^{i}(s,Z_{s})+\rho^{i}v^{i}(s,Z_{s})\geq
f^{i}(Z_{s}),\mbox{ for all }s\in[t,T]\,\mathbb{P}\,a.s.$ (4.4)

Let us now consider the second term: by the definition of
$\mathcal{M}^{i}v^{i}$, we have:

$\displaystyle v^{i}(\tau_{n}^{i},Z_{\tau_{n}^{i-}})$ $\displaystyle\geq$
$\displaystyle{\cal
M}^{i}v^{i}(\tau_{n}^{i},X_{\tau_{n}^{i-}},Y^{i}_{\tau_{n}^{i-}},Y^{j}_{\tau_{n}^{i-}})$
(4.5) $\displaystyle=$
$\displaystyle\sup_{\zeta_{n}^{i}\in[\zeta_{min},\zeta_{max}]}\Big{\\{}v^{i}(\tau_{n}^{i},X_{\tau_{n}^{i-}},\Gamma^{i}(Y^{i}_{\tau^{n}_{i-}},\zeta_{n}^{i-}),Y^{j}_{\tau_{n}^{i-}})-\phi^{i}(Y^{i}_{(\tau_{n}^{i-})},\zeta_{n}^{i})\Big{\\}}$
$\displaystyle\geq$
$\displaystyle{v^{i}(\tau_{n}^{i},X_{\tau_{n}^{i-}},\Gamma^{i}(Y^{i}_{\tau_{n}^{i-}},\zeta_{n}^{i}),Y^{i}_{\tau_{n}^{i-}})-\phi^{i}(Y^{i}_{\tau_{n}^{i-}},\zeta_{n}^{i})}$
$\displaystyle=$
$\displaystyle{v^{i}(\tau_{n}^{i},Z_{\tau_{n}^{i}})-\phi^{i}(Z_{\tau_{n}^{i-}},\zeta_{n}^{i})}.$

For the third term, we have $({\cal
M}^{j}v^{j}-v^{j})(\tau_{n}^{j},Z_{\tau_{n}^{j-}})=0$, hence, by the
definition of $\mathcal{H}^{i}v^{i}$, we have:

$\displaystyle v^{i}(\tau_{n}^{j},Z_{(\tau_{n}^{j-})})$ $\displaystyle=$
$\displaystyle{\cal
H}^{i}v^{i}(\tau_{n}^{j},X_{\tau_{n}^{j-}},Y^{i}_{\tau_{n}^{j-}},Y^{j}_{\tau_{n}^{j-}})$
(4.6) $\displaystyle=$ $\displaystyle
v^{i}(\tau_{n}^{j},X_{\tau_{n}^{j-}},Y^{i}_{\tau_{n}^{j-}},\Gamma^{j}(Y^{j}_{\tau_{n}^{j-}},\zeta_{n}^{j}))$
$\displaystyle=$ $\displaystyle v^{i}(\tau_{n}^{j},Z_{\tau_{n}^{j}}).$

We focus on the fourth term which is related to the simultaneous interventions
case of the two players i.e for time interventions which are in $IT^{c}$ , we
have:

$\displaystyle v^{i}(\tau_{n}^{i},Z_{\tau_{n}^{i-}})$ (4.7) $\displaystyle=$
$\displaystyle{\cal H}^{i}v^{i}(\tau_{n}^{i},Z_{\tau_{n}^{i-}})$
$\displaystyle=$ $\displaystyle{\cal M}^{i}{\cal
H}^{i}v^{i}(\tau_{n}^{i},X_{\tau_{n}^{i-}},Y^{i}_{\tau_{n}^{i-}},Y^{j}_{\tau_{n}^{i-}})$
$\displaystyle=$
$\displaystyle\sup_{\zeta_{n}^{i}\in[\zeta_{min},\zeta_{max}]}\Big{\\{}v^{i}(\tau_{n}^{i},X_{\tau_{n}^{i-}},\Gamma^{i}(Y^{i}_{\tau_{n}^{i-}},\zeta_{n}^{i-}),\Gamma^{j}(Y^{j}_{\tau_{n}^{j-}},\zeta_{n}^{j}))-\phi^{i}(Y^{i}_{(\tau_{n}^{i-})},\zeta_{n}^{i})\Big{\\}}$
$\displaystyle\geq$
$\displaystyle{v^{i}(\tau_{n}^{i},X_{\tau_{n}^{i-}},\Gamma^{i}(Y^{i}_{\tau_{n}^{i-}},\zeta_{n}^{i}),\Gamma^{i}(Y^{i}_{\tau_{n}^{i-}},\zeta_{n}^{j}))-\phi^{i}(Y^{i}_{\tau_{n}^{i-}},\zeta_{n}^{i})}$
$\displaystyle=$
$\displaystyle{v^{i}(\tau_{n}^{i},Z_{\tau_{n}^{i}})-\phi^{i}(Y^{i}_{\tau_{n}^{i-}},\zeta_{n}^{i})}.$

By (4), and the inequalities (4.4)-(4.6), it follows that:

$\displaystyle v^{i}(t,z)$ $\displaystyle\geq$
$\displaystyle\mathbb{E}\big{[}\int^{\tau_{r}\wedge
T}_{t}e^{-\rho^{i}(s-t)}f^{i}(Z_{u})du-\sum_{t\leq{\tau_{n}^{i}}<{\tau_{r}\wedge
T}}e^{-\rho^{i}(\tau_{n}^{i}-t)}\phi^{i}(Y^{i}_{\tau_{n}^{i-}},\zeta_{n}^{i})$
$\displaystyle+$ $\displaystyle e^{-\rho^{i}(\tau_{r}\wedge
T-t)}{v^{i}}(\tau_{r}\wedge T,Z_{\tau_{r}\wedge T})\big{]}.$

As $f^{i}(z)\leq K^{i}x$ (see Remark 2.1), $\phi^{i}$ is bounded ( See
Assumption ${\cal H}(i)$), $v^{i}$ satisfies linear growth (see inequality
(3.1)), and since $\alpha^{i}\in{\cal A}^{i}_{t}$, we can pass to the limit as
$r\rightarrow\infty$ and use the dominated convergence theorem. it yields
that:

$\displaystyle v^{i}(t,z)$ $\displaystyle\geq$ $\displaystyle
J^{i}(t,z,{\alpha}^{i},\alpha^{j}).$

Step 2: $v^{i}(t,z)=J^{i}(t,z,\alpha^{i*},\alpha^{j*})$, where
$(\alpha^{i*},\alpha^{j*})$ are defined by (4.1)-(4.2)
We argue as in Step 1, but here all the inequalities are equalities by the
properties of $\alpha^{i*}.$
$\Box$

##  5 Viscosity characterization

It is well known that the theory of viscosity solutions is a powerful tool to
characterize the value function as a solution in a weaker sense of the
associated Hamilton Jacobi Bellman equation. The couple of value functions are
not known to be continuous a priori and so we shall work with the notion of
discontinuous viscosity solutions. For a locally bounded function $u$ on
$[0,T]\times\bar{\cal S}$, which is the case of the couple of value functions
$(v^{1},v^{2})$, we denote by $\underline{u}$ (resp. $\overline{u}$) the lower
semi-continuous (LSC) (resp. upper semi-continuous (USC)) envelope of $u$. We
recall that in general, $\underline{u}$ $\leq$ $u$ $\leq$ $\overline{u}$, and
that $u$ is LSC iff $u$ $=$ $\underline{u}$, $u$ is UCS iff $u$ $=$
$\overline{u}$, and $u$ is continuous iff $\underline{u}$ $=$ $\overline{u}$
($=$ $u$). We denote by $LSC([0,T)\times\bar{\cal S})$ (resp.
$USC([0,T)\times\bar{\cal S})$) the set of lsc (resp. usc) functions on
$[0,T)\times\bar{\cal S}$.
We work with the suitable notion of constrained viscosity solutions. The use
of constrained viscosity solutions was initiated by [19] for first-order
equations and applied in stochastic control problems arising in optimal
investment problems in [20]. The subsolution viscosity is satisfied in
$[0,T)\times\bar{\cal S}$ and the supersolution viscosity is satisfied in
$[0,T)\times{\cal S}$.

###### Definition 5.1

(i) Let ${\cal O}\subset\bar{\cal S}$. A locally bounded function $u^{i}$ on
$[0,T]\times\bar{\cal S}$ is a viscosity subsolution of (2.11)-(2.12) in
$[0,T)\times{\cal O}$ if for all $(\bar{t},\bar{z})\in[0,T)\times{\cal O}$ and
$\varphi^{i}\in C^{1,2}([0,T)\times\bar{\cal S})$ s.t.
$(\overline{u}^{i}-\varphi^{i})(\bar{t},\bar{z})=0$ and $(\bar{t},\bar{z})$ is
a maximum of $\overline{u}^{i}-\varphi^{i}$ on $[0,T)\times{\cal O}$, we have

$\displaystyle\overline{u}^{i}-\max\\{{\cal M}^{i}{\cal
H}^{i}\overline{u}^{i},{\cal H}^{i}\overline{u}^{i}\\}$ $\displaystyle\leq$
$\displaystyle 0\quad\hbox{in}\quad\overline{{\cal I}}^{i}$ (5.1)
$\displaystyle\min\\{-\displaystyle\frac{\partial\varphi^{i}}{\partial
t}-{\cal L}\varphi^{i}+\rho^{i}\varphi^{i}-f^{i},\overline{u}^{i}-{\cal
M}^{i}\overline{u}^{i}\\}$ $\displaystyle\leq$ $\displaystyle
0\quad\hbox{in}\quad{\cal I}^{i}$ (5.2)

(ii) Let ${\cal O}\subset{\cal S}$. A locally bounded function $u^{i}$ on
$[0,T]\times\bar{\cal S}$ is a viscosity supersolution of (2.11)-(2.12) in
$[0,T)\times{\cal O}$ if for all $(\bar{t},\bar{z})\in[0,T)\times{\cal O}$ and
$\varphi^{i}\in C^{1,2}([0,T)\times\bar{\cal S})$ s.t.
$(\underline{u}^{i}-\varphi^{i})(\bar{t},\bar{z})=0$ and $(\bar{t},\bar{z})$
is a minimum of $\underline{u}^{i}-\varphi^{i}$ on $[0,T)\times{\cal O}$, we
have

$\displaystyle\underline{u}^{i}-\max\\{{\cal M}^{i}{\cal
H}^{i}\underline{u}^{i},{\cal H}^{i}\underline{u}^{i}\\}$ $\displaystyle\geq$
$\displaystyle 0\quad\hbox{in}\quad\overline{{\cal I}}^{i}$ (5.3)
$\displaystyle\min\\{-\displaystyle\frac{\partial\varphi^{i}}{\partial
t}-{\cal L}\varphi^{i}+\rho^{i}\varphi^{i}-f^{i},\underline{u}^{i}-{\cal
M}^{i}{\underline{u}}^{i}\\}$ $\displaystyle\geq$ $\displaystyle
0\quad\hbox{in}\quad{\cal I}^{i}$ (5.4)

(iii) A function $u^{i}$ on $[0,T]\times\bar{\cal S}$ is a constrained
viscosity solution of (2.11)-(2.12) in $[0,T)\times\bar{\cal S}$ if it is a
viscosity subsolution of (2.11)-(2.12) in $[0,T)\times\bar{\cal S}$ and a
viscosity supersolution of (2.11)-(2.12) in $[0,T)\times{\cal S}$.

###### Remark 5.1

There is an equivalent formulation of viscosity solutions, which is useful for
proving uniqueness results, see [8] :

(i) Let ${\cal O}$ $\subset$ $\bar{\cal S}$. A function $u^{i}$ $\in$
$USC([0,T)\times\bar{\cal S})$ is viscosity subsolution of (2.11)-(2.12) in
$[0,T)\times{\cal O}$ if

$\displaystyle\min$ $\displaystyle\\{$ $\displaystyle\overline{u}^{i}-{\cal
M}^{i}{\cal H}^{i}\overline{u}^{i},\overline{u}^{i}-{\cal
H}^{i}\overline{u}^{i}\\}\leq 0\,\hbox{in}\,[0,T)\times{\cal
O}\cap\overline{{\cal I}}^{i},$ (5.5) $\displaystyle\min$ $\displaystyle\\{$
$\displaystyle-s_{0}-\mu
xs_{1}-\frac{\sigma^{2}x^{2}}{2}M_{11}+\rho^{i}u^{i}-f^{i},u^{i}-{\cal
M}^{i}u^{i}\\}\leq 0\,\hbox{in}\,[0,T)\times{\cal O}\cap{{\cal I}}^{i},$ (5.6)

where $(s_{0},s=(s_{k})_{1\leq k\leq 3},M=(M_{k_{1}k_{2}})_{1\leq
k_{1},k_{2}\leq 3})$ $\in$ $\bar{J}^{2,+}u^{i}(t,z)$.

(ii)Let ${\cal O}$ $\subset$ ${\cal S}$. A function $u^{i}$ $\in$
$USC([0,T)\times\bar{\cal S})$ is viscosity supersolution of (2.11)-(2.12) in
$[0,T)\times{\cal O}$ if

$\displaystyle\min$ $\displaystyle\\{\overline{u}^{i}-{\cal M}^{i}{\cal
H}^{i}\overline{u}^{i},\overline{u}^{i}-{\cal H}^{i}\overline{u}^{i}\\}\geq
0\,\hbox{in}\,[0,T)\times{\cal O}\cap\overline{{\cal I}}^{i},$ (5.7)
$\displaystyle\min$ $\displaystyle\\{-s_{0}-\mu
xs_{1}-\frac{\sigma^{2}x^{2}}{2}M_{11}+\rho^{i}u^{i}-f^{i},u^{i}-{\cal
M}^{i}u^{i}\\}\geq 0\,\hbox{in}\,[0,T)\times{\cal O}\cap{{\cal I}}^{i},$ (5.8)

where $(s_{0},s=(s_{k})_{1\leq k\leq 3},M=(M_{k_{1}k_{2}})_{1\leq
k_{1},k_{2}\leq 3})$ $\in$ $\bar{J}^{2,-}u^{i}(t,z)$.

(iii) A locally bounded function $u^{i}$ on $[0,T)\times\bar{\cal S}$ is a
constrained viscosity solution to (2.11)-(2.12) if $\bar{u}^{i}$ satisfies
(5.5)-(5.6) in $[0,T)\times\bar{\cal S}$, where $(s_{0},s,M)$ $\in$
$\bar{J}^{2,+}\bar{u}^{i}(t,z)$, and $\underline{u}^{i}$ satisfies (5.7)-(5.8)
in $[0,T)\times{\cal S}$, where $(s_{0},s,M)$ $\in$
$\bar{J}^{2,-}\underline{u}^{i}(t,z)$.

Here $J^{2,+}u(t,z)$ is the parabolic second order superjet defined by :

$\displaystyle
J^{2,+}u^{i}(t,z)\;=\;\\{(s_{0},s,M)\in\mathbb{R}\times\mathbb{R}^{3}\times\mathbb{S}^{3}~{}:$
$\displaystyle\;\;\;\limsup_{\tiny\begin{array}[]{c}(t^{\prime},z^{\prime})\rightarrow(t,z)\\\
(t^{\prime},z^{\prime})\in[0,T)\times{\cal
S}\end{array}}\frac{u^{i}(t^{\prime},z^{\prime})-u^{i}(t,z)-s_{0}(t^{\prime}-t)-s.(z^{\prime}-z)-\frac{1}{2}(z^{\prime}-z).M(z^{\prime}-z)}{|t^{\prime}-t|+|z^{\prime}-z|^{2}}\;\leq\;0\\},$

$\mathbb{S}^{3}$ is the set of symmetric $3\times 3$ matrices,
$\bar{J}^{2,+}\bar{u}^{i}(t,z)$ is its closure :

$\displaystyle\bar{J}^{2,+}u^{i}(t,x)$ $\displaystyle=$
$\displaystyle\left\\{(s_{0},s,M)\;=\;\lim_{m\rightarrow\infty}(s_{0}^{m},s^{m},M^{m})\;\;\;\mbox{
with }(s_{0}^{m},s^{m},M^{m})\in J^{2,+}u^{i}(t_{m},z_{m})\right.$
$\displaystyle\;\;\;\left.\mbox{ and
}\;\lim_{m\rightarrow\infty}(t_{m},z_{m},u^{i}(t_{m},z_{m}))\;=\;(t,z,u^{i}(t,z))\right\\},$

and $J^{2,-}u^{i}(t,z)$ $=$ $-J^{2,+}(-u^{i})(t,z)$, $\bar{J}^{2,-}u(t,z)$ $=$
$-\bar{J}^{2,+}(-u^{i})(t,z)$.

The dynamic programming principle (DPP) is the key tool to characterize the
couple of value functions $(v^{i},v^{j})$ defined by (2.10) as a viscosity
solution of the system (2.11)-(2.12). We will use the weak version of the
dynamic programming principle (WDPP) developed by Bouchard and Touzi [7] (See
Theorem 3.5 and Corollary 3.6). In fact, the DPP is intuitive and its proof
requires that the value function is measurable on the first stage and the
delicate measurable selection theorem holds. The WDPP is a powerful tool to
avoid these technical points. Under the assumptions of independence,
causality, stability under concatenation and consistency with deterministic
initial data, which could be easily checked in our case, we have the following
result.

###### Proposition 5.1

Let Assumption $({\cal H})$ holds. Then,
(i) for all $(\alpha^{i},\alpha^{j})\in{\cal A}^{i}_{0}\times{\cal
A}^{j}_{0}$, we have $J^{i}(.,.,\alpha^{i},\alpha^{j})\in
LSC([0,T)\times\bar{\cal S})$.
(ii) for any stopping time $\theta\in{\cal T}_{t,T}$ (set of stopping times
valued in $[t,T]$) we have the following inequalities:

$\displaystyle v^{i}(t,z)$ $\displaystyle\leq$
$\displaystyle\displaystyle\sup_{\alpha^{i}\in{\cal
A}^{i}_{t}}\mathbb{E}[\int_{t}^{\theta}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{s})ds$
(5.10)
$\displaystyle-\sum_{t\leq\tau_{k}^{i}<\theta}e^{-\rho^{i}(\tau_{k}^{i}-t)}\phi^{i}(Y^{i,t,y^{i},\alpha^{i}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})+e^{-\rho^{i}(\theta-t)}{\bar{v}}^{i}(\theta,Z^{t,z,\alpha^{i},\alpha^{j}}_{\theta})],$
$\displaystyle v^{i}(t,z)$ $\displaystyle\geq$
$\displaystyle\displaystyle\sup_{\alpha^{i}\in{\cal
A}^{i}_{t}}\mathbb{E}[\int_{t}^{\theta}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{s})ds$
(5.11)
$\displaystyle-\sum_{t\leq\tau_{k}^{i}<\theta}e^{-\rho^{i}(\tau_{k}^{i}-t)}\phi_{i}(Y^{i,t,y^{i},\alpha^{i}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})+e^{-\rho^{i}(\theta-t)}{\underline{v}}^{i}(\theta,Z^{t,z,\alpha^{i},\alpha^{j}}_{\theta})].$

_Proof:_ (i) Let $(\alpha^{i},\alpha^{j})\in{\cal A}^{i}_{0}\times{\cal
A}^{j}_{0}$. From Remark 2.1 and (3.1), the random variable
$\int_{t_{n}}^{T}e^{-\rho^{i}(s-t_{n})}f^{i}(Z^{t_{n},z_{n},\alpha^{i},\alpha^{j}}_{s})ds+e^{-\rho^{i}(T-t_{n})}{g}^{i}(Z^{t_{n},z_{n},\alpha^{i},\alpha^{j}}_{T})$
is bounded from below by an integrable random variable. By applying Fatou’s
lemma, we obtain

$\displaystyle\displaystyle\liminf_{\stackrel{{\scriptstyle(t_{n},z_{n})\rightarrow(t,z)}}{{t_{n}\in[0,T),\,z_{n}\in(0,+\infty)^{3}}}}$
$\displaystyle\mathbb{E}$
$\displaystyle[\int_{t_{n}}^{T}e^{-\rho^{i}(s-t_{n})}f^{i}(Z^{t_{n},z_{n},\alpha^{i},\alpha^{j}}_{s})ds+e^{-\rho^{i}(T-t_{n})}{g}^{i}(Z^{t_{n},z_{n},\alpha^{i},\alpha^{j}}_{T})]$
$\displaystyle\geq$ $\displaystyle\mathbb{E}$
$\displaystyle[\int_{t}^{T}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{s})ds+e^{-\rho^{i}(T-t)}{g}^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{T})].$

Under Assumption $({\cal H})(i)$, and since $\alpha^{i}\in{\cal D}_{0}^{i}$,
the sequence
$\left(-\displaystyle\sum_{t_{n}\leq\tau_{k}^{i}<T}e^{-\rho^{i}(\tau_{k}^{i}-t_{n})}\phi_{i}(Y^{i,t_{n},y_{n}^{i},\alpha^{i}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})\right)_{n}$
is uniformly integrable, and so we obtain

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}[-\sum_{t_{n}\leq\tau_{k}^{i}<T}e^{-\rho^{i}(\tau_{k}^{i}-t_{n})}\phi^{i}(Y^{i,t_{n},y_{n}^{i},\alpha^{i}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})]=\mathbb{E}[-\sum_{t\leq\tau_{k}^{i}<T}e^{-\rho^{i}(\tau_{k}^{i}-t)}\phi_{i}(Y^{i,t,y^{i},\alpha^{i}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})].$

It yields that

$\underline{J}^{i}(t,z,\alpha^{i},\alpha^{j})=\displaystyle\liminf_{\stackrel{{\scriptstyle(t_{n},z_{n})\rightarrow(t,z)}}{{t_{n}\in[0,T),\,z_{n}\in(0,+\infty)^{3}}}}J^{i}(t_{n},z_{n},\alpha^{i},\alpha^{j})\geq
J^{i}(t,z,\alpha^{i},\alpha^{j}).$

This shows that $J^{i}(.,.,\alpha^{i},\alpha^{j})\in LSC([0,T)\times\bar{\cal
S})$.
(ii) Since the strategy $\alpha^{i}$ is independent of ${\cal F}_{t}$, and
following [7] (See their Remark 5.2), one can prove

$\displaystyle v^{i}(t,z):=\displaystyle\sup_{\alpha^{i}\in{\cal
A}^{i}_{0}}J^{i}(t,z,\alpha^{i},\alpha^{j})\quad i\neq j,$

which means that one could get rid of the dependence of time $t$ in the set
${\cal A}^{i}_{t}$.
The proof of inequality (5.10) is then similar to [7].
We fix $R>0$ and a control $(\alpha^{i},\alpha^{j})\in{\cal
A}^{i}_{0}\times{\cal A}^{j}_{0}$. We consider the stopping time
$\tau^{R}:=\inf\\{s\geq t\mbox{ s.t.
}|Z^{t,z,\alpha^{i},\alpha^{j}}_{s}-z|>R\\}$. Then
$Z^{t,z,\alpha^{i},\alpha^{j}}1_{[t,\theta\wedge\tau^{R})}$ is $\L^{\infty}$
bounded. By Corollary 3.6 in [7], we have

$\displaystyle v^{i}(t,z)$ $\displaystyle\geq$
$\displaystyle\mathbb{E}[\int_{t}^{\theta\wedge\tau^{R}}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{s})ds-\displaystyle\sum_{t\leq\tau_{k}^{i}<\theta\wedge\tau^{R}}e^{-\rho^{i}(\tau^{i}_{k}-t)}\phi^{i}(Y^{i,t,y^{i},\alpha^{i}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})$
$\displaystyle+e^{-\rho^{i}(\theta\wedge\tau^{R}-t)}{\underline{v}}^{i}(\theta\wedge\tau^{R},Z^{t,z,\alpha^{i},\alpha^{j}}_{\theta\wedge\tau^{R}})].$

Using Assumption $({\cal H})(i)$, Remark 2.1, inequality (3.1), and since
$(\alpha^{i},\alpha^{j})\in{\cal D}_{0}^{i}\times{\cal D}_{0}^{j}$, Fatou’s
lemma implies

$\displaystyle\displaystyle\liminf_{R\longrightarrow\infty}$
$\displaystyle\mathbb{E}[\int_{t}^{\theta\wedge\tau^{R}}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{s})ds-\displaystyle\sum_{t\leq\tau_{k}^{i}<\theta\wedge\tau^{R}}e^{-\rho^{i}(\tau^{i}_{k}-t)}\phi^{i}(Y^{i,t,y^{i},\alpha^{i}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})$
$\displaystyle+e^{-\rho^{i}(\theta\wedge\tau^{R}-t)}{\underline{v}}^{i}(\theta\wedge\tau^{R},Z^{t,z,\alpha^{i},\alpha^{j}}_{\theta\wedge\tau^{R}})]$
$\displaystyle\geq$
$\displaystyle\mathbb{E}[\int_{t}^{\theta}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{s})ds-\displaystyle\sum_{t\leq\tau_{k}^{i}<\theta}e^{-\rho^{i}(\tau_{k}^{i}-t)}\phi^{i}(Y^{i,t,y^{i},\alpha^{i}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})$
$\displaystyle+e^{-\rho^{i}(\theta-t)}{\underline{v}}^{i}(\theta,Z^{t,z,\alpha^{i},\alpha^{j}}_{\theta})].$

As $(\alpha^{i},\alpha^{j})$ is chosen arbitrary, we obtain (5.11).  $\Box$

In the following lemma, we give some auxiliary results on the intervention
operators ${\cal M}^{i}$ and ${\cal H}^{i}$ which will be useful later.

###### Lemma 5.1

Let $u^{i}$ be a locally bounded function on $[0,T)\times\bar{\cal S}$.

(i) If $u^{i}$ is lsc, then ${\cal M}^{i}u^{i}$ is lsc.

(ii) If $u^{i}$ is usc, then ${\cal M}^{i}u^{i}$ is usc.

(iii) If $u^{i}$ is lsc (resp. usc), then ${\cal H}^{i}u^{i}$ is lsc (resp.
usc).

_Proof:_ We prove only the first and the second statement.
(i) Fix some $(t,z)$ $\in$ $[0,T)\times\bar{\cal S}$ and let $(t_{n},z_{n})$
be a sequence in $[0,T)\times\bar{\cal S}$ converging to $(t,z)$ and s.t.
${\cal M}^{i}u^{i}(t_{n},z_{n})$ converges to ${\underline{{\cal
M}^{i}u^{i}}}(t,z)$. Then, using also the lower semi-continuity of $u^{i}$, we
have:

$\displaystyle{\cal M}^{i}u^{i}(t,z)$ $\displaystyle=$
$\displaystyle\sup_{\zeta\in[\zeta_{min},\zeta_{max}]}u^{i}(t,x,\Gamma^{i}(y^{i},\zeta),y^{j}))\;\leq\;\sup_{\zeta\in[\zeta_{min},\zeta_{max}]}\liminf_{n\rightarrow\infty}u^{i}(t_{n},x_{n},\Gamma^{i}(y_{n}^{i},\zeta),y_{n}^{j})$
$\displaystyle\leq$
$\displaystyle\liminf_{n\rightarrow\infty}\sup_{\zeta\in[\zeta_{min},\zeta_{max}]}u^{i}(t_{n},x_{n},\Gamma^{i}(y_{n}^{i},\zeta),y_{n}^{j})\;\leq\;\lim_{n\rightarrow\infty}{\cal
M}^{i}u^{i}(t_{n},z_{n})\;=\;\underline{{\cal M}^{i}u^{i}}(t,z),$

which shows the lower semi-continuity of ${\cal M}^{i}u^{i}$.

(ii) Fix some $(t,z)$ $\in$ $[0,T)\times{\cal S}$ and let
$(t_{n},z_{n})_{n\geq 1}$ be a sequence in $[0,T)\times{\cal S}$ converging to
$(t,z)$ when $n$ goes to infinity. Since $u^{i}$ is usc, for each $n$ $\geq$
$1$, by selection measurable theorem, there exists a sequence
$(\hat{\zeta}_{n})_{n\geq 1}$ with $\hat{\zeta}_{n}$ $\in$
$[\zeta_{min},\zeta_{max}]$ such that:

$\displaystyle{\cal M}^{i}u^{i}(t_{n},z_{n})$ $\displaystyle=$ $\displaystyle
u^{i}(t_{n},x_{n},\Gamma^{i}(y^{i}_{n},\hat{\zeta}_{n}),y_{n}^{j}),\;\;\;\forall
n\geq 1.$

As $[\zeta_{min},\zeta_{max}]$ is a compact, the sequence
$(\hat{\zeta}_{n})_{n\geq 1}$ converges, up to a subsequence, to some
$\hat{\zeta}$ $\in$ $[\zeta_{min},\zeta_{max}]$. Therefore, we get :

$\displaystyle{\cal
M}^{i}u^{i}(t,z)\;\geq\;u^{i}(t,x,\Gamma^{i}(y^{i},\hat{\zeta}),y^{j})$
$\displaystyle\geq$
$\displaystyle\limsup_{n\rightarrow\infty}u^{i}(t_{n},x_{n},\Gamma^{i}(y^{i}_{n},\hat{\zeta}_{n}),y^{j}_{n})\;=\;\limsup_{n\rightarrow\infty}{\cal
M}^{i}u^{i}(t_{n},z_{n}),$

which proves that ${\cal M}^{i}u^{i}$ is usc.
(iii) The proof is similar to (i) and (ii).
$\Box$
The following technical lemma is needed to prove that the value functions
$(v^{1},v^{2})$ are viscosity solutions of the associated QVIs system.

###### Lemma 5.2

For all $(t,z)\in{{\cal I}}^{i}$ and $i\in\\{1,2\\}$ we have:

$\displaystyle\underline{v}^{i}(t,z)-{\cal M}^{i}\underline{v}^{i}(t,z)$
$\displaystyle\geq$ $\displaystyle 0.$ (5.12)

For all $(t,z)\in\overline{{\cal I}}^{i}$ and $i\in\\{1,2\\}$ we have:

$\displaystyle\min\\{\underline{v}^{i}(t,z)-{\cal M}^{i}{\cal
H}^{i}\underline{v}^{i}(t,z),\underline{v}^{i}(t,z)-{\cal
H}^{i}\underline{v}^{i}(t,z)\\}$ $\displaystyle\geq$ $\displaystyle 0,$ (5.13)
$\displaystyle\min\\{\overline{v}^{i}(t,z)-{\cal M}^{i}{\cal
H}^{i}\overline{v}^{i}(t,z),\overline{v}^{i}(t,z)-{\cal
H}^{i}\overline{v}^{i}(t,z)\\}$ $\displaystyle\leq$ $\displaystyle 0.$ (5.14)

_Proof:_ We fix $i\in\\{1,2\\}.$ Let $\alpha^{i}$ an admissible control
associated to the player $i$ where she chooses to intervene immediately at
time $t$ with an arbitrary size $\zeta^{i}\in[\zeta_{min},\zeta_{max}]$. By
applying the WDPP (5.11), for all $(t,z)\in[0,T)\times{\cal S}$, we have that:

$\displaystyle v^{i}(t,z)$ $\displaystyle\geq$
$\displaystyle\underline{v}^{i}(t,x,\Gamma^{i}(y^{i},\zeta^{i}),y^{j})-\phi^{i}(z,\zeta^{i}).$

As $\phi^{i}$ is continuous (See Assumption $({\cal H})(ii)$), the right hand
side of the last inequality is lsc, and so we deduce that:

$\displaystyle\underline{v}^{i}(t,z)$ $\displaystyle\geq$
$\displaystyle\underline{v}^{i}(t,x,\Gamma^{i}(y^{i},\zeta^{i}),y^{j})-\phi^{i}(z,\zeta^{i}).$

From the arbitrariness of $\zeta^{i}$, we deduce

$\displaystyle\underline{v}^{i}(t,z)$ $\displaystyle\geq$ $\displaystyle{\cal
M}^{i}\underline{v}^{i}(t,z),$

and so the inequality (5.12) is proved.
We fix $(t,z)\in\overline{{\cal I}}^{i}$. Two cases are possible: the player
$j$ decides to intervene or the two players $i$ and $j$ intervene
simultaneously. In the first case, we denote by $\zeta^{j}$ the size of
intervention of player $j$. Then, by applying the WDPP (5.11), we have

$\displaystyle
v^{i}(t,z)\geq\underline{v}^{i}(t,x,y^{i},\Gamma^{j}(y^{j},\zeta^{j}))={\cal
H}^{i}\underline{v}^{i}(t,z).$

As the right hand side of the last inequality is lsc, we deduce that

$\displaystyle\underline{v}^{i}(t,z)\geq{\cal H}^{i}\underline{v}^{i}(t,z).$
(5.15)

In the second case, if the players $i$ and $j$ decide to intervene in $(t,z)$,
then, by applying the WDPP (5.11), for all $(t,z)\in[0,T)\times{\cal S}$, we
have that

$\displaystyle v^{i}(t,z)$ $\displaystyle\geq$
$\displaystyle\underline{v}^{i}(t,x,\Gamma^{i}(y^{i},\zeta^{i}),\Gamma^{j}(y^{j},\zeta^{j}))-\phi^{i}(z,\zeta^{i}).$

Arguing as above, we deduce

$\displaystyle\underline{v}^{i}(t,z)$ $\displaystyle\geq$ $\displaystyle{\cal
M}^{i}{\cal H}^{i}\underline{v}^{i}(t,z).$ (5.16)

From (5.15) and (5.16), we deduce that (5.13). It remains to prove (5.14).
From the WDPP (5.10), if player $j$ decides to intervene, we obtain

$\displaystyle{v}^{i}(t,z)\leq{\cal H}^{i}\overline{v}^{i}(t,z).$

As the right hand side of the obove inequality is usc, then

$\displaystyle{\overline{v}}^{i}(t,z)\leq{\cal H}^{i}\overline{v}^{i}(t,z).$
(5.17)

If the two players intervene simultanously, then by using (5.10) and arguing
as above, we obtain

$\displaystyle\overline{v}^{i}(t,z)\leq{{\cal M}}^{i}{{\cal
H}}^{i}\overline{v}^{i}(t,z).$ (5.18)

From (5.17) and (5.18), we deduce (5.14).  $\Box$
Our main result of the section is the following.

###### Theorem 5.1

The function $v^{i},\;i\in\\{1,2\\}$ defined by (2.10) is a constrained
viscosity solution of (2.11)-(2.12) in $[0,T)\times{\cal S}$.

_Proof of supersolution property on $[0,T)\times{\cal S}$:_
We fix $i\in\\{1,2\\}$. Let $(\bar{t},\bar{z})\in[0,T)\times{\cal S}$ and
$\varphi^{i}\in C^{1,2}([0,T)\times\bar{\cal S})$ s.t.
$(\underline{v}^{i}-\varphi^{i})(\bar{t},\bar{z})=0$ and $(\bar{t},\bar{z})$
is a minimum of $\underline{v}^{i}-\varphi^{i}$ on $[0,T)\times{\cal S}$. Two
cases are possible: $(\bar{t},\bar{z})\in{\bar{\cal I}}^{i}$ or
$(\bar{t},\bar{z})\in{{\cal I}}^{i}$. If $(\bar{t},\bar{z})\in{\bar{\cal
I}}^{i}$, then from Lemma 5.2, we have
$\displaystyle\min\\{\underline{v}^{i}(\bar{t},\bar{z})-{\cal M}^{i}{\cal
H}^{i}\underline{v}^{i}(\bar{t},\bar{z}),\underline{v}^{i}(\bar{t},\bar{z})-{\cal
H}^{i}\underline{v}^{i}(\bar{t},\bar{z})\\}\geq 0$ (See inequality (5.13)). If
$(\bar{t},\bar{z})\in{{\cal I}}^{i}$, then we have
$\underline{v}^{i}(\bar{t},\bar{z})-{\cal
M}^{i}\underline{v}^{i}(\bar{t},\bar{z})\geq 0$ (See inequality (5.12)). It
remains to show that

$\displaystyle-\displaystyle\frac{\partial\varphi^{i}}{\partial
t}(\bar{t},\bar{z})-{\cal
L}\varphi^{i}(\bar{t},\bar{z})+\rho^{i}\varphi^{i}(\bar{t},\bar{z})-f^{i}(\bar{z})\geq
0,$ (5.19)

From the definition of $\underline{v}^{i}$, there exists a sequence
$(t_{m},z_{m})_{m\geq 1}\in{\cal S}$ s.t. $(t_{m},z_{m})$ and
$v^{i}(t_{m},z_{m})$ converge respectively to $(\bar{t},\bar{z})$ and
$\underline{v}^{i}(\bar{t},\bar{z})$ as $m$ goes to infinity. By continuity of
$\varphi^{i}$, we also have that $\gamma_{m}$ $:=$
$v^{i}(t_{m},z_{m})-\varphi^{i}(t_{m},z_{m})$ converges to $0$ as $m$ goes to
infinity. We denote by $\tau_{1,m}^{j*}$ the first optimal interversion of
player $j$ when the state process $Z$ starts from $z_{m}$ at time $t_{m}$. Two
cases are possible $\tau_{1,m}^{j*}=t_{m}$ or $\tau_{1,m}^{j*}>t_{m}$. If the
first case occurs, then the player $j$ makes an intervention immediately. By
applying the WDPP (5.11), $v^{i}(t_{m},z_{m})\geq{\cal
H}^{i}\underline{v}^{i}(t_{m},z_{m})$. As $t_{m}$ converges to $\bar{t}$ as
$m$ goes to infinity, and since ${\cal H}^{i}\underline{v}^{i}$ is lsc (See
Lemma 5.1 (iii)), we obtain $\underline{v}^{i}(\bar{t},\bar{z})\geq{\cal
H}^{i}\underline{v}^{i}(\bar{t},\bar{z})$. This means that
$(\bar{t},\bar{z})\in{\bar{\cal I}}^{i}$, which is false, and so we must have
$\tau_{1,m}^{j*}>t_{m}$. As $\bar{t}<T$ and $\bar{z}\in{\cal S}$, for $m$
large enough, one could have $t_{m}$ $<$ $T$ and there exists $\delta$ $>$ $0$
such $B(z_{m},\delta/2):=\\{z\mbox{ s.t
}|z-z_{m}|\leq\frac{\delta}{2}\\}\subset{\cal S}$. Let us then consider the
admissible control in ${\cal A}^{i}_{t_{m}}$ with no impulse until the first
exit time $\tau_{m}$, from the ball $B(z_{m},\delta/2)$ before
$T\wedge\tau_{1,m}^{j*}$ of the associated state process $Z_{s}$, defined by:

$\displaystyle\tau_{m}$ $\displaystyle:=$ $\displaystyle\inf\left\\{s\geq
t_{m}~{}:|Z_{s}^{t_{m},z_{m}}-z_{m}|\geq\delta/2\right\\}\wedge
T\wedge\tau_{1,m}^{j*}.$

Consider also a strictly positive sequence $(h_{m})_{m}$ s.t. $h_{m}$ and
$\gamma_{m}/h_{m}$ converge to zero as $m$ goes to infinity. By using the WDPP
(5.11) for $v_{i}(t_{m},z_{m})$ and $\hat{\tau}_{m}$ $:=$
$\tau_{m}\wedge(t_{m}+h_{m})$, we get :

$\displaystyle v^{i}(t_{m},z_{m})$ $\displaystyle=$
$\displaystyle\gamma_{m}+\varphi^{i}(t_{m},z_{m})$ $\displaystyle\geq$
$\displaystyle\mathbb{E}[\int_{t_{m}}^{\hat{\tau}_{m}}e^{-\rho^{i}(s-t_{m})}f^{i}(Z^{t_{m},z_{m}}_{s})ds+e^{-\rho^{i}(\hat{\tau}_{m}-t_{m})}\underline{v}^{i}(\hat{\tau}_{m},Z_{\hat{\tau}_{m}}^{t_{m},z_{m}})]$
$\displaystyle\geq$
$\displaystyle\mathbb{E}[\int_{t_{m}}^{\hat{\tau}_{m}}e^{-\rho^{i}(s-t_{m})}f^{i}(Z^{t_{m},z_{m}}_{s})ds+e^{-\rho^{i}(\hat{\tau}_{m}-t_{m})}\varphi^{i}(\hat{\tau}_{m},Z_{\hat{\tau}_{m}}^{t_{m},z_{m}})].$

Applying Itô’s formula to $\varphi^{i}(s,Z_{s}^{t_{m},z_{m}})$ between $t_{m}$
and $\hat{\tau}_{m}$ and noting that the integrant of the stochastic integral
term is bounded, we obtain by taking expectation :

$\displaystyle\frac{\gamma_{m}}{h_{m}}+\mathbb{E}\left[\frac{1}{h_{m}}\int_{t_{m}}^{\hat{\tau}_{m}}e^{-\rho^{i}(s-t_{m})}\left(-\displaystyle\frac{\partial\varphi^{i}}{\partial
t}-{\cal
L}\varphi^{i}+\rho^{i}\varphi^{i}-f^{i}\right)(s,Z_{s}^{t_{m},z_{m}})ds\right]$
$\displaystyle\geq$ $\displaystyle 0.$ (5.20)

By continuity a.s. of $Z_{s}^{t_{m},z_{m}}$, we have for $m$ large enough,
$\hat{\tau}_{m}$ $=$ $t_{m}+h_{m}$, and so by the mean-value theorem, the
random variable inside the expectation in (5.20) converges a.s. to
$(-\displaystyle\frac{\partial\varphi^{i}}{\partial t}-{\cal
L}\varphi^{i}+\rho^{i}\varphi^{i}-f^{i})(\bar{t},\bar{z})$ as $m$ goes to
infinity. Since this random variable is also bounded by a constant independent
of $m$, we conclude by the dominated convergence theorem and we obtain (5.19).
_Proof of subsolution property on $[0,T)\times\bar{\cal S}$:_
We fix $i\in\\{1,2\\}$ and $(\bar{t},\bar{z})\in[0,T)\times\bar{\cal S}$. If
$(\bar{t},\bar{z})\in{\bar{\cal I}}^{i}$, then from Lemma 5.2, we have
$\displaystyle\min\\{\overline{v}^{i}(\bar{t},\bar{z})-{\cal M}^{i}{\cal
H}^{i}\overline{v}^{i}(\bar{t},\bar{z}),\overline{v}^{i}(\bar{t},\bar{z})-{\cal
H}^{i}\overline{v}^{i}(\bar{t},\bar{z})\\}\leq 0$ (See inequality (5.14)). If
$(\bar{t},\bar{z})\in{{\cal I}}^{i}$, we consider $\varphi^{i}$ $\in$
$C^{1,2}([0,T)\times\bar{\cal S})$ s.t. $\overline{v}^{i}(\bar{t},\bar{z})$
$=$ $\varphi^{i}(\bar{t},\bar{z})$ and $\varphi^{i}$ $\geq$ $\overline{v}^{i}$
on $\bar{\cal S}$. If $\overline{v}^{i}(\bar{t},\bar{z})$ $\leq$ ${\cal
M}^{i}\overline{v}^{i}(\bar{t},\bar{z})$ then the subsolution inequality holds
trivially. Consider now the case where $\overline{v}^{i}(\bar{t},\bar{z})$ $>$
${\cal M}^{i}\overline{v}^{i}(\bar{t},\bar{z})$ and argue by contradiction by
assuming on the contrary that

$\displaystyle\bar{\eta}\;:=\;-\displaystyle\frac{\partial\varphi^{i}}{\partial
t}(\bar{t},\bar{z})-{\cal
L}\varphi^{i}(\bar{t},\bar{z})+\rho^{i}\varphi^{i}(\bar{t},\bar{z})-f^{i}(\bar{z})$
$\displaystyle>$ $\displaystyle 0.$

By continuity of $f^{i}$ ( see equation(2.5)), $\varphi^{i}$ and its
derivatives, and since $(\bar{t},\bar{z})\in[0,T)\times\bar{\cal S}$, there
exists some $\delta_{0}$ $>$ $0$ s.t. $\bar{t}+\delta_{0}$ $<$ $T$ and for all
$0<\delta\leq\delta_{0}$ and for all
$(t,z)\in((\bar{t}-\delta)_{+},\bar{t}+\delta)\times(B(\bar{z},\delta)\cap\bar{\cal
S})$, we have:

$\displaystyle-\displaystyle\frac{\partial\varphi^{i}}{\partial t}(t,z)-{\cal
L}\varphi^{i}(t,z)+\rho^{i}\varphi^{i}(t,z)-f^{i}(z)>\frac{\bar{\eta}}{2}.$
(5.21)

From the definition of $\overline{v}_{i}$, there exists a sequence
$(t_{m},z_{m})_{m\geq 1}$ $\in$
$((\bar{t}-\delta/2)_{+},\bar{t}+\delta/2)\times
B(\bar{z},\delta/2)\cap\bar{\cal S}$ s.t. $(t_{m},z_{m})$ and
$v^{i}(t_{m},z_{m})$ converge respectively to $(\bar{t},\bar{z})$ and
$\overline{v}_{i}(\bar{t},\bar{z})$ as $m$ goes to infinity. By continuity of
$\varphi^{i}$, we also have that $\gamma_{m}$ $:=$
$v^{i}(t_{m},z_{m})-\varphi^{i}(t_{m},z_{m})$ converges to $0$ as $m$ goes to
infinity. By the WDPP (5.10), given $m$ $\geq$ $1$, for any stopping time
$\tau$ valued in $[t_{m},T]$, we have

$\displaystyle v^{i}(t_{m},z_{m})$ $\displaystyle\leq$
$\displaystyle\displaystyle\sup_{\alpha^{i}\in{\cal
A}^{i}_{t_{m}}}\mathbb{E}[\int_{t_{m}}^{\tau}e^{-\rho^{i}(s-t_{m})}f^{i}(Z^{t_{m},z,\alpha^{i},\alpha^{j}}_{s})ds$
(5.22) $\displaystyle-$
$\displaystyle\sum_{t_{m}\leq\tau_{k}^{i}<\tau}e^{-\rho^{i}(\tau_{k}^{i}-t_{m})}\phi^{i}(Y^{i,t_{m},y^{i}_{m},\alpha^{i}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})+e^{-\rho^{i}(\tau-
t_{m})}{\bar{v}}^{i}(\tau,Z^{t_{m},z_{m},\alpha^{i},\alpha^{j}}_{\tau})]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}[\int_{t_{m}}^{\tau}e^{-\rho^{i}(s-t_{m})}f^{i}(\hat{Z}_{s}^{m})ds-\sum_{t_{m}\leq\tau_{m,k}^{i}<\tau}e^{-\rho^{i}(\tau_{m,k}^{i}-t_{m})}\phi^{i}(\hat{Y}^{i,m}_{(\tau_{m,k}^{i})^{-}},\zeta_{m,k}^{i})$
$\displaystyle+$ $\displaystyle e^{-\rho^{i}(\tau-
t_{m})}{\bar{v}}^{i}(\tau,\hat{Z}_{\tau}^{m})]+\frac{1}{m}.$

Here $\hat{Z}^{m}$ is the state process, starting from $z_{m}$ at $t_{m}$, and
controlled by $(\alpha_{m}^{i},\alpha_{m}^{j})$ an $\frac{1}{m}-\mbox{optimal
strategy}$.

We define $\bar{\tau}_{m}$ $:$ $=$
$\tau_{m,1}^{i}\wedge\tau_{\delta}^{m}\wedge\tau_{m,1}^{j}$ where,

$\displaystyle\tau_{\delta}^{m}$ $\displaystyle=$
$\displaystyle\inf\left\\{s\geq
t_{m}~{}:\hat{Z}_{s}^{m}\notin{\mathring{B}}(z_{m},\delta/2)\cap\bar{\cal
S}\right\\}\wedge(t_{m}+\delta/2)$

is the first exit time before $t_{m}+\delta/2$ of $\hat{Z}^{m}$ from the
intersection between open ball $\mathring{B}(z_{m},\delta/2)$ and $\bar{\cal
S}$. We claim that $\tau_{m,1}^{i}>t_{m}$. In fact, if $\tau_{m,1}^{i}=t_{m}$,
the player $i$ makes an intervention immediately, then

$\displaystyle
v^{i}(t_{m},z_{m})\leq{\bar{v}}^{i}(t_{m},x_{m},\Gamma^{i}(y^{i}_{m},\zeta_{m,k}^{i}),y^{j}_{m})-\phi^{i}(y^{i}_{m},\zeta_{m,k}^{i})+\frac{1}{m}.$

Sending m to infinity and taking the supremum over all admissible
interventions, we obtain ${\bar{v}}^{i}(\bar{t},\bar{z})\leq{\cal
M}^{i}{\bar{v}}^{i}(\bar{t},\bar{z})$ which is false in this particular case.
We claim that $\tau_{m,1}^{j}>t_{m}$. In fact if $\tau_{m,1}^{j}=t_{m}$, the
player $j$ makes an intervention immediately, then

$\displaystyle
v^{j}(t_{m},z_{m})\leq{\bar{v}}^{j}(t_{m},x_{m},y^{i}_{m},\Gamma^{j}(y^{j}_{m},\zeta_{m,k}^{j}))-\phi^{j}(y^{j}_{m},\zeta_{m,k}^{j})+\frac{1}{m}.$

Sending m to infinity, we deduce that ${\bar{v}}^{j}(\bar{t},\bar{z})\leq{\cal
M}^{j}{\bar{v}}^{j}(\bar{t},\bar{z})$ and so $(\bar{t},\bar{z})\in{\bar{\cal
I}}^{i}$ which is false. As the process $\hat{Z}^{m}$ is càdlàg, we have
$\tau_{\delta}^{m}>t_{m}$ $P$ a.s. Then we deduce that $\bar{\tau}^{m}>t_{m}$
$P$ a.s. As the filtration $({\cal F}_{t})_{0\leq t\leq T}$ is a Brownian
filtration, the stopping time $\bar{\tau}^{m}$ is predictable and so there
exists a sequence of stopping times $(\tau_{n,m})_{n}$ s.t.
$t_{m}<\bar{\tau}_{n,m}<\bar{\tau}^{m}$,
$\bar{\tau}_{n,m}\uparrow\bar{\tau}^{m}$, when $n$ goes to infinity. We fix
$n$ and $m$, we choose $\tau=\bar{\tau}_{n,m}$ in (5.22). Then, we get:

$\displaystyle v^{i}(t_{m},z_{m})$ $\displaystyle\leq$
$\displaystyle\mathbb{E}[\int_{t_{m}}^{\bar{\tau}_{n,m}}e^{-\rho^{i}(s-t_{m})}f^{i}(\hat{Z}_{s}^{m})ds$
$\displaystyle+e^{-\rho^{i}(\bar{\tau}_{n,m}-t_{m})}v^{i}(\bar{\tau}_{n,m},\hat{Z}^{m}_{\bar{\tau}_{n,m}})]+\frac{1}{m}.$

Now, since $\overline{v}_{i}$ $\leq$ $\varphi_{i}$, we obtain:

$\displaystyle\varphi^{i}(t_{m},z_{m})+\gamma_{m}$ $\displaystyle\leq$
$\displaystyle\mathbb{E}[\int_{t_{m}}^{\bar{\tau}_{n,m}}e^{-\rho^{i}(s-t_{m})}f^{i}(\hat{Z}_{s}^{m})ds$
$\displaystyle+e^{-\rho^{i}(\bar{\tau}_{n,m}-t_{m})}\varphi^{i}(\bar{\tau}_{n,m},\hat{Z}_{\bar{\tau}_{n,m}})]+\frac{1}{m}.$

By applying Itô’s formula to $\varphi^{i}(s,\hat{Z}_{s}^{m})$ between $t_{m}$
and $\bar{\tau}_{n,m}$, and using (5.21), we then get:

$\displaystyle\gamma_{m}$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\left[\int_{t_{m}}^{\bar{\tau}_{n,m}}e^{-\rho^{i}(s-t_{m})}\left(\displaystyle\frac{\partial\varphi^{i}}{\partial
t}+{\cal
L}\varphi^{i}-\rho^{i}\varphi^{i}+f^{i}\right)(s,\hat{Z}_{s}^{m})ds\right]$
$\displaystyle\leq$
$\displaystyle-\frac{\bar{\eta}}{2\rho^{i}}(1-\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{n,m}-t_{m})}]).$

This implies that:
$\lim_{n,m\rightarrow\infty}\mathbb{E}[e^{-\rho^{i}\bar{\tau}_{n,m}}]=e^{-\rho^{i}\bar{t}}$,
and

$\displaystyle\lim_{m\rightarrow\infty}\mathbb{E}[e^{-\rho^{i}\bar{\tau}_{m}}]=e^{-\rho^{i}\bar{t}},$

where the last equality is deduced by using monotone convergence theorem.
Then, along a subsequence, one could have
$\displaystyle\lim_{m\rightarrow\infty}\bar{\tau}_{m}=\bar{t}$ a.s. Applying
the WDPP (5.10) for $\tau=\bar{\tau}_{m}$, we obtain:

$\displaystyle v^{i}(t_{m},z_{m})$ (5.24) $\displaystyle\leq$
$\displaystyle\mathbb{E}[\int_{t_{m}}^{\bar{\tau}_{m}}e^{-\rho^{i}(s-t_{m})}f^{i}(\hat{Z}_{s}^{m})ds]+\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}\bar{v}^{i}(\bar{\tau}_{m},\hat{Z}^{m}_{\bar{\tau}_{m}})1_{\tau_{\delta}^{m}<\tau_{m,1}^{i}\wedge\tau_{m,1}^{j}}]$
$\displaystyle+$
$\displaystyle\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}\bar{v}^{i}(\bar{\tau}_{m},\hat{X}^{m}_{\bar{\tau}_{m}},\hat{Y}^{i,m}_{\bar{\tau}_{m}},\Gamma^{j}(\hat{Y}^{j,m}_{\bar{\tau}_{m}^{-}},\zeta_{m,1}^{j}))1_{\tau_{m,1}^{j}\leq\tau_{m,1}^{i}\wedge\tau_{\delta}^{m},\,\tau_{m,1}^{j}\neq\tau_{m,1}^{i}}]$
$\displaystyle+$
$\displaystyle\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}(\bar{v}^{i}(\bar{\tau}_{m},\hat{X}^{m}_{\bar{\tau}_{m}},\Gamma^{i}(\hat{Y}^{i,m}_{\bar{\tau}_{m}^{-}},\zeta_{m,1}^{i}),\Gamma^{j}(\hat{Y}^{j,m}_{\bar{\tau}_{m}^{-}},\zeta_{m,1}^{j}))$
$\displaystyle-$
$\displaystyle\phi^{i}(\hat{Z}^{m}_{(\tau_{m,1}^{i})^{-}},\zeta_{m,1}^{i}))1_{\tau_{m,1}^{i}=\tau_{m,1}^{j}\leq\tau_{\delta}^{m}}]$
$\displaystyle+$
$\displaystyle\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}(\bar{v}^{i}(\bar{\tau}_{m},\hat{X}^{m}_{\bar{\tau}_{m}},\Gamma^{i}(\hat{Z}^{m}_{\bar{\tau}^{m,-}},\zeta_{m,1}^{i}),\hat{Y}^{j,m}_{\bar{\tau}_{m}}-\phi^{i}(\hat{Z}^{m}_{(\tau_{m,1}^{i})^{-}},\zeta_{m,1}^{i}))$
$\displaystyle
1_{\tau_{m,1}^{i}\leq\tau_{\delta}^{m}\wedge\tau_{m,1}^{j},\tau_{m,1}^{i}\neq\tau_{m,1}^{j}}]+\frac{1}{m}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}[\int_{t_{m}}^{\bar{\tau}_{m}}e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}f^{i}(\hat{Z}_{s}^{m})ds]+\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}\bar{v}^{i}(\bar{\tau}_{m},\hat{Z}^{m}_{\bar{\tau}_{m}})1_{\tau_{\delta}^{m}<\tau_{m,1}^{i}\wedge\tau_{m,1}^{j}}]$
$\displaystyle+$
$\displaystyle\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}{\cal
H}^{i}\bar{v}^{i}(\bar{\tau}_{m},\hat{Z}^{m}_{\bar{\tau}_{m}^{-}})1_{\tau_{m,1}^{j}\leq\tau_{m,1}^{i}\wedge\tau_{\delta}^{m},\,\tau_{m,1}^{j}\neq\tau_{m,1}^{i}}]$
$\displaystyle+$
$\displaystyle\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}{\cal M}^{i}{\cal
H}^{i}\bar{v}^{i}(\bar{\tau}_{m},\hat{Z}^{m}_{\bar{\tau}_{m}^{-}})1_{\tau_{m,1}^{i}=\tau_{m,1}^{j}\leq\tau_{\delta}^{m}}]$
$\displaystyle+$
$\displaystyle\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}{\cal
M}^{i}\bar{v}^{i}(\bar{\tau}_{m},\hat{Z}^{m}_{\bar{\tau}_{m}^{-}})1_{\tau_{m,1}^{i}\leq\tau_{\delta}^{m}\wedge\tau_{m,1}^{j},\tau_{m,1}^{i}\neq\tau_{m,1}^{j}}]+\frac{1}{m},$

which implies,

$\displaystyle v^{i}(t_{m},z_{m})$ $\displaystyle\leq$
$\displaystyle\sup_{\tiny\begin{array}[]{c}|t^{\prime}-\bar{t}|<\delta\\\
|z^{\prime}-\bar{z}|<\delta\end{array}}f^{i}(z^{\prime})\frac{(1-\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}])}{\rho^{i}}$
(5.27) $\displaystyle+$
$\displaystyle\sup_{\tiny\begin{array}[]{c}|t^{\prime}-\bar{t}|<\delta\\\
|z^{\prime}-\bar{z}|<\delta\end{array}}\bar{v}^{i}(t^{\prime},z^{\prime})\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}1_{\tau_{\delta}^{m}<\tau_{m,1}^{i}\wedge\tau_{m,1}^{j}}]$
(5.30) $\displaystyle+$
$\displaystyle\sup_{\tiny\begin{array}[]{c}|t^{\prime}-\bar{t}|<\delta\\\
|z^{\prime}-\bar{z}|<\delta\end{array}}{\cal
H}^{i}\bar{v}^{i}(t^{\prime},z^{\prime})\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}1_{\tau_{m,1}^{j}\leq\tau_{m,1}^{i}\wedge\tau_{\delta}^{m},\,\tau_{m,1}^{j}\neq\tau_{m,1}^{i}}]$
(5.33) $\displaystyle+$
$\displaystyle\sup_{\tiny\begin{array}[]{c}|t^{\prime}-\bar{t}|<\delta\\\
|z^{\prime}-\bar{z}|<\delta\end{array}}{\cal M}^{i}{\cal
H}^{i}\bar{v}^{i}(t^{\prime},z^{\prime})\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}1_{\tau_{m,1}^{j}=\tau_{m,1}^{i}\leq\tau_{\delta}^{m}}]$
(5.36) $\displaystyle+$
$\displaystyle\sup_{\tiny\begin{array}[]{c}|t^{\prime}-\bar{t}|<\delta\\\
|z^{\prime}-\bar{z}|<\delta\end{array}}{\cal
M}^{i}\bar{v}^{i}(t^{\prime},z^{\prime})\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}1_{\tau_{m,1}^{i}\leq\tau_{\delta}^{m}\wedge\tau_{m,1}^{j},\,\tau_{m,1}^{i}\neq\tau_{m,1}^{j}}]+\frac{1}{m}.$
(5.39)

Since

$\displaystyle\displaystyle\lim_{m\rightarrow\infty}\bar{\tau}_{m}=\displaystyle\lim_{m\rightarrow\infty}\tau_{m,1}^{j}\wedge\tau_{m,1}^{i}\wedge\tau_{\delta}^{m}=\bar{t}\,\,\mbox{
a.s.},$ (5.40)

by the dominated convergence theorem, we obtain:

$\displaystyle\displaystyle\lim_{m\rightarrow\infty}\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}]=1$
(5.41)

For the second term in the right hand side of (5.27), we define
$A_{m}:=\\{\tau_{\delta}^{m}<\tau_{m,1}^{i}\wedge\tau_{m,1}^{j}\\}$ and
$A:=\cap_{m\geq 0}\cup_{n\geq m}A_{n}$. We assume that $\mathbb{P}(A)>0$ and
we consider $\omega\in A$. As
$\displaystyle\lim_{m\rightarrow\infty}\tau_{\delta}^{m}(\omega)=\bar{t}$ and
$\displaystyle\lim_{m\rightarrow\infty}x_{m}=\bar{x}$, then

$\displaystyle\displaystyle\lim_{m\rightarrow\infty}\hat{X}_{\tau_{\delta}^{m}(\omega)}^{m}(\omega)=\displaystyle\lim_{m\rightarrow\infty}x_{m}e^{(\mu-\frac{\sigma^{2}}{2})(\tau_{\delta}^{m}-t_{m})+\sigma(W_{\tau_{\delta}^{m}}(\omega)-W_{t_{m}}(\omega))}=\bar{x}.$

As the processes $\hat{Y}^{i,m}$ and $\hat{Y}^{j,m}$ are constant on
$[t_{m},\tau_{\delta}^{m}(\omega)]$ and
$\displaystyle\lim_{m\rightarrow\infty}(y^{i}_{m},y^{j}_{m})=(\bar{y}^{i},\bar{y}^{j})$,
then
$\displaystyle\lim_{m\rightarrow\infty}\hat{Y}_{\tau_{\delta}^{m}(\omega)}^{i,m}(\omega)=\displaystyle\lim_{m\rightarrow\infty}y^{i}_{m}=\bar{y}^{i}$,
and
$\displaystyle\lim_{m\rightarrow\infty}\hat{Y}_{\tau_{\delta}^{m}(\omega)}^{j,m}(\omega)=\displaystyle\lim_{m\rightarrow\infty}y^{j}_{m}=\bar{y}^{j}$.
This shows that
$\displaystyle\lim_{m\rightarrow\infty}\hat{Z}_{\tau_{\delta}^{m}(\omega)}^{m}(\omega)=\bar{z}$.
On the other hand, from the definition of $\tau_{\delta}^{m}$, we have
$\hat{Z}_{\tau_{\delta}^{m}(\omega)}^{m}(\omega)\notin
B(z_{m},\frac{\delta}{2})$ i.e
$|\hat{Z}_{\tau_{\delta}^{m}(\omega)}^{m}(\omega)-z_{m}|>\frac{\delta}{2}$.
Sending $m$ to infinity, we obtain $0>\frac{\delta}{2}$ which is false, and so
$\mathbb{P}(A)=0$. On the other hand,
$0\leq\mathbb{P}(A_{m})\leq\mathbb{P}(\cup_{n\geq m}A_{n})$, and
$\mathbb{P}(\cup_{n\geq m}A_{n})\downarrow P(A)$ whem $m$ goes to infinity,
then $\mathbb{P}(A_{m})$ goes to zero when $m$ goes to infinity. By the
dominated convergence theorem, we obtain:

$\displaystyle\displaystyle\lim_{m\rightarrow\infty}\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}1_{\tau_{\delta}^{m}<\tau_{m,1}^{i}\wedge\tau_{m,1}^{j}}]=0.$
(5.42)

For the third term in the right hand side of (5.27), we consider the set
$B_{m}:=\\{\tau_{m,1}^{j}\leq\tau_{m,1}^{i}\wedge\tau_{\delta}^{m},\,\tau_{m,1}^{j}\neq\tau_{m,1}^{i}\\}$.
We define $B:=\cap_{m\geq 0}\cup_{n\geq m}B_{n}$. We assume that
$\mathbb{P}(B)>0$ and we consider $\omega\in B$. The player $j$ is the first
to make an intervention. As the process $\hat{Z}_{.}^{m}$ is right continuous,
then $\tau_{m,1}^{j}(\omega)>\bar{t}$. Since
$\displaystyle\lim_{m\rightarrow\infty}\tau_{m,1}^{j}(\omega)=\bar{t}$, and
$\hat{Z}_{.}^{m}$ is right continuous then
$\displaystyle\lim_{m\rightarrow\infty}\hat{Z}_{\tau_{m,1}^{j-}(\omega)}^{m}(\omega)=\bar{z}$.
It yields that $(\bar{t},\bar{z})\in\bar{\cal I}^{i}$ which is false. Then, we
have $\mathbb{P}(B)=0$. Arguing as above, by the dominated convergence
theorem, we obtain:

$\displaystyle\displaystyle\lim_{m\rightarrow\infty}\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}1_{\tau_{m,1}^{j}<\tau_{m,1}^{i}\wedge\tau_{\delta}^{m}}]=0.$
(5.43)

For the fourth term in the right hand side of (5.27), we consider the set
$C_{m}:=\\{\tau_{m,1}^{j}=\tau_{m,1}^{i}\leq\tau_{\delta}^{m}\\}$. We define
$C:=\cap_{m\geq 0}\cup_{n\geq m}C_{n}$. Arguing as above, one could prove that
$\mathbb{P}(C)=0$ and by the dominated convergence theorem, we obtain:

$\displaystyle\displaystyle\lim_{m\rightarrow\infty}\mathbb{E}[e^{-\rho^{i}(\bar{\tau}_{m}-t_{m})}1_{\tau_{m,1}^{j}=\tau_{m,1}^{i}\leq\tau_{\delta}^{m}}]=0.$
(5.44)

Sending $m$ to infinity, from (5.27),(5.41)-(5.44) we deduce that

$\displaystyle\overline{v}^{i}(\bar{t},\bar{z})$ $\displaystyle\leq$
$\displaystyle\sup_{\tiny\begin{array}[]{c}|t^{\prime}-\bar{t}|<\delta\\\
|z^{\prime}-\bar{z}|<\delta\end{array}}{\cal
M}^{i}\bar{v}^{i}(t^{\prime},z^{\prime})$

Sending $\delta$ to $0$, by Lemma 5.1, we have

$\displaystyle\overline{v}_{i}(\bar{t},\bar{z})$ $\displaystyle\leq$
$\displaystyle{\cal M}_{i}\overline{v}_{i}(\bar{t},\bar{z}).$

As $\overline{v}^{i}(\bar{t},\bar{z})$ $>$ ${\cal
M}^{i}\overline{v}^{i}(\bar{t},\bar{z})$, we obtain a contradiction. This
shows that the subsolution property is satisfied.  $\Box$

##  6 Comparison principle of viscosity solutions

We turn to uniqueness result by proving a comparison principle for
discontinuous viscosity solutions to the QVIs (2.11)-(2.12). The comparison
theorem is based on the Ishii technique [13] to produce a strict supersolution
and we adapt arguments to handle the nonlocal operators. The idea is to build
a test function so that the maximum associated with the strict supersolution
is not attained on the boundary. Furthermmore, to construct a strict
supersolution, we consider an auxiliary problem by adding a fixed cost
$\kappa$ $>$ $0$ to the intervention operator ${\cal H}^{i}$. This means that
for an intervention of size $\zeta^{j}$ made by player $j$, the operator
${\cal H}^{i}_{\kappa}$ is defined by:

$\displaystyle{\cal
H}_{\kappa}^{i}h(t,z):=h(t,x,y^{i},\Gamma^{j}(y^{j},\zeta^{j}))-\kappa,$

for all locally bounded function $h:[0,T]\times S\longrightarrow{\mathbb{R}}$.
We define the value function of the perturbed problem $v^{i}_{\kappa}$ by:

$\displaystyle v^{i}_{\kappa}(t,z):=\displaystyle\sup_{\alpha^{i}\in{\cal
A}^{i}_{t}}J_{\kappa}^{i}(t,z,\alpha^{i},\alpha^{j}),$

where

$\displaystyle J_{\kappa}^{i}(t,z,\alpha^{i},\alpha^{j})$ $\displaystyle:=$
$\displaystyle\mathbb{E}[\int_{t}^{T}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{s})ds-\sum_{t\leq\tau_{k}^{i}<T}e^{-\rho^{i}(\tau_{k}^{i}-t)}\phi^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{(\tau_{k}^{i})^{-}},\zeta_{k}^{i})$
$\displaystyle-$
$\displaystyle\kappa\sum_{t\leq\tau_{k}^{j}<T}e^{-\rho^{i}(\tau_{k}^{j}-t)}+e^{-\rho^{i}(T-t)}g^{i}(Z^{t,z,\alpha^{i},\alpha^{j}}_{T})],\,(t,z)\in[0,T]\times\bar{\cal
S}$

One can prove that the value function $v^{i}_{\kappa}$ satisfies the growth
condition (3.1), and is a constrained viscosity solution in
$[0,T)\times\bar{\cal S}$ of the following system of QVIs:

$\displaystyle\min\\{v^{i}_{\kappa}-{\cal M}^{i}{\cal
H}_{\kappa}^{i}v^{i}_{\kappa},v^{i}_{\kappa}-{\cal
H}_{\kappa}^{i}v^{i}_{\kappa}\\}$ $\displaystyle=$ $\displaystyle
0\quad\hbox{in}\quad\overline{{\cal I}}^{i}$ (6.1)
$\displaystyle\min\\{-\displaystyle\frac{\partial v^{i}_{\kappa}}{\partial
t}-{\cal L}v^{i}_{\kappa}+\rho^{i}v^{i}_{\kappa}-f^{i},v^{i}_{\kappa}-{\cal
M}^{i}v^{i}_{\kappa}\\}$ $\displaystyle=$ $\displaystyle
0\quad\hbox{in}\quad{\cal I}^{i},$ (6.2)

with terminal condition:

$\displaystyle v^{i}_{\kappa}(T,z)$ $\displaystyle=$ $\displaystyle
g^{i}(z)\quad\hbox{in }\quad\\{T\\}\times\bar{\cal S},$ (6.3)

and boundary conditions:

$\displaystyle v_{\kappa}^{i}(t,z)=\begin{cases}0&\quad\text{if
}(t,z)\in[0,T)\times\partial^{y^{1}}{\cal S}\cup\partial^{y^{2}}{\cal S},\\\
-\frac{x}{2}(\frac{e^{(\mu-\rho^{i})(T-t)}-1}{\mu-\rho^{i}}+e^{(\mu-\rho^{i})(T-t)})&\quad\text{if
}(t,z)\in[0,T)\times\partial^{x}{\cal S}.\end{cases}$ (6.4)

###### Remark 6.1

The proof of viscosity property for $v^{i}_{\kappa}$, $i\in\\{1,2\\}$, follows
the same lines of arguments as for $v^{i}$, $i\in\\{1,2\\}$. For the growth
condition (3.1), similar arguments holds as for the lower bound (see
inequality (3.2)) and for the upper bound (see inequality (3.3)), since the
number of interventions of each player is uniformly integrable and so
satisfies $\displaystyle\sup_{\alpha^{i}\in{\cal D}_{0}^{i}}\mathbb{E}[{\cal
N}_{T}(\alpha^{i})]<\infty$.

The first result of this section is a convergence result.

###### Proposition 6.1

For $i\in\\{1,2\\}$, the sequence $(v^{i}_{\kappa})_{\kappa}$ is
nonincreasing, and converges pointwise on $[0,T]\times\bar{\cal S}$ towards
$v^{i}$ as $\kappa$ goes to zero.

_Proof:_ Notice that for any $0<\kappa_{1}\leq\kappa_{2}$,
$(t,z)\in[0,T)\times\bar{\cal S}$ and $(\alpha^{i},\alpha^{j})\in{\cal
A}_{t}^{i}\times{\cal A}_{t}^{j}$, we have

$\displaystyle J_{\kappa_{1}}^{i}(t,z,\alpha^{i},\alpha^{j})\geq
J_{\kappa_{2}}^{i}(t,z,\alpha^{i},\alpha^{j}).$

As $\alpha^{i}$ is arbitrary and $\alpha^{j}$ is the best response againt
$\alpha^{i}$, we deduce that $v_{\kappa_{1}}^{i}(t,z)\geq
v_{\kappa_{2}}^{i}(t,z)$. This shows that the sequence $(v^{i}_{\kappa})$ is
nonincreasing, and is upper-bounded by the value function $v^{i}$, so that

$\displaystyle\lim_{\kappa\downarrow 0}v_{\kappa}^{i}(t,z)$
$\displaystyle\leq$ $\displaystyle
v^{i}(t,z),\;\;\;\forall(t,z)\in[0,T]\times\bar{\cal S}.$ (6.5)

For the converse inequality, we fix some point $(t,z)$ $\in$
$[0,T]\times\bar{\cal S}$. From the representation (2.10) of $v^{i}(t,z)$,
there exists for any $n$ $\geq$ $1$, an $1/n$-optimal control denoted by
$\alpha_{n}^{i}$ et $\alpha_{n}^{j}$ the best response against
$\alpha_{n}^{i}$ such that:

$\displaystyle J^{i}(t,z,\alpha_{n}^{i},\alpha_{n}^{j})\geq
v^{i}(t,z)-\frac{1}{n}.$ (6.6)

On the other hand, we have
$J^{i}(t,z,\alpha_{n}^{i},\alpha_{n}^{j})=J_{\kappa}^{i}(t,z,\alpha_{n}^{i},\alpha_{n}^{j})+\kappa\mathbb{E}[\sum_{t\leq\tau_{k}^{j}<T}e^{-\rho^{i}(\tau_{k}^{j}-t)}]$.
From (6.6), we deduce that:

$\displaystyle
J_{\kappa}^{i}(t,z,\alpha_{n}^{i},\alpha_{n}^{j})+\kappa\mathbb{E}[\sum_{t\leq\tau_{k}^{j}<T}e^{-\rho^{i}(\tau_{k}^{j}-t)}]\geq
v^{i}(t,z)-\frac{1}{n}.$

Taking the supremum over all admissible strategies, we obtain:

$\displaystyle v_{\kappa}^{i}(t,z)+\kappa\displaystyle\sup_{\alpha^{j}\in{\cal
A}^{j}_{t}}\mathbb{E}[\sum_{t\leq\tau_{k}^{j}<T}e^{-\rho^{i}(\tau_{k}^{j}-t)}]\geq
v^{i}(t,z)-\frac{1}{n}$

By Sending $\kappa$ to zero and $n$ to infinity, and since $\alpha^{j}\in{\cal
D}_{t}^{j}$, we obtain

$\displaystyle\lim_{\kappa\downarrow 0}v_{\kappa}^{i}(t,z)$
$\displaystyle\geq$ $\displaystyle v^{i}(t,z).$ (6.7)

The result follows from (6.6) and (6.7).  $\Box$
Thanks to the following proposition, we produce a suitable strict viscosity
supersolution, which will allow us to compare a viscosity supersolution and a
viscosity subsolution on $[0,T)\times{\cal S}$.

###### Proposition 6.2

Let $w_{\kappa}^{i},\;i\in\\{1,2\\}$ be a lower semi-continuous viscosity
supersolution on $[0,T)\times{\cal S}$ of the QVIs (6.1)-(6.2). For all
$(t,z)\in[0,T)\times{\cal S}$, we consider

$\displaystyle
h^{i}(t,z)=\left(A^{i}+B^{i}x^{2}+C^{i}\log(y^{i}+1)+D^{i}\log(y^{j}+1)\right)e^{\rho(T-t)},$

where

$\displaystyle\rho>(2\mu+\sigma^{2}-\rho^{i})_{+},\,A^{i}>\frac{K^{i}}{\rho^{i}+\rho},\,B^{i}>\frac{K^{i}}{\rho^{i}+\rho-2\mu-\sigma^{2}},$
$\displaystyle\frac{C^{\phi}_{1}}{\lambda\zeta_{max}}e^{-\rho
T}>C^{i}>0,\mbox{ and }\frac{\kappa}{\lambda\zeta_{max}}e^{-\rho T}>D^{i}>0.$
(6.8)

Then, for each $\gamma\in(0,1)$,
$w_{\kappa}^{i,\gamma}=(1-\gamma)w_{\kappa}^{i}+\gamma g^{i}$ is a viscosity
supersolution of

$\displaystyle\left\\{\begin{array}[]{rlll}\min\\{w_{\kappa}^{i,\gamma}-{\cal
M}^{i}{\cal H}^{i}_{\kappa}w_{\kappa}^{i,\gamma},w_{\kappa}^{i,\gamma}-{\cal
H}^{i}_{\kappa}w_{\kappa}^{i,\gamma}\\}&\geq&\gamma\eta^{{}^{\prime}}\quad\hbox{in}\quad\overline{{\cal
I}}^{i},\\\ \min\\{-\displaystyle\frac{\partial
w_{\kappa}^{i,\gamma}}{\partial t}-{\cal
L}w_{\kappa}^{i,\gamma}+\rho^{i}w_{\kappa}^{i,\gamma}-f^{i},w_{\kappa}^{i,\gamma}-{\cal
M}^{i}w_{\kappa}^{i,\gamma}\\}&\geq&\gamma\eta\quad\hbox{in}\quad\mathcal{I}^{i},\\\
\end{array}\right.$ (6.11)

where $\eta$ and $\eta^{{}^{\prime}}$ are positive constants.

_Proof:_

Let $(t,z)\in{\cal I}^{i}$ and $\gamma\in(0,1).$ We fix
$\zeta^{i}\in[\zeta_{min},\zeta_{max}]$, then

$\displaystyle
h^{i}(t,z)-\left(h^{i}(t,x,\Gamma^{i}(z^{i},\zeta^{i}),y^{j})-\phi^{i}(y^{i},\zeta^{i})\right)$
$\displaystyle=$ $\displaystyle
h^{i}(t,z)-\left(A^{i}+B^{i}x^{2}+C^{i}\log(y^{i}e^{\lambda\zeta^{i}}+1)+D^{i}\log(y^{j}+1)\right)e^{\rho(T-t)}+\phi^{i}(y^{i},\zeta^{i})$
$\displaystyle\geq$ $\displaystyle
h^{i}(t,z)-\left(A^{i}+B^{i}x^{2}+C^{i}\log((y^{i}e^{\lambda\zeta_{max}}+e^{\lambda\zeta_{max}})+D^{i}\log(y^{j}+1)\right)e^{\rho(T-t)}+C^{\phi}_{1}$
$\displaystyle\geq$ $\displaystyle
h^{i}(t,z)-\left(A^{i}+B^{i}x^{2}+C^{i}\log(e^{\lambda\zeta_{max}}(y^{i}+1))+D^{i}\log(y^{j}+1)\right)e^{\rho(T-t)}+C^{\phi}_{1}$
$\displaystyle\geq$ $\displaystyle
h^{i}(t,z)-\left(A^{i}+B^{i}x^{2}+C^{i}\log(y^{i}+1)+C^{i}\log(e^{\lambda\zeta_{max}})+D^{i}\log(y^{j}+1)\right)e^{\rho(T-t)}+C^{\phi}_{1}$
$\displaystyle\geq$ $\displaystyle
h^{i}(t,z)-h^{i}(t,z)-C^{i}\lambda\zeta_{max}e^{\rho(T-t)}+C^{\phi}_{1}$
$\displaystyle\geq$
$\displaystyle-C^{i}\lambda\zeta_{max}e^{\rho(T-t)}+C^{\phi}_{1},$

where the first inequality is obtained since $\phi^{i}$ satisfies ${\cal
H}_{1}$ and $\zeta_{max}$ is positive. From the choice of $C^{i}$ (See (6.2))
and from the arbitrariness of $\zeta^{i}$, we deduce that:

$\displaystyle
h^{i}(t,z)-\mathcal{M}^{i}h^{i}(t,z)\geq-C^{i}{\lambda\zeta_{max}}e^{\rho
T}+C^{\phi}_{1}:=\eta_{1}>0.$ (6.13)

Moreover, as $w^{i}$ is lsc and a supersolution of (2.12), we have

$\displaystyle w_{\kappa}^{i}-{\cal M}^{i}w_{\kappa}^{i}\geq 0.$ (6.14)

Combining (6.13), (6.14) and using the convexity of the operator${\cal M}$, we
obtain

$\displaystyle w_{\kappa}^{i,\gamma}(t,z)-{\cal
M}^{i}w_{\kappa}^{i,\gamma}(t,z)$ $\displaystyle\geq$
$\displaystyle(1-\gamma)(w_{\kappa}^{i}(t,z)-\mathcal{M}^{i}w_{\kappa}^{i}(t,z))+\gamma(h^{i}(t,z)-{\cal
M}^{i}h^{i}(t,z))$ (6.15) $\displaystyle\geq$ $\displaystyle\gamma\eta_{1}>0$

From the definition of $h^{i}$, for all $(t,z)\in{\cal I}^{i}$, we have

$\displaystyle-\frac{\partial h^{i}}{\partial
t}(t,z)-\mathcal{L}h^{i}(t,z)+\rho^{i}h^{i}(t,z)-f^{i}(z)$
$\displaystyle=(\rho^{i}+\rho)h^{i}(t,z)-\mu
x\frac{\partial{h^{i}}}{\partial{x}}(t,z)-\frac{\sigma^{2}}{2}x^{2}\frac{\partial^{2}{h^{i}}}{\partial{x}^{2}}(t,z)-f^{i}(z)$
$\displaystyle=(\rho^{i}+\rho)h^{i}(t,z)-(2\mu+\sigma^{2})B^{i}x^{2}e^{\rho(T-t)}-f^{i}(z)$
$\displaystyle\geq(\rho^{i}+\rho)(A^{i}+B^{i}x^{2}+C^{i}log(y^{i}+1)+D^{i}\log(y^{j}+1))e^{\rho(T-t)}-(2\mu+\sigma^{2})B^{i}x^{2}e^{\rho(T-t)}-K^{i}x$
$\displaystyle\geq
x^{2}\left(B^{i}e^{\rho(T-t)}(\rho^{i}+\rho-2\mu-\sigma^{2})-K^{i}\right)+\left((\rho^{i}+\rho)A^{i}e^{\rho(T-t)}-K^{i}\right)$
$\displaystyle+C^{i}e^{\rho(T-t)}(\rho^{i}+\rho)log(y^{i}+1)+D^{i}e^{\rho(T-t)}(\rho^{i}+\rho)log(y^{j}+1)$
$\displaystyle\geq
x^{2}\left(B^{i}(\rho^{i}+\rho-2\mu-\sigma^{2})-K^{i}\right)+\left((\rho^{i}+\rho)A^{i}-K^{i}\right)$
$\displaystyle+C^{i}(\rho^{i}+\rho)log(y^{i}+1)+D^{i}(\rho^{i}+\rho)log(y^{j}+1),$

where the first and the second inequalities are obtained since $f^{i}(z)\leq
K^{i}x\leq K^{i}(1+x^{2})$. Thanks to the choice of $A^{i}$, $B^{i}$ and
$\rho$ (See (6.2)), we have

$\displaystyle-\frac{\partial h^{i}}{\partial
t}(t,z)-\mathcal{L}h^{i}(t,z)-\rho^{i}h^{i}(t,z)-f^{i}(z)\geq(\rho^{i}+\rho)A^{i}-K^{i}=:\eta_{2}>0.$
(6.16)

Thus we obtain

$\displaystyle-\displaystyle\frac{\partial w_{\kappa}^{i,\gamma}}{\partial
t}-{\cal
L}w_{\kappa}^{i,\gamma}+\rho^{i}w_{\kappa}^{i,\gamma}-f^{i}\geq\gamma\eta_{2}\mbox{
in the viscosity sense.}$ (6.17)

Combining (6.15) and (6.17), we have

$\displaystyle min\\{-\displaystyle\frac{\partial
w_{\kappa}^{i,\gamma}}{\partial t}-{\cal
L}w_{\kappa}^{i,\gamma}+\rho^{i}w_{\kappa}^{i,\gamma}-f^{i},w_{\kappa}^{i,\gamma}-{\cal
M}^{i}w_{\kappa}^{i,\gamma}\\}\geq\gamma\eta\quad\hbox{in}\quad{{\cal
I}}^{i},$ (6.18)

where $\eta:=\min\\{\eta_{1},\eta_{2}\\}>0$ and the QVI (6.18) should be
interpreted in the viscosity sense.
Let $(t,z)\in\bar{\cal I}^{i}$ and $\gamma\in(0,1).$ From the definition of
$h^{i}$, we have:

$\displaystyle h^{i}(t,z)-{\cal H}_{\kappa}^{i}h^{i}(t,z)$ $\displaystyle=$
$\displaystyle
h^{i}(t,z)-\left(h^{i}(t,x,y^{i},\Gamma^{j}(z^{j},\zeta^{j}))-\kappa\right)$
$\displaystyle=$ $\displaystyle
h^{i}(t,z)-\left(A^{i}+B^{i}x^{2}+C^{i}\log(y^{i}+1)+D^{i}\log(y^{j}e^{\lambda\zeta^{j}}+1)\right)e^{\rho(T-t)}+\kappa$

From the choice of $D^{i}$ (See (6.2)), we deduce that:

$\displaystyle h^{i}(t,z)-{\cal
H}_{\kappa}^{i}h^{i}(t,z)\geq-D^{i}{\lambda\zeta_{max}}e^{\rho
T}+\kappa:=\eta^{{}^{\prime}}_{1}>0.$ (6.19)

On the other hand, for $(t,z)\in\bar{\cal I}^{i}$, we have:

$\displaystyle h^{i}(t,z)-{\cal M}^{i}{\cal H}_{\kappa}^{i}h^{i}(t,z)$
$\displaystyle=$ $\displaystyle
h^{i}(t,z)-\left(h^{i}(t,x,\Gamma^{i}(z^{i},\zeta^{i}),\Gamma^{j}(z^{j},\zeta^{j}))-\phi^{i}(y^{i},\zeta^{i})-\kappa\right)$
$\displaystyle=$ $\displaystyle
h^{i}(t,z)-\left(A^{i}+B^{i}x^{2}+C^{i}\log(y^{i}e^{\lambda\zeta^{i}}+1)+D^{i}\log(y^{j}e^{\lambda\zeta^{j}}+1)\right)e^{\rho(T-t)}+\phi^{i}(y^{i},\zeta^{i})+\kappa$
$\displaystyle\geq$
$\displaystyle-C^{i}\lambda\zeta_{max}e^{\rho(T-t)}+C^{\phi}_{1}-D^{i}\lambda\zeta_{max}e^{\rho(T-t)}+\kappa,$

From the choices of $C^{i}$ and $D^{i}$ (See (6.2)), we deduce that:

$\displaystyle h^{i}(t,z)-{\cal M}^{i}{\cal
H}_{\kappa}^{i}h^{i}(t,z)\geq-C^{i}{\lambda\zeta_{max}}e^{\rho
T}+C^{\phi}_{1}-D^{i}{\lambda\zeta_{max}}e^{\rho
T}+\kappa:=\eta^{{}^{\prime}}_{2}>0.$ (6.20)

From (6.19) and (6.20), we deduce that:

$\displaystyle\displaystyle\min\\{h^{i}-{\cal M}^{i}{\cal
H}_{\kappa}^{i}h^{i},h^{i}-{\cal
H}_{\kappa}^{i}h^{i}\\}\geq\eta^{{}^{\prime}}:=\displaystyle\min\\{\eta^{{}^{\prime}}_{1},\eta^{{}^{\prime}}_{2}\\}\quad\hbox{in}\quad\overline{{\cal
I}}^{i}.$

As $w_{\kappa}^{i}$ is lsc and a viscosity supersolution of (2.11), then

$\displaystyle\displaystyle\min\\{w_{\kappa}^{i}-{\cal M}^{i}{\cal
H}_{\kappa}^{i}w_{\kappa}^{i},w_{\kappa}^{i}-{\cal
H}_{\kappa}^{i}w_{\kappa}^{i}\\}\geq 0\quad\hbox{in}\quad\overline{{\cal
I}}^{i}.$

We deduce that:

$\displaystyle\displaystyle\min\\{w_{\kappa}^{i,\gamma}-{\cal M}^{i}{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\gamma},w_{\kappa}^{i,\gamma}{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}\\}\geq\gamma\eta^{{}^{\prime}}\quad\hbox{in}\quad\overline{{\cal
I}}^{i}.$ (6.21)

The result follows from (6.18) and (6.21).  $\Box$

The following result is a comparison theorem. It states that we can compare a
viscosity subsolution to (6.1)-(6.2) in $[0,T)\times{\cal S}$ and a viscosity
supersolution to (6.1)-(6.2) in $[0,T)\times{\cal S}$, provided that we can
compare them at the terminal date but also at the corner lines $D_{0}$ of the
solvency boundary. We have to handle with the difficulties coming from the
boundary. We adapt the argument in Barles [4] (See Theorem 4.5) which needs a
smooth boundary. Akian et al. [1] proved a comparison theorem where the
Hamilton Jacobi Bellman Variational inequality is satisfied up the boundary.
The non regularity at some points of the boundary is studied in Ly Vath et al.
[14]. They proved a comparison theorem on the solvency region except the non
regular part of the boundary. In our case, we can not compare a subsolution
and a supersolution at the corner lines.

###### Theorem 6.1

We fix $i\in\\{1,2\\}$. Suppose that $u_{\kappa}^{i}\in
USC([0,T)\times\bar{\cal S})$ is a viscosity subsolution to (6.1)-(6.2) in
$[0,T)\times\bar{\cal S}$ and $w_{\kappa}^{i}\in LSC([0,T)\times\bar{\cal S})$
is a viscosity supersolution to (6.1)-(6.2) in $[0,T)\times{\cal S}$. We
assume that $u^{i}_{\kappa}$ and $w^{i}_{\kappa}$ satisfy the growth condition
(3.1) and

$\displaystyle
u_{\kappa}^{i}(t,z)\leq\displaystyle\liminf_{(t^{{}^{\prime}},z^{{}^{\prime}})\longrightarrow(t,z)}w_{\kappa}^{i}(t^{{}^{\prime}},z^{{}^{\prime}}),\,\,\forall(t,z)\in[0,T)\times
D_{0},$ (6.22) $\displaystyle
u_{\kappa}^{i}(T,z):=\displaystyle\limsup_{\tiny\begin{array}[]{c}(t,z^{\prime})\rightarrow(T,z)\\\
t<T,z^{\prime}\in{\cal
S}\end{array}}u_{\kappa}^{i}(t^{{}^{\prime}},z^{{}^{\prime}})\leq
w_{\kappa}^{i}(T,z):=\displaystyle\liminf_{\tiny\begin{array}[]{c}(t,z^{\prime})\rightarrow(T,z)\\\
t<T,z^{\prime}\in{\cal
S}\end{array}}w_{\kappa}^{i}(t^{{}^{\prime}},z^{{}^{\prime}})\,\,\forall
z\in\bar{\cal S}.$ (6.27)

Then,

$\displaystyle u_{\kappa}^{i}\leq w_{\kappa}^{i}\,\,\mbox{ on }[0,T]\times\cal
S.$

_Proof:_ In order to prove the comparison principle, it suffices to show that
for all $\gamma\in(0,1):$

$\displaystyle\sup_{z\in{\cal S}}(u_{\kappa}^{i}-w_{\kappa}^{i,\gamma})\leq
0,$

where $w_{\kappa}^{i,\gamma}$ is a strict supersolution satisfying (6.11). We
obtain the required result by letting $\gamma$ to 0. We argue by
contradiction. We suppose that

$\displaystyle\mu^{i}:=\sup_{z\in{\cal
S}}(u_{\kappa}^{i}-w_{\kappa}^{i,\gamma})>0.$ (6.28)

We define $w^{i}_{\kappa}$ for $i\in\\{1,2\\}$ on $[0,T)\times\partial{\cal
S}$ by :

$\displaystyle w_{\kappa}^{i}(t,z)$ $\displaystyle=$
$\displaystyle\liminf_{\tiny\begin{array}[]{c}(t^{\prime},z^{\prime})\rightarrow(t,z)\\\
(t^{\prime},z^{\prime})\in\bar{[}0,T)\times{\cal
S}\end{array}}w_{\kappa}^{i}(t^{\prime},z^{\prime}),\;\;\;\forall(t,z)\in[0,T)\times\partial{\cal
S}.$ (6.31)

As $u_{\kappa}^{i}$ is usc and $w_{\kappa}^{i,\gamma}$ is lsc, then
$u_{\kappa}^{i}-w_{\kappa}^{i,\gamma}$ is usc. By the choice of the strict
supersolution, we have
$\displaystyle\lim_{|x|,|y^{i}|,|y^{j}|\longrightarrow\infty}u^{i}(t,z)-w^{i,\gamma}(t,z)=-\infty$.
Hence, the set $\displaystyle\operatorname*{argmax}_{[0,T]\times\bar{\cal
S}}(u_{\kappa}^{i}-w_{\kappa}^{i,\gamma})$ is non empty. As
$u_{\kappa}^{i}(T,z)-w_{\kappa}^{i,\gamma}(T,z)\leq 0$ and
$u_{\kappa}^{i}(t,z)-w_{\kappa}^{i,\gamma}(t,z)\leq 0$ for all
$(t,z)\in[0,T)\times D_{0}$, there exists an open set ${\cal K}$ with compact
closure $\bar{\cal K}$ s.t.

$\displaystyle\displaystyle\operatorname*{argmax}_{[0,T]\times\bar{\cal
S}}(u_{\kappa}^{i}-w_{\kappa}^{i,\gamma})\subset[0,T)\times((\bar{\cal
S}\setminus D_{0})\cap{\cal K}).$

We choose $(t_{0},z_{0})\in[0,T)\times(\bar{\cal S}\setminus D_{0}\cap{\cal
K})$ s.t.

$\displaystyle\mu^{i}=u_{\kappa}^{i}(t_{0},z_{0})-w_{\kappa}^{i,\gamma}(t_{0},z_{0}).$
(6.32)

We distinguish two cases:
$\bullet$ Case 1. :$z_{0}\in\partial{\cal S}\setminus D_{0}\cap{\cal K}$. From
(6.31), there exists a sequence $(t_{n},z_{n})_{n\geq 1}$ in
$[0,T)\times({\cal S}\cap{\cal K})$ converging to $(t_{0},z_{0})$ s.t.
$w_{\kappa}^{i,\gamma}(t_{n},z_{n})$ tends to
$w_{\kappa}^{i,\gamma}(t_{0},z_{0})$ when $n$ goes to infinity. We then set
$\beta_{n}$ $:=$ $|t_{n}-t_{0}|$, $\varepsilon_{n}$ $:=$ $|z_{n}-z_{0}|$ and
consider the function $\Phi_{n}^{i}$ defined on $[0,T]^{2}\times(\bar{\cal
S}\cap\bar{\cal K})^{2}$ by :

$\displaystyle\Phi_{n}^{i}(t,t^{\prime},z,z^{\prime})$ $\displaystyle=$
$\displaystyle
u_{\kappa}^{i}(t,z)-w_{\kappa}^{i,\gamma}(t^{\prime},z^{\prime})-\varphi^{i}_{n}(t,t^{\prime},z,z^{\prime})$
(6.33) $\displaystyle\varphi_{n}^{i}(t,t^{\prime},z,z^{\prime})$
$\displaystyle=$
$\displaystyle|t-t_{0}|^{2}+|z-z_{0}|^{4}+\frac{|t-t^{\prime}|^{2}}{2\beta_{n}}+\frac{|z-z^{\prime}|^{2}}{2\varepsilon_{n}}+\left(\frac{d(z^{\prime})}{d(z_{n})}-1\right)^{4}.$

Here $d(z)$ denotes the distance from $z$ to $\partial{\cal S}$. It is known
that for $z_{0}\notin D_{0}$, there exists an open neighborhood of $z_{0}$ in
which the distance $d(.)$ is twice differentiable with bounded derivatives.
Such property fails when $z_{0}$ belongs to the corner lines i.e. $z_{0}\in
D_{0}$ (See [11]).
Since $\Phi_{n}^{i}$ is usc on the compact set $[0,T]^{2}\times(\bar{\cal
S}\cap\bar{\cal K})^{2}$, there exists
$(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}})$
$\in$ $[0,T]^{2}\times(\bar{\cal S}\cap\bar{\cal K})^{2}$ that attains its
maximum on $[0,T]^{2}\times(\bar{\cal S}\cap\bar{\cal K})^{2}$. We define

$\displaystyle\mu_{n}^{i}$ $\displaystyle:=$
$\displaystyle\sup_{[0,T]^{2}\times(\bar{\cal S}\cap{\cal
K})^{2}}\Phi_{n}^{i}(t,t^{\prime},z,z^{\prime})\;=\;\Phi_{n}^{i}(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}}).$

Moreover, there exists a subsequence, also denoted
$(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}})_{n\geq
1}$, converging to
$(\hat{t}_{0},\hat{t}_{0}^{{}^{\prime}},\hat{z}_{0},\hat{z}_{0}^{{}^{\prime}})$
$\in$ $[0,T]^{2}\times(\bar{\cal S}\cap\bar{\cal K})^{2}$. Since
$\Phi_{n}^{i}(t_{0},t_{n},z_{0},z_{n})$ $\leq$
$\mu^{i}_{n}=\Phi_{n}^{i}(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}})$,
we have :

$\displaystyle
u_{\kappa}^{i}(t_{0},z_{0})-w_{\kappa}^{i,\gamma}(t_{n},z_{n})-\frac{1}{2}\left(|t_{n}-t_{0}|+|z_{n}-z_{0}|\right)$
(6.34) $\displaystyle\leq$
$\displaystyle\mu^{i}_{n}\;=\;u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})-w_{\kappa}^{i,\gamma}(\hat{t}_{n}^{\prime},\hat{z}_{n}^{\prime})-\left(|\hat{t}_{n}-t_{0}|^{2}+|\hat{z}_{n}-z_{0}|^{4}\right)-R_{n}$
(6.35) $\displaystyle\leq$ $\displaystyle
u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})-w_{\kappa}^{i,\gamma}(\hat{t}_{n}^{\prime},\hat{z}_{n}^{\prime})-\left(|\hat{t}_{n}-t_{0}|^{2}+|\hat{z}_{n}-z_{0}|^{4}\right),$
(6.36)

where we set

$\displaystyle R_{n}$ $\displaystyle:=$
$\displaystyle\frac{|\hat{t}_{n}-\hat{t}_{n}^{\prime}|^{2}}{2\beta_{n}}+\frac{|\hat{z}_{n}-\hat{z}_{n}^{\prime}|^{2}}{2\varepsilon_{n}}+\left(\frac{d(\hat{z}_{n}^{\prime})}{d(z_{n})}-1\right)^{4}.$

As $u_{\kappa}^{i}$, $w_{\kappa}^{i,\gamma}$ are bounded on
$[0,T]\times\bar{\cal S}\cap\bar{\cal K}$, then inequality (6.35) implies the
boundedness of $(R_{n})_{n\geq 1}$. This yields that, there exists a
subsequence, also denoted $(R_{n})_{n\geq 1}$, which is convergent. As
$\beta_{n}$ and $\varepsilon_{n}$ go to $0$ when $n$ goes to infinity, we must
have

$\displaystyle\hat{t}_{0}\,=\,\hat{t}_{0}^{\prime}\mbox{ and
}\hat{z}_{0}\,=\,\hat{z}_{0}^{\prime}.$ (6.37)

As $u^{i}$ is upper semi-continuous and $w_{\kappa}^{i,\gamma}$ is lower semi-
continuous, then, by sending $n$ to infinity into (6.34) and (6.36), we obtain

$\displaystyle\mu^{i}=u_{\kappa}^{i}(t_{0},z_{0})-w_{\kappa}^{i,\gamma}(t_{0},z_{0})\leq
u_{\kappa}^{i}(\hat{t}_{0},\hat{z}_{0})-w_{\kappa}^{i,\gamma}(\hat{t}_{0},\hat{z}_{0})-|\hat{t}_{0}-t_{0}|^{2}-|\hat{z}_{0}-z_{0}|^{4}\leq\mu^{i},$

where the last inequality is deduced from the definition of $\mu^{i}$. It
yields that:

$\displaystyle\hat{t}_{0}\;=\hat{t}_{0}^{\prime}\;=\;t_{0},$
$\displaystyle\hat{z}_{0}\;=\hat{z}_{0}^{\prime}\;=\;z_{0}.$ (6.38)

Using again (6.34)-(6.35)-(6.36) and sendig $n$ to infinity, we deduce that:

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mu^{i}_{n}=\mu^{i}.$
(6.39)

From equality (6.35), we have
$\displaystyle\lim_{n\rightarrow\infty}\mu^{i}_{n}=\mu^{i}-\displaystyle\lim_{n\rightarrow\infty}R_{n}$,
then we deduce that:

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}R_{n}=0$ (6.40)

In particular, for $n$ large enough , we have $\hat{t}_{n}$,
$\hat{t}_{n}^{\prime}$ $<$ $T$ (since $t_{0}$ $<$ $T$). Besides as
$\frac{d(\hat{z}_{n}^{\prime})}{d(z_{n})}\rightarrow 1$ when $n$ goes to
infinity, we have $d(\hat{z}_{n}^{\prime})\geq\frac{d(z_{n})}{2}$ for $n$
large enough. It yields that $\hat{z}_{n}^{\prime}\in{\cal S}$ and
consequently $\hat{z}_{n}$ $\in$ ${\cal S}$.
Applying Ishii’s lemma (see Theorem 8.3 in [8]) to
$(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}})$
$\in$ $[0,T)\times[0,T)\times({\cal S}\cap{\cal V}_{0})\times({\cal
S}\cap{\cal V}_{0})$ that attains the maximum of $\Phi_{n}^{i}$ in (6.33),
where ${\cal V}_{0}$ is a neighborhood of $z_{0}$. We get the existence of two
$3\times 3$ symmetric matrices ${M}$ and ${N}$ s.t.:

$\displaystyle(s_{0},s,M)\quad$
$\displaystyle\in\quad\bar{J}^{2,+}u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n}),$
$\displaystyle(q_{0},q,N)\quad$
$\displaystyle\in\quad\bar{J}^{2,-}w_{\kappa}^{i,\gamma}(\hat{t}_{n}^{\prime},\hat{z}_{n}^{\prime}),$

where

$\displaystyle\begin{pmatrix}M&0\\\ 0&-N\end{pmatrix}\leq
D^{2}_{z,z^{\prime}}\varphi_{n}^{i}(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}})+{\varepsilon_{n}}(D^{2}_{z,z^{\prime}}\varphi_{n}^{i}(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}}))^{2}$
(6.41) $\displaystyle s_{0}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\partial\varphi_{n}^{i}}{\partial
t}(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}})=2(\hat{t}_{n}-t_{0})+\frac{(\hat{t}_{n}-\hat{t}^{{}^{\prime}}_{n})}{\beta_{n}},$
$\displaystyle q_{0}$ $\displaystyle=$
$\displaystyle-\frac{\partial\varphi_{n}^{i}}{\partial
t^{{}^{\prime}}}(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}})=\frac{(\hat{t}_{n}-\hat{t}^{{}^{\prime}}_{n})}{\beta_{n}},$
$\displaystyle s$ $\displaystyle=$ $\displaystyle(s_{k})_{1\leq k\leq
3}=D_{z}{\varphi_{n}^{i}}(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}})=4(\hat{z}_{n}-z_{0})|\hat{z}_{n}-z_{0}|^{2}+\frac{(\hat{z}_{n}-\hat{z}^{{}^{\prime}}_{n})}{\varepsilon_{n}},$
$\displaystyle q$ $\displaystyle=$ $\displaystyle(q_{k})_{1\leq k\leq
3}=-D_{z^{{}^{\prime}}}\varphi_{n}^{i}(\hat{t}_{n},\hat{t}_{n}^{{}^{\prime}},\hat{z}_{n},\hat{z}_{n}^{{}^{\prime}})=\frac{(\hat{z}_{n}-\hat{z}^{{}^{\prime}}_{n})}{\varepsilon_{n}}-\frac{4}{d(z_{n})}\left(\frac{d(\hat{z}^{{}^{\prime}}_{n})}{d(z_{n})}-1\right)^{3}Dd(\hat{z}^{{}^{\prime}}_{n}),$

and

$\displaystyle
D^{2}_{z,z^{\prime}}\varphi_{n}^{i}(\hat{t}_{n},\hat{t}_{n}^{\prime},\hat{z}_{n},\hat{z}_{n}^{\prime})=\begin{pmatrix}\frac{1}{\varepsilon_{n}}I_{3}+P_{n}&-\frac{1}{\varepsilon_{n}}I_{3}\\\
-\frac{1}{\varepsilon_{n}}I_{3}&\frac{1}{\varepsilon_{n}}I_{3}+Q_{n}\par\end{pmatrix},$

where

$\displaystyle P_{n}$ $\displaystyle=$ $\displaystyle
4|\hat{z}_{n}-z_{0}|^{2}I_{3}+8(\hat{z}_{n}-z_{0})(\hat{z}_{n}-z_{0})^{\scriptscriptstyle{\intercal}},$
$\displaystyle Q_{n}$ $\displaystyle=$ $\displaystyle
12\left(\frac{d(\hat{z}^{\prime}_{n})}{d(\hat{z}_{n})}-1\right)^{2}\frac{1}{d(\hat{z}_{n})^{2}}Dd(\hat{z}^{\prime}_{n})Dd(\hat{z}^{\prime}_{n})^{\scriptscriptstyle{\intercal}}+\frac{4}{d(z_{n})}\left(\frac{d(\hat{z}^{\prime}_{n})}{d(\hat{z}_{n})}-1\right)^{3}D^{2}d(\hat{z}^{\prime}_{n}).$

Here ⊺ denotes the transpose operator. By writing the viscosity subsolution
property of $u_{\kappa}^{i}$ and the viscosity supersolution property of
$w_{\kappa}^{i,\gamma},$ we have the following inequalities:

$\displaystyle\min$
$\displaystyle\\{-s_{0}-\mu\hat{x}_{n}s_{1}-\frac{\sigma^{2}{\hat{x}_{n}}^{2}}{2}M_{11}-f^{i}(\hat{z}_{n})+\rho^{i}u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n}),$
$\displaystyle u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})-{\cal
M}^{i}u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})\\}\leq
0\quad\hbox{if}\quad(\hat{t}_{n},\hat{z}_{n})\in{{\cal I}}^{i},$
$\displaystyle\min$
$\displaystyle\\{-q_{0}-\mu{\hat{x}}_{n}^{{}^{\prime}}q_{1}-\frac{\sigma^{2}{\hat{x}}_{n}^{{}^{\prime}2}}{2}N_{11}-f^{i}(\hat{z}_{n}^{{}^{\prime}})+\rho^{i}u_{\kappa}^{i}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n}),$
$\displaystyle
w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n})-{\cal
M}^{i}w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n})\\}\geq\gamma\eta\quad\hbox{if}\quad({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n})\in{{\cal
I}}^{i},$ $\displaystyle\min$
$\displaystyle\\{u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})-{\cal M}^{i}{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n}),$ $\displaystyle
u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})-{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})\\}\leq
0\quad\hbox{if}\quad(\hat{t}_{n},\hat{z}_{n})\in\overline{{\cal I}}^{i},$
$\displaystyle\min$
$\displaystyle\\{w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n})-{\cal
M}^{i}{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n}),$
$\displaystyle
w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n})-{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}\\}\geq\gamma\eta^{{}^{\prime}}\quad\hbox{if}\quad({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n})\in\overline{{\cal
I}}^{i}.$

We then distinguish four subcases:
Subcase 1: $u_{\kappa}^{i}({\hat{t}}_{n},{\hat{z}}_{n})-{\cal
M}^{i}u_{\kappa}^{i}({\hat{t}}_{n},{\hat{z}}_{n})\leq 0$ in (6).
From the definition of
$(\hat{t}_{n},\hat{t}_{n}^{\prime},\hat{z}_{n},\hat{z}_{n}^{\prime}),$ we
have:

$\displaystyle\mu^{i}_{n}$ $\displaystyle\leq$ $\displaystyle
u_{\kappa}^{i}({\hat{t}}_{n},{\hat{z}}_{n})-w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n})$
$\displaystyle\leq$ $\displaystyle{\cal
M}^{i}u_{\kappa}^{i}({\hat{t}}_{n},{\hat{z}}_{n})-{\cal
M}^{i}w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}_{n}^{{}^{\prime}})-\gamma\eta,$

where the last inequality is deduced from (6). Now, letting $n$ going to
$\infty,$ and using (6.39), we obtain: we obtain :

$\displaystyle\mu^{i}$ $\displaystyle\leq$
$\displaystyle\limsup_{n\rightarrow\infty}{\cal
M}^{i}u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})-\liminf_{n\rightarrow\infty}{\cal
M}^{i}w_{\kappa}^{i,\gamma}(\hat{t}_{n}^{\prime},\hat{z}_{n}^{\prime})-\gamma\eta$
$\displaystyle\leq$ $\displaystyle{\cal
M}^{i}u_{\kappa}^{i}(t_{0},z_{0})-{\cal
M}^{i}w_{\kappa}^{i,\gamma}(t_{0},z_{0})-\gamma\eta,$

where we used the upper-semicontinuity of ${\cal M}^{i}u_{\kappa}^{i}$ and the
lower-semicontinuity of ${\cal M}^{i}w_{\kappa}^{i,\gamma}$ (see Lemma 5.1).
As $u_{\kappa}^{i}$ is usc, there exists
$\zeta^{i}\in[\zeta_{min},\zeta_{max}]$ s.t.
${\cal M}^{i}u_{\kappa}^{i}(t_{0},z_{0})$ $=$
$u_{\kappa}^{i}(t_{0},x_{0},\Gamma^{i}(y^{i}_{0},\zeta^{i}),y^{j}_{0})-\phi^{i}(y^{i}_{0},\zeta^{i})$.
We then get

$\displaystyle\mu^{i}$ $\displaystyle\leq$ $\displaystyle{\cal
M}^{i}u_{\kappa}^{i}(t_{0},z_{0})-{\cal
M}^{i}w_{\kappa}^{i,\delta}(t_{0},z_{0})-\gamma\eta$ $\displaystyle\leq$
$\displaystyle
u_{\kappa}^{i}(t_{0},x_{0},\Gamma^{i}(y^{i}_{0},\zeta^{i}),y^{j}_{0})-w_{\kappa}^{i,\delta}(t_{0},x_{0},\Gamma^{i}(y^{i}_{0},\zeta^{i}),y^{j}_{0})-\gamma\eta\;\leq\;\mu^{i}-\gamma\eta,$

which is obviously a contradiction.
Subcase 2:
$-s_{0}-\mu\hat{x}_{n}s_{1}-\frac{\sigma^{2}{\hat{x}_{n}}^{2}}{2}M_{11}-f^{i}(\hat{z}_{n})+\rho^{i}u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})\leq
0$ in (6). From (6), we have
$-q_{0}-\mu{\hat{x}}_{n}^{{}^{\prime}}q_{1}-\frac{\sigma^{2}{\hat{x}}_{n}^{{}^{\prime}2}}{2}N_{11}-f^{i}(\hat{z}_{n}^{{}^{\prime}})+\rho^{i}u_{\kappa}^{i}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n})\geq\gamma\eta,$
which implies in this case

$\displaystyle-$
$\displaystyle(s_{0}-q_{0})-\mu(\hat{x}_{n}s_{1}-{\hat{x}}_{n}^{{}^{\prime}}q_{1})-\frac{\sigma^{2}}{2}(M_{11}{\hat{x}}^{2}_{n}-N_{11}{\hat{x}}^{{}^{\prime}2}_{n})-f^{i}(\hat{z}_{n})+f^{i}(\hat{z}_{n}^{{}^{\prime}})$
(6.46) $\displaystyle+$
$\displaystyle\rho^{i}(u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})-u_{\kappa}^{i}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n}))\leq-\gamma\eta.$

We have that

$\displaystyle s_{0}-q_{0}=2(\hat{t}_{n}-t_{0}).$

Since $\hat{t}_{n}$ goes to $t_{0}$ when $n$ goes to infinity, we deduce that
$s_{0}-q_{0}$ goes to zero when $n$ goes to infinity. The second term of
(6.46) is expressed as folows:

$\displaystyle\hat{x}_{n}s_{1}-{\hat{x}}_{n}^{{}^{\prime}}q_{1}=4\hat{x}_{n}(\hat{x}_{n}-x_{0})|\hat{z}_{n}-z_{0}|^{2}+4\frac{(\hat{x}_{n}-\hat{x}^{{}^{\prime}}_{n})^{2}}{\varepsilon_{n}}+4{\hat{x}}_{n}^{{}^{\prime}}\frac{D_{x}d(\hat{z}^{{}^{\prime}}_{n})}{d(\hat{z}_{n})}\left(\frac{d(\hat{z}^{{}^{\prime}}_{n})}{d(z_{n})}-1\right)^{3},$

where $D_{x}d(\hat{z}^{{}^{\prime}}_{n})$ is the first component of
$Dd(\hat{z}^{{}^{\prime}}_{n})$, which is known to be continuous on $\bar{\cal
S}\setminus D_{0}$. Using (6.38) and (6.40), we deduce that
$\hat{x}_{n}s_{1}-{\hat{x}}_{n}^{{}^{\prime}}q_{1}$ goes to 0 when $n$ goes to
infinity. Moreover, from (6.41), we have

$\frac{\sigma^{2}}{2}(M_{11}{\hat{x}}_{n}^{2}-N_{11}{\hat{x}}_{n}^{{}^{\prime}2})\leq\Upsilon_{n}$

where

$\displaystyle\Upsilon_{n}=\Lambda_{n}\left(D_{z,z^{\prime}}^{2}\varphi_{n}(\hat{t}_{n},\hat{t}^{\prime}_{n},\hat{z}_{n},\hat{z}^{\prime}_{n})+\varepsilon_{n}\left(D_{z,z^{\prime}}^{2}\varphi_{n}^{i}(\hat{t}_{n},\hat{t}^{{}^{\prime}}_{n},\hat{z}_{n},\hat{z}^{{}^{\prime}}_{n})\right)^{2}\right)\Lambda_{n}^{{}^{\scriptscriptstyle{\intercal}}},$

and

$\displaystyle\Lambda_{n}=(\frac{\sigma}{\sqrt{2}}{\hat{x}}_{n},0,0,\frac{\sigma}{\sqrt{2}}{\hat{x}}^{{}^{\prime}}_{n},0,0)$

After some straightforward calculation, we then get :

$\displaystyle\Upsilon_{n}$ $\displaystyle=$ $\displaystyle
3\frac{(\hat{x}_{n}-\hat{x}^{\prime}_{n})^{2}}{\varepsilon_{n}}+\Lambda_{n}\left(\left(\begin{array}[]{cc}3P_{n}&-P_{n}-Q_{n}\\\
-P_{n}-Q_{n}&3Q_{n}\end{array}\right)+\varepsilon_{n}\left(\begin{array}[]{cc}P_{n}^{2}&0\\\
0&Q_{n}^{2}\end{array}\right)\right)\Lambda_{n}^{{}^{\scriptscriptstyle{\intercal}}},$

which converges also to zero from (6.38) and (6.40). We have that
$\Upsilon_{n}$ goes to zero when $n$ goes to infinity. Sending $n$ goes to
infinity in inequality (6.46), we obtain the required contradiction:
$0\leq-\gamma\eta<0.$
Subcase 3: $u^{i}({\hat{t}}_{n},{\hat{z}}_{n})-{\cal
H}_{\kappa}^{i}u^{i}({\hat{t}}_{n},{\hat{z}}_{n})\leq 0$ in (6).
From the definition of
$(\hat{t}_{n},\hat{t}_{n}^{\prime},\hat{z}_{n},\hat{z}_{n}^{\prime}),$ we have

$\displaystyle\mu^{i}_{n}$ $\displaystyle\leq$ $\displaystyle
u_{\kappa}^{i}({\hat{t}}_{n},{\hat{z}}_{n})-w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n})$
$\displaystyle\leq$ $\displaystyle{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}({\hat{t}}_{n},{\hat{z}}_{n})-{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}_{n}^{{}^{\prime}})-\gamma\eta^{{}^{\prime}},$

where the last inequality is deduced from (6). Now, using the upper-
semicontinuity of ${\cal H}_{\kappa}^{i}u_{\kappa}^{i}$ and the lower-
semicontinuity of ${\cal H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}$ (see Lemma 5.1
(iii)), letting $n$ going to $\infty,$ and using (6.39), we obtain :

$\displaystyle\mu^{i}$ $\displaystyle\leq$
$\displaystyle\limsup_{n\rightarrow\infty}{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})-\liminf_{n\rightarrow\infty}{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}(\hat{t}_{n}^{\prime},\hat{z}_{n}^{\prime})-\gamma\eta^{{}^{\prime}}$
$\displaystyle\leq$ $\displaystyle{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}(t_{0},z_{0})-{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}(t_{0},z_{0})-\gamma\eta\;\leq\;\mu^{i}-\gamma\eta^{{}^{\prime}},$

which is obviously a contradiction.
Subcase 4: $u_{\kappa}^{i}({\hat{t}}_{n},{\hat{z}}_{n})-{\cal M}^{i}{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}({\hat{t}}_{n},{\hat{z}}_{n})\leq 0$ in (6).
From the definition of
$(\hat{t}_{n},\hat{t}_{n}^{\prime},\hat{z}_{n},\hat{z}_{n}^{\prime}),$ we have

$\displaystyle\mu^{i}_{n}$ $\displaystyle\leq$ $\displaystyle
u_{\kappa}^{i}({\hat{t}}_{n},{\hat{z}}_{n})-w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}^{{}^{\prime}}_{n})$
$\displaystyle\leq$ $\displaystyle{\cal M}^{i}{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}({\hat{t}}_{n},{\hat{z}}_{n})-{\cal M}^{i}{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}({\hat{t}}_{n}^{{}^{\prime}},{\hat{z}}_{n}^{{}^{\prime}})-\gamma\eta^{{}^{\prime}},$

where the last inequality is deduced from (6). Now, letting $n$ going to
$\infty,$ and using (6.39), we obtain:

$\displaystyle\mu^{i}$ $\displaystyle\leq$
$\displaystyle\limsup_{n\rightarrow\infty}{\cal M}^{i}{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}(\hat{t}_{n},\hat{z}_{n})-\liminf_{n\rightarrow\infty}{\cal
M}^{i}{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}(\hat{t}_{n}^{\prime},\hat{z}_{n}^{\prime})-\gamma\eta$
$\displaystyle\leq$ $\displaystyle{\cal M}^{i}{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}(t_{0},z_{0})-{\cal M}^{i}{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}(t_{0},z_{0})-\gamma\eta^{{}^{\prime}},$

where we used the upper-semicontinuity of ${\cal M}^{i}{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}$ and the lower-semicontinuity of ${\cal
M}^{i}{\cal H}_{\kappa}^{i}w_{\kappa}^{i,\gamma}$ (see Lemma 5.1). As
$u_{\kappa}^{j}$ is usc, there exists $\zeta^{j}\in[\zeta_{min},\zeta_{max}]$
s.t. ${\cal M}^{j}u_{\kappa}^{i}(t_{0},z_{0})$ $=$
$u_{\kappa}^{i}(t_{0},x_{0},y^{i}_{0},\Gamma^{j}(y^{j}_{0},\zeta^{j}))$, and
then ${\cal
H}_{\kappa}^{i}u_{\kappa}^{i}(t_{0},z_{0})=u_{\kappa}^{i}(t_{0},x_{0},y^{i}_{0},\Gamma^{j}(y^{j}_{0},\zeta^{j}))-\kappa$.
As $u_{\kappa}^{i}$ is usc, there exists
$\zeta^{i}\in[\zeta_{min},\zeta_{max}]$ s.t. ${\cal M}^{i}{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}(t_{0},z_{0})$ $=$
$u_{\kappa}^{i}(t_{0},x_{0},\Gamma^{i}(y^{i}_{0},\zeta^{i}),\Gamma^{j}(y^{j}_{0},\zeta^{j}))-\phi(y^{i}_{0},\zeta^{i})-\kappa$.
We then get

$\displaystyle\mu^{i}$ $\displaystyle\leq$ $\displaystyle{\cal M}^{i}{\cal
H}_{\kappa}^{i}u_{\kappa}^{i}(t_{0},z_{0})-{\cal M}^{i}{\cal
H}_{\kappa}^{i}w_{\kappa}^{i,\delta}(t_{0},z_{0})-\gamma\eta$
$\displaystyle\leq$ $\displaystyle
u_{\kappa}^{i}(t_{0},x_{0},\Gamma^{i}(y^{i}_{0},\zeta^{i}),\Gamma^{j}(y^{j}_{0},\zeta^{j}))-w_{\kappa}^{i,\delta}(t_{0},x_{0},\Gamma^{i}(y^{i}_{0},\zeta^{i}),\Gamma^{j}(y^{j}_{0},\zeta^{j}))-\gamma\eta^{{}^{\prime}}\;\leq\;\mu^{i}-\gamma\eta,$

which is obviously a contradiction.
$\bullet$ Case 2. : $z_{0}\in{\cal S}\cap{\cal K}$ We consider the function

$\displaystyle\Phi_{n}^{i}(t,z,z^{\prime})$ $\displaystyle=$ $\displaystyle
u^{i}(t,z)-w^{i,\gamma}(t,z^{\prime})-\varphi^{i,\gamma}_{n}(t,z,z^{\prime})$
$\displaystyle\varphi^{i,\gamma}_{n}(t,z,z^{\prime})$ $\displaystyle=$
$\displaystyle|t-t_{0}|^{2}+|z-z_{0}|^{4}+\frac{n}{2}|z-z^{\prime}|^{2},$

for $n$ $\geq$ $1$, and to take a maximum
$(\tilde{t}_{n},\tilde{z}_{n},\tilde{z}_{n}^{\prime})$ of $\Phi_{n}^{i}$. We
then show that the sequence
$(\tilde{t}_{n},\tilde{z}_{n},\tilde{z}_{n}^{\prime})_{n\geq 1}$ converges to
$(t_{0},z_{0},z_{0})$ as $n$ goes to infinity and we apply Ishii’s lemma to
get the required contradiction.

$\Box$
The condition (6.22) in the comparison theorem that must be satisfied is not
obvious to check for $v^{i}$, $i\in\\{1,2\\}$. To circumvent this difficulty,
we introduce the following function:

$\displaystyle{\hat{v}}^{i}(t,z):=\mathbb{E}[\int_{t}^{T}e^{-\rho^{i}(s-t)}f^{i}(Z^{t,z}_{s})ds+e^{-\rho^{i}(T-t)}g^{i}(Z^{t,z}_{T})],\,\forall\,(t,z)\in[0,T)\times\bar{\cal
S},$ (6.48)

where $(Z^{t,z}_{s})_{t\leq s\leq T}$ is the state process associated with the
no impulse strategy. The next result shows that the value function
$\hat{v}^{i}$ is continuous on the part $[0,T)\times D_{0}$ of the solvency
region.

###### Proposition 6.3

For all $i\in\\{1,2\\}$, we have:

$\displaystyle\lim_{(t^{{}^{\prime}},z^{{}^{\prime}})\longrightarrow(t,z)}\hat{v}^{i}(t^{{}^{\prime}},z^{{}^{\prime}})=\hat{v}^{i}(t,z)\,\mbox{
for all }\,(t,z)\in[0,T)\times D_{0},$ (6.49)

and ${\hat{v}}^{i}$ is a classical solution of (6.1)-(6.2) in a neighbourhood
of $[0,T)\times D_{0}$.

_Proof:_ We fix $i\in\\{1,2\\}$ and $(t,z)\in[0,T)\times D_{0}$. We consider
${\mathring{B}}((t,z),\delta)\cap\bar{\cal S}$ a neighbourhood of $(t,z)$,
where
${\mathring{B}}((t,z),\delta):=\\{(t^{{}^{\prime}},z^{{}^{\prime}})\mbox{ s.t.
}|z-z^{{}^{\prime}}|+|t-t^{{}^{\prime}}|<\delta\\}$ and $\delta$ is a positive
constant.
Step 1: We prove the continuity property (6.49).

From the definition of the process
$(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{s})_{t^{{}^{\prime}}\leq s\leq T}$ with
no impulse strategy, we have:

$\displaystyle Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{s}$
$\displaystyle=(X^{t^{{}^{\prime}},x^{{}^{\prime}}}_{s},Y^{1,t^{{}^{\prime}},y^{{}^{\prime}1}}_{s},Y^{2,t^{{}^{\prime}},y^{{}^{\prime}2}}_{s})=(X^{t^{{}^{\prime}},x^{{}^{\prime}}}_{s},y^{{}^{\prime}1},y^{{}^{\prime}2})=(x^{{}^{\prime}}e^{(\mu-\frac{\sigma^{2}}{2})(s-t^{{}^{\prime}})+\sigma(W_{s}-W_{t^{{}^{\prime}}})},y^{{}^{\prime}1},y^{{}^{\prime}2})$
$\displaystyle\longrightarrow$
$\displaystyle(X^{t,x}_{s},y^{1},y^{2})dt\otimes d\mathbb{P},\mbox{ when
}(t^{{}^{\prime}},z^{{}^{\prime}})\longrightarrow(t,z).$

We fix $p>1$. From Remark 2.1, for $h^{i}=f^{i},g^{i}$, we have
$|h^{i}(z)|\leq C^{i}(1+x)$, which implies

$\displaystyle\mathbb{E}[\left(\int_{t^{{}^{\prime}}}^{T}f^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{s})ds+g^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{T})\right)^{p}]$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbb{E}[\left(\int_{t^{{}^{\prime}}}^{T}f^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{s})ds\right)^{p}]+\mathbb{E}[g^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{T})^{p}]\right)$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}[\int_{t^{{}^{\prime}}}^{T}f^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{s})^{p}ds](T-t^{{}^{\prime}})^{p-1}+\mathbb{E}[g^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{T})^{p}])$
$\displaystyle\leq$ $\displaystyle
C(1+T^{p})(1+\mathbb{E}[\displaystyle\sup_{s\in[t^{{}^{\prime}},T]}|X_{s}^{t^{{}^{\prime}},x^{{}^{\prime}}}|^{p}]),$

where the second inequality is obtained by using Hölder inequality and $C$ is
a generic constant independent of $(t^{{}^{\prime}},z^{{}^{\prime}})$. From
the definition of the wholesale price and for
$(t^{{}^{\prime}},z^{{}^{\prime}})\in{\mathring{B}}((t,z),\delta)$, we have

$\displaystyle\mathbb{E}[\displaystyle\sup_{s\in[t^{{}^{\prime}},T]}|X_{s}^{t^{{}^{\prime}},z^{{}^{\prime}}}|^{p}]\leq
C|x^{{}^{\prime}}|^{p}\leq C(1+|x|^{p}).$ (6.51)

From (6)-(6.51), we deduce

$\displaystyle\mathbb{E}[\left(\int_{t^{{}^{\prime}}}^{T}f^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{s})ds+g^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{T})\right)^{p}]\leq
C(1+|x|^{p}),$

where $C$ is a positive constant which depends on $\delta$. This shows the
boundedness of
$\left(\int_{t^{{}^{\prime}}}^{T}f^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{s})ds+g^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{T})\right)$
in $L^{p}(\mathbb{P})$ for $p>1$, which implies the uniform integrability of
$\left(\int_{t^{{}^{\prime}}}^{T}f^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{s})ds+g^{i}(Z^{t^{{}^{\prime}},z^{{}^{\prime}}}_{T})\right)_{t^{{}^{\prime}},z^{{}^{\prime}}}$.
It yields that
${\hat{v}}^{i}(t^{{}^{\prime}},z^{{}^{\prime}})\longrightarrow{\hat{v}}^{i}(t,z)$
when $(t^{{}^{\prime}},z^{{}^{\prime}})\longrightarrow(t,z)$.
Step 2: We show that ${\hat{v}}^{i}$ is the solution of the IQV (2.11) on
${\mathring{B}}((t,z),\delta^{{}^{\prime}})\cap\bar{\cal S}$ for some
$\delta^{{}^{\prime}}$. From Equation (6.48), and since the process
$Z^{t^{{}^{\prime}},z^{{}^{\prime}}}$ is Log-normal, then ${\hat{v}^{i}}$ is
regular and we have:

$\displaystyle-\displaystyle\frac{\partial\hat{v}^{i}}{\partial
t}(t^{{}^{\prime}},z^{{}^{\prime}})-{\cal
L}{\hat{v}}^{i}(t^{{}^{\prime}},z^{{}^{\prime}})+\rho^{i}{\hat{v}}^{i}(t^{{}^{\prime}},z^{{}^{\prime}})-f^{i}(z^{{}^{\prime}})=0,$

on ${\mathring{B}}((t,z),\delta)\cap\bar{\cal S}$. It remains to prove that
${\hat{v}}^{i}(t^{{}^{\prime}},z^{{}^{\prime}})>{\cal
M}^{i}{\hat{v}}^{i}(t^{{}^{\prime}},z^{{}^{\prime}})$ in a neighbourhood of
$(t,z)$. We argue by contradiction. We assume that for all $\delta>0$, there
exists
$(t^{{}^{\prime\prime}},z^{{}^{\prime\prime}})\in{\mathring{B}}((t,z),\delta)\cap\bar{\cal
S}$ such that
${\hat{v}}^{i}(t^{{}^{\prime\prime}},z^{{}^{\prime\prime}})\leq{\cal
M}^{i}{\hat{v}}^{i}(t^{{}^{\prime\prime}},z^{{}^{\prime\prime}})$. Sending
$\delta$ towards 0, using Lemma 5.1, and the continuity of $\hat{v}^{i}$ w.r.t
$(t,z)$, we obtain:

$\displaystyle{\hat{v}^{i}}(t,z)\leq\displaystyle\limsup_{(t^{{}^{\prime\prime}},z^{{}^{\prime\prime}})\longrightarrow(t,z)}{\cal
M}^{i}{\hat{v}^{i}}(t^{{}^{\prime\prime}},z^{{}^{\prime\prime}})\leq{\cal
M}^{i}{\hat{v}^{i}}(t,z).$ (6.52)

On the other hand, as $(t,z)\in[0,T)\times D_{0}$, we have

$\displaystyle\hat{v}^{i}(t,z)=\begin{cases}0&\quad\text{ if
}(t,z)\in[0,T)\times\partial^{y^{1}}{\cal S}\cup\partial^{y^{2}}{\cal S},\\\
-\frac{x}{2}(\frac{e^{(\mu-\rho^{i})(T-t)}-1}{\mu-\rho^{i}}+e^{(\mu-\rho^{i})(T-t)})&\quad\text{
if }(t,z)\in[0,T)\times\partial^{x}{\cal S}.\end{cases}$

By straight forward computation, we have ${\hat{v}}^{i}(t,z)>{\cal
M}^{i}{\hat{v}^{i}}(t,z)$, which means that there is no intervention in
$(t,z)$, and so the contradiction is obtained. Symmetrically, In a
neighbourhood of $(t,z)\in[0,T)\times D_{0}$, we have
${\hat{v}}^{j}(t^{{}^{\prime}},z^{{}^{\prime}})>{\cal
M}^{j}{\hat{v}^{j}}(t^{{}^{\prime}},z^{{}^{\prime}})$ for $j\neq i$, which
means, against the no intervention startegy of player $i$, the best reponse of
player $j$ is also not to make an intervention. This shows that there exists a
neighborhood of $[0,T)\times D_{0}$ which is included in the continuation
region and ${\hat{v}}^{i}$ is a regular solution of (6.2) in this
neighbourhood.
$\Box$

###### Proposition 6.4

(i)For all $i\in\\{1,2\\}$, we have:

$\displaystyle\overline{v}_{\kappa}^{i}(t,z)\leq\displaystyle\liminf_{(t^{{}^{\prime}},z^{{}^{\prime}})\longrightarrow(t,z)}\underline{v}_{\kappa}^{i}(t^{{}^{\prime}},z^{{}^{\prime}}),\,\,\forall(t,z)\in[0,T)\times
D_{0},$ (6.53)

(ii)For all $i\in\\{1,2\\}$, we have:

$\displaystyle\overline{v}_{\kappa}^{i}(t,z)=\underline{v}_{\kappa}^{i}(t,z)=v_{\kappa}^{i}(t,z),\,\,\forall(t,z)\in[0,T)\times
D_{0},$ (6.54)

_Proof:_ Step 1: We proved that the value function $v_{\kappa}^{i}$ is a
viscosity subsolution of (6.1)-(6.2) in $[0,T)\times\bar{\cal S}$, then for
all $({t},{z})\in[0,T)\times D_{0}$ and $\varphi^{i}\in
C^{1,2}([0,T)\times\bar{\cal S})$ s.t.
$(\overline{v}_{\kappa}^{i}-\varphi^{i})({t},{z})=0$ and $({t},{z})$ is a
maximum of $\overline{v}_{\kappa}^{i}-\varphi^{i}$ on $[0,T)\times D_{0}$, we
have

$\displaystyle\min\\{-\displaystyle\frac{\partial\varphi^{i}}{\partial
t}({t},{z})-{\cal
L}\varphi^{i}({t},{z})+\rho^{i}\varphi^{i}({t},{z})-f^{i}({z}),\overline{v}_{\kappa}^{i}({t},{z})-{\cal
M}^{i}\overline{v}_{\kappa}^{i}({t},{z})\\}\leq 0\,\hbox{in}\,[0,T)\times
D_{0}.$

As $(t,z)\in[0,T)\times D_{0}$, by straight forward computation, we have
${v}_{\kappa}^{i}(t,z)>{\cal M}^{i}{v^{i}_{\kappa}}(t,z)$, where
$v^{i}_{\kappa}$ is given by equation (6.4), which means that there is no
intervention in $(t,z)$. It yields that

$\displaystyle-\displaystyle\frac{\partial\varphi^{i}}{\partial t}(t,z)-{\cal
L}\varphi^{i}(t,z)+\rho^{i}\varphi^{i}(t,z)-f^{i}(z)\leq 0,\,\mbox{ in
}[0,T)\times D_{0}.$ (6.55)

For the terminal condition, we take

$\displaystyle{\varphi^{i}}(T,z)\leq g^{i}(z).$ (6.56)

Tow cases are possible:
$\star$ First case: $z\in\partial^{y^{j}}{\cal S}\cup\partial^{y^{i}}{\cal
S}$, inequalities (6.55)-(6.56) become

$\displaystyle-\displaystyle\frac{\partial\varphi^{i}}{\partial
t}(t,z)+\rho^{i}\varphi^{i}(t,z)\leq 0,\,\mbox{ in
}[0,T)\times\partial^{y^{j}}{\cal S}\cup\partial^{y^{i}}{\cal S},$

and

$\displaystyle{\varphi^{i}}(T,z)\leq 0=v_{\kappa}^{i}(T,z)=\hat{v}^{i}(T,z).$

which implies ${\varphi^{i}}(t,z)\leq 0=\hat{v}^{i}(t,z)$ for all
$(t,z)\in[0,T)\times\partial^{y^{j}}{\cal S}\cup\partial^{y^{i}}{\cal S}$
$\star$ Second case: $z\in\partial^{x}{\cal S}$, inequalities (6.55)-(6.56)
become

$\displaystyle-\displaystyle\frac{\partial\varphi^{i}}{\partial t}(t,z)-{\cal
L}\varphi^{i}(t,z)+\rho^{i}\varphi^{i}(t,z)-f(t,z)\leq 0,\,\mbox{ in
}[0,T)\times\partial^{x}{\cal S}\setminus{(0,0,0)},$ (6.57)

and

$\displaystyle{\varphi^{i}}(T,z)\leq 0.$ (6.58)

For the boundary condition, we take

$\displaystyle{\varphi^{i}}(t,0)\leq
0=v_{\kappa}^{i}(t,0)=\hat{v}^{i}(t,0),\,\mbox{ in }[0,T).$ (6.59)