#
On a switching control problem with càdlàg costs

Said Hamadène111Département de Mathématiques, Equipe Statistique, et
Processus, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans, Cedex
9, France<EMAIL_ADDRESS>Héctor Jasso-Fuentes222Departamento de
Matemáticas. CINVESTAV-IPN. A. Postal 14-740, Ciudad de México, 07000, México.
{hjasso<EMAIL_ADDRESS>333Corresponding author. Yamid A.
Osorio-Agudelo222Departamento de Matemáticas. CINVESTAV-IPN. A. Postal 14-740,
Ciudad de México, 07000, México. {hjasso<EMAIL_ADDRESS>
###### Abstract

This work addresses a switching control problem under which the cost
associated with the changes of regimes is allowed to have discontinuities in
time. Our main contribution is to show several characterizations of the
optimal cost function as well as the existence of $\varepsilon$-optimal
control policies. As a by-product, we also study the existence and uniqueness
of solutions of a system of backward stochastic differential equations whose
barriers (or obstacles) are discontinuous (in fact of càdlàg type) and depend
itself on the unknown solution. At the last part of the paper, we study the
case when an underlying diffusion is part of the dynamic of the system. In
this special case, the optimal payoff becomes a weak solution of the HJB
system of PDEs with obstacles which is of quasi-variational type. This paper
is somehow a continuation of the papers [8, 17] that consider continuous
costs.

2010 Mathematics Subject Classification: 60G40, 93E20, 62P20, 91B99
Keywords and phrases: _Switching control, $\varepsilon$-optimal strategies,
backward stochastic differential equations, viscosity solutions._

##  1 Introduction

Among the family of optimal control problems we can highlight those whose
control is applied on the discontinuities to the dynamic. A special type of
these problems is the so-called _optimal multiple switching problems_
consisting in configuring the state of system according to doing changes of
_regimes_ (a.k.a. _configurations_) allowed for the controller. The times on
which these changes are triggered are also part of the control, so the
controller needs to apply a sequence, say $(\tau_{n},\xi_{n})$ such that at
time $\tau_{n}$, he/she changes the state from the regime $\xi_{n-1}$ to
$\xi_{n}$, $n\geq 1$. The objective for him/her is to find an optimal sequence
like the one above that maximizes a certain total payoff.

This class of problems has been studied in the literature by several authors.
For instance, Carmona and Ludkovski [5] study this kind of problems in order
to find management optimal strategies with the purpose to release a power
plant that converts natural gas into electricity and hence to sell this
commodity in the market. Doucet and Ristic [9] apply the switching control
theory to problems of target tracking that are commonly used in aerospace and
electronic systems. Trigeorgis [28, 29] relates this type of problems to real
option theory. Perhaps the most studied switching control problem is when only
two-modes are considered. Several authors have put attention on this type of
problems (see e.g., Brekke and Oksendal [3, 4], Hamadène and Jeanblanc [16],
Duckworth and Zervos [10], among others).

During the last decade, the switching control problem has been extensively
studied by several authors including [5, 6, 8, 16, 18, 19, 27], etc. (see also
the references therein).

However all the aforementioned papers consider the cases where the switching
costs are continuous. To the best of our knowledge the case where the
switching costs are discontinuous has not been considered yet. This is the
main objective of this work.

In this paper, as for the continuous switching costs, we show that the optimal
payoffs are given by either a solution of a system of reflected BSDEs with
obstacles depending on the solution or equivalently a system of processes
expressed through their corresponding Snell envelopes. This solution is
discontinuous in time. On the other hand, while an optimal strategy may not
exist, a nearly optimal strategy of switching always exists. Finally in the
Markovian framework of randomness, the previous system provides a viscosity
solution in weak sense of the Hamilton-Jacobi-Bellman system of PDEs
associated with the switching problem. This paper is somehow the extension of
the references Djehiche et. al. [8] and Hamadène and Morlais [17] when the
switching costs are of càdlàg type.

The rest of the paper is organized as follows: After this introductory part,
in section 2 we introduce our switching problem and provide a verification
theorem that is very common in control theory. In section 3, we present the
existence and uniqueness of the solution for the system of RBSDEs with
interconnected càdlàg obstacles and whose coefficients (drift and obstacles)
depend on the unknown solution. Finally, in Section 4 we show that under a
Markovian framework i.e. the dynamic of the system is also governed by a
underlying diffusion process, our unique solution, obtained in section 3,
provides a weak viscosity solution for a system of a quasi-variational
inequality with interconnected obstacles. $\blacksquare$

###  1.1 Notation and terminology

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a fixed probability space and
$B=\left(B_{t}\right)_{t\leq T}$ a $d$-dimensional Brownian motion with
completed natural filtration $\left(\mathcal{F}_{t}:=\sigma\left\\{B_{s},s\leq
t\right\\}\right)_{t\leq T}$ thus it satisfies the usual conditions, i.e., it
is right continuous and complete. Associated to $\mathbb{P}$, we denote by
$\mathbb{E}$ its respective expectation.

Next let us consider the following elements:

  * •

$\left|\cdot\right|$ will denote the Euclidean norm in $\mathbb{R}^{l}$, for
some appropriate $l\in\mathbb{N}$.

  * •

Given $\theta\in[0,T]$, $L^{2}(\mathcal{F}_{\theta})$ is the set of random
variables $\xi$, $\mathcal{F}_{\theta}$-measurable and such that
$\mathbb{E}\big{[}\left|\xi\right|^{2}\big{]}<\infty$.

  * •

$\mathcal{P}$ denotes the $\sigma$-algebra on $\left[0,T\right]\times\Omega$
of $(\mathcal{F}_{t})_{t\leq T}$-progressively measurable sets.

  * •

$\mathcal{H}^{2,l}$ denotes the set of $\mathcal{P}$-measurable processes
$w=\left(w_{t}\right)_{t\leq T}$ with values in $\mathbb{R}^{l}$ such that
$\left\|w\right\|_{\mathcal{H}^{2,l}}:=\mathbb{E}\big{[}\int^{T}_{0}\left|w_{s}\right|^{2}ds\big{]}^{\frac{1}{2}}<\infty$.
If $l=1$, then we will simply write $\mathcal{H}^{2,1}=\mathcal{H}^{2}$.

  * •

$\mathscr{S}^{2}$ stands for the set of $\mathcal{P}$-measurable, càdlàg,
$\mathbb{R}$-valued processes $w=\left(w_{t}\right)_{t\leq T}$ such that
$\left\|w\right\|_{\mathscr{S}^{2}}:=\\{\mathbb{E}\big{[}\sup_{t\leq
T}\left|w_{t}\right|^{2}\big{]}\\}^{\frac{1}{2}}<\infty$.

  * •

A random variable $\tau$ defined on $\Omega$ and valued in
$\mathbb{R}_{+}\cup\left\\{+\infty\right\\}$ is called a stopping time with
respect to the filtration $(\mathcal{F}_{t})_{t\leq T}$, or simply an
$\mathcal{F}_{t}$-stopping time, if for all $t\in\mathbb{R}_{+}$
$\left\\{\omega|\tau(\omega)\leq t\right\\}\in\mathcal{F}_{t}$.

  * •

$\mathcal{I}=\left\\{1,\ldots,q\right\\}$ denotes the set of indexes so-called
set of configurations, while the notation $\mathcal{I}^{-i}$ means
$\mathcal{I}-\left\\{i\right\\}$.

  * •

The notation $D^{2}_{xx}\phi$ and $D_{x}\phi$ denote the Hessian matrix and
the gradient vector of the function $\phi$, respectively.

##  2 Model definition and preliminary results

Consider the stochastic processes $\psi_{i}\in\mathcal{H}^{2}$,
$i\in\mathcal{I}$, and $g_{ik}\in\mathscr{S}^{2}$, $i\in\mathcal{I}$ and
$k\in\mathcal{I}^{-i}$, together with a sequence

$\mathcal{S}=\left(\tau_{n},\xi_{n}\right)_{n\geq 0}$ (2.1)

of non-decreasing $\mathcal{F}$-stopping times $\tau_{n}$, and random
variables $\xi_{n}$ which are $\mathcal{F}_{\tau_{n}}$-measurable with values
in $\mathcal{I}=\left\\{1,\ldots,q\right\\}$, such that $\tau_{0}=0$,
$\xi_{0}=i$ for some $i\in\mathcal{I}$.

Together with these elements, define the functional $J$ as follows:

$J(\mathcal{S},i)=\mathbb{E}\Bigg{[}\sum^{\infty}_{n=0}\displaystyle\int^{\tau_{n+1}}_{\tau_{n}}\psi_{\xi_{n}}\left(s\right)ds-\displaystyle\sum^{\infty}_{n=1}g_{\xi_{n-1}\xi_{n}}(\tau_{n})\mathbf{1}_{\left[\tau_{n}<T\right]}\Bigg{]},$
(2.2)

with $\psi_{\xi_{n}}:=\psi_{j}$, when $\xi_{n}=j$; and the same reasoning
applies to $g_{\xi_{n-1}\xi_{n}}$, i.e., $g_{\xi_{n-1}\xi_{n}}=g_{ik}$ if
$\xi_{n-1}=i$ and $\xi_{n}=k$.

######  Definition 2.1.

A sequence $\mathcal{S}=\left(\tau_{n},\xi_{n}\right)_{n\geq 0}$ defined as in
(2.1) is called a strategy or switching control policy for the controller.
Furthermore, we say that a strategy $\mathcal{S}$ is admissible if it
satisfies the following condition:

$\mathbb{P}\left[\tau_{n}<T,\hskip 2.84544pt\forall n\geq 0\right]=0.$

For each $i=1,\ldots,q$, denote by $\mathcal{A}_{i}$ the set of admissible
strategies with the property of $\tau_{0}=0$, $\xi_{0}=i$.

The processes $\psi_{i}$ and $g_{ik}$ are usually called the payoff rate per
unit of the time and the switching cost, respectively. We will impose a
condition to the processes $g_{ik}$, $i\in\mathcal{I}$, $k\in\mathcal{I}^{-i}$
that will be considered throughout this paper

######  Assumption A.

There exists a constant $\gamma>0$ such that the processes $g_{ik}\geq\gamma$
$\mathbb{P}$-a.s.

A finite horizon switching control problem with $q$-modes and initial
configuration $\xi_{0}=i$ for $i\in\mathcal{I}$, consists in finding an
admissible sequence $S^{*}=\left(\tau^{*}_{n},\xi^{*}_{n}\right)_{n\geq 0}$
$\in\mathcal{A}_{i}$ such that

$J(\mathcal{S}^{*},i)=\sup\limits_{\mathcal{S}\in\mathcal{A}_{i}}J(\mathcal{S},i)=:J^{*}(i),$
(2.3)

where $J$ is the functional defined in (2.2).

There is also a weaker formulation of what we understand for optimal strategy,
namely, we say that $\mathcal{S}^{*}\in\mathcal{A}_{i}$ is
$\varepsilon$-optimal strategy if for all $\varepsilon>0$, we have

$J(\mathcal{S}^{*},i)\geq J^{*}(i)-\varepsilon.$

We first provide an existence result of $q$-interconnected processes, which
will be useful later on.

######  Theorem 2.2.

Consider $q$ processes $\psi_{i}\in\mathcal{H}^{2}$, $i\in\mathcal{I}$ and
$q(q-1)$ processes $g_{ik}\in\mathscr{S}^{2}$, $i\in\mathcal{I}$,
$k\in\mathcal{I}^{-i}$. Then, under Assumption (A), there exist $q$
$\mathbb{R}-$valued càdlàg processes
$\left(Y^{i}_{\cdot}:=(Y^{i}_{t}\right)_{t\leq
T},i=1,\ldots,q)\in(\mathscr{S}^{2})^{q}$ satisfying: $\forall i\in{\cal I}$

$\mathbb{P}-a.s.,\,\,\forall t\leq
T,\,\,Y^{i}_{t}=\operatorname*{ess\,sup}_{\tau\geq
t}\mathbb{E}\bigg{[}\displaystyle\int^{\tau}_{t}\psi_{i}\left(s\right)ds+\max_{k\in\mathcal{I}^{-i}}\big{(}Y^{k}_{\tau}-g_{ik}(\tau)\big{)}\mathbf{1}_{\left[\tau<T\right]}\Big{|}\mathcal{F}_{t}\bigg{]}.$
(2.4)

_Proof._ For $i\in\mathcal{I}$, and any $0\leq t\leq T$, use the sequence
$(Y^{i,n}_{t})_{n\geq 0}$ defined by:

$Y^{i,0}_{t}=\mathbb{E}\bigg{[}\displaystyle\int^{T}_{t}\psi_{i}\left(s\right)ds\Big{|}\mathcal{F}_{t}\bigg{]},$

and for $n\geq 1$,

$Y^{i,n}_{t}=\operatorname*{ess\,sup}_{\tau\geq
t}\mathbb{E}\bigg{[}\displaystyle\int^{\tau}_{t}\psi_{i}\left(s\right)ds+\max_{k\in\mathcal{I}^{-i}}\big{(}Y^{k,n-1}_{\tau}-g_{ik}(\tau)\big{)}\mathbf{1}_{\left[\tau<T\right]}\Big{|}\mathcal{F}_{t}\bigg{]}.$
(2.5)

First note that the process $(Y^{k,0}_{t})_{t\leq T}$ is continuous for all
$k\in\mathcal{I}$. Next since the process $g_{ik}$ is càdlàg,
$(Y^{i,1}_{t})_{t\leq T}$ is also a càdlàg process, and thus by an induction
procedure we have that for all $n\geq 1$, $Y^{i,n}_{t}$ is càdlàg too.

Let us prove now that, for $i\in\mathcal{I}$, the sequence
$(Y^{i,n}_{\cdot})_{n\geq 0}$ converges increasingly and pointwisely
$\mathbb{P}$-a.s. for any $0\leq t\leq T$ and in the norm $\mathcal{H}^{2}$ to
a càdlàg process $Y^{i}_{\cdot}$. To begin with, for any $n\geq 1$ let us
define $\mathcal{A}^{i,n}_{t}=\left\\{\mathcal{S}=(\tau_{m},\xi_{m})_{m\geq
0}:\xi_{0}=i,\tau_{0}=t\hskip 8.5359pt\textrm{and}\hskip
8.5359pt\tau_{n+1}=T\right\\}$, and let us prove that for $N$ fixed,
$Y^{i,N}_{\cdot}$ can be characterized by

$Y^{i,N}_{t}=\operatorname*{ess\,sup}_{\mathcal{S}\in\mathcal{A}^{i,N}_{t}}\mathbb{E}\Bigg{[}\sum^{N}_{j=0}\displaystyle\int^{\tau_{j+1}}_{\tau_{j}}\psi_{\xi_{j}}\left(s\right)ds-\displaystyle\sum^{N-1}_{j=0}g_{\xi_{j}\xi_{j+1}}(\tau_{j+1})\mathbf{1}_{\left[\tau_{j+1}<T\right]}\bigg{|}\mathcal{F}_{t}\Bigg{]}.$
(2.6)

Since the processes $g_{ik}$ for $i,k\in\mathcal{I}$ are càdlàg, it is not
obvious to use the same procedure as given in Djehiche. et al. [8] Proposition
3-(ii). In contrast, we shall consider the sequence of $\varepsilon$-stopping
times $(\tau^{\varepsilon}_{n})_{n\geq 0}$ given by follows:
$\tau^{\varepsilon}_{0}:=t$,

$\tau^{\varepsilon}_{1}:=\inf\Big{\\{}s\geq
t:Y^{i,N}_{s}\leq\max_{k\in\mathcal{I}^{-i}}\big{(}Y_{s}^{k,N-1}-g_{ik}\left(s\right)\big{)}+\frac{\varepsilon}{2}\Big{\\}}\wedge
T$

and for $2\leq n\leq N$,

$\begin{array}[]{ll}\tau^{\varepsilon}_{n}:=\inf\Big{\\{}s\geq\tau^{\varepsilon}_{n-1}:Y^{\hat{\xi}_{n-1},N-n+1}_{s}\leq\max\limits_{k\in\mathcal{I}^{-\hat{\xi}_{n-1}}}\big{(}Y_{s}^{k,N-n}-g_{\hat{\xi}_{n-1}k}(s)\big{)}+\frac{\varepsilon}{2^{n}}\Big{\\}}\wedge
T.&\\\ &\\\ \tau^{\varepsilon}_{N+1}:=T,\end{array}$

where

  * •

$\hat{\xi}_{0}:=i$,
$\hat{\xi}_{1}:=\arg\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{k,N-1}_{\tau^{\varepsilon}_{1}}-g_{ik}(\tau^{\varepsilon}_{1})\right\\}$

and for $n\geq 2$,

  * •

$\hat{\xi}_{n}=\arg\max\limits_{k\in\mathcal{I}^{-\hat{\xi}_{n-1}}}\left\\{Y^{k,N-n}_{\tau^{\varepsilon}_{n}}-g_{\hat{\xi}_{n-1}k}(\tau^{\varepsilon}_{n})\right\\}.$

Note that by (2.5) the process $(Y^{i,N}_{s}+\int^{s}_{t}\psi_{i}(r)dr)_{t\leq
s\leq\tau^{\varepsilon}_{1}}$ is a super-martingale. Hence, if its Doob-Meyer
decomposition is given by $(M_{s}-K_{s})_{t\leq s\leq\tau^{\varepsilon}_{1}}$
(recall that $M$ is a martingale and $K$ a non-decreasing process), then by
definition of $\tau^{\varepsilon}_{1}$, we have that $K_{s}=0$ for
$s\in[t,\tau^{\varepsilon}_{1}]$, i.e.,
$(Y^{i,N}_{s}+\int^{s}_{t}\psi_{i}(r)dr)_{t\leq s\leq\tau^{\varepsilon}_{1}}$
is a martingale. Therefore,

$\begin{array}[]{ll}Y^{i,N}_{t}&=\mathbb{E}\bigg{[}Y^{i,N}_{\tau^{\varepsilon}_{1}}+\displaystyle\int^{\tau^{\varepsilon}_{1}}_{t}\psi_{i}(r)dr\Big{|}\mathcal{F}_{t}\bigg{]}\\\
\\\
&\leq\mathbb{E}\bigg{[}\max\limits_{k\in\mathcal{I}^{-i}}\left(Y_{\tau^{\varepsilon}_{1}}^{k,N-1}-g_{ik}\left(\tau^{\varepsilon}_{1}\right)\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{1}<T\right]}+\frac{\varepsilon}{2}+\displaystyle\int^{\tau^{\varepsilon}_{1}}_{t}\psi_{i}(r)dr\Big{|}\mathcal{F}_{t}\bigg{]}\\\
\\\
&=\mathbb{E}\bigg{[}\left(Y_{\tau^{\varepsilon}_{1}}^{\hat{\xi}_{1},N-1}-g_{i\hat{\xi}_{1}}\left(\tau^{\varepsilon}_{1}\right)\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{1}<T\right]}+\frac{\varepsilon}{2}+\displaystyle\int^{\tau^{\varepsilon}_{1}}_{t}\psi_{i}(r)dr\Big{|}\mathcal{F}_{t}\bigg{]}\\\
\\\
&=\mathbb{E}\bigg{[}\displaystyle\int^{\tau^{\varepsilon}_{1}}_{t}\psi_{i}(r)dr-
g_{i\hat{\xi}_{1}}\left(\tau^{\varepsilon}_{1}\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{1}<T\right]}+Y_{\tau^{\varepsilon}_{1}}^{\hat{\xi}_{1},N-1}\mathbf{1}_{\left[\tau^{\varepsilon}_{1}<T\right]}\Big{|}\mathcal{F}_{t}\bigg{]}+\frac{\varepsilon}{2}.\end{array}$
(2.7)

Analogously, taking:

$\tau^{\varepsilon}_{2}=\inf\Big{\\{}s\geq\tau^{\varepsilon}_{1},Y^{\hat{\xi}_{1},N-1}_{s}\leq\max_{k\in\mathcal{I}^{-\hat{\xi}_{1}}}\big{(}Y_{s}^{k,N-2}-g_{\hat{\xi}_{1}k}(s)\big{)}+\frac{\varepsilon}{4}\Big{\\}}\wedge
T$

we have again that
$(Y^{\hat{\xi}_{1},N-1}_{s}+\int^{s}_{\tau^{\varepsilon}_{1}}\psi_{\hat{\xi}_{1}}(r)dr)_{\tau^{\varepsilon}_{1}\leq
s\leq\tau^{\varepsilon}_{2}}$ is a martingale. Arguing similarly as above, we
have

$\begin{array}[]{ll}Y^{\hat{\xi}_{1},N-1}_{\tau^{\varepsilon}_{1}}&=\mathbb{E}\bigg{[}Y^{\hat{\xi}_{1},N-1}_{\tau^{\varepsilon}_{2}}+\displaystyle\int^{\tau^{\varepsilon}_{2}}_{\tau^{\varepsilon}_{1}}\psi_{\hat{\xi}_{1}}(r)dr\Big{|}\mathcal{F}_{\tau^{\varepsilon}_{1}}\bigg{]}\\\
\\\
&\leq\mathbb{E}\bigg{[}\max\limits_{k\in\mathcal{I}^{-\hat{\xi}_{1}}}\left(Y_{\tau^{\varepsilon}_{2}}^{k,N-2}-g_{\hat{\xi}_{1}k}\left(\tau^{\varepsilon}_{2}\right)\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{2}<T\right]}+\frac{\varepsilon}{4}+\displaystyle\int^{\tau^{\varepsilon}_{2}}_{\tau^{\varepsilon}_{1}}\psi_{\hat{\xi}_{1}}(r)dr\Big{|}\mathcal{F}_{\tau^{\varepsilon}_{1}}\bigg{]}\\\
\\\
&=\mathbb{E}\bigg{[}\left(Y_{\tau^{\varepsilon}_{2}}^{\hat{\xi}_{2},N-2}-g_{\hat{\xi}_{1}\hat{\xi}_{2}}\left(\tau^{\varepsilon}_{2}\right)\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{2}<T\right]}+\frac{\varepsilon}{4}+\displaystyle\int^{\tau^{\varepsilon}_{2}}_{\tau^{\varepsilon}_{1}}\psi_{\hat{\xi}_{1}}(r)dr\Big{|}\mathcal{F}_{\tau^{\varepsilon}_{1}}\bigg{]}\\\
\\\
&=\mathbb{E}\bigg{[}\displaystyle\int^{\tau^{\varepsilon}_{2}}_{\tau^{\varepsilon}_{1}}\psi_{\hat{\xi}_{1}}(r)dr-
g_{\hat{\xi}_{1}\hat{\xi}_{2}}\left(\tau^{\varepsilon}_{2}\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{2}<T\right]}+Y_{\tau^{\varepsilon}_{2}}^{\hat{\xi}_{2},N-2}\mathbf{1}_{\left[\tau^{\varepsilon}_{2}<T\right]}\Big{|}\mathcal{F}_{\tau^{\varepsilon}_{1}}\bigg{]}+\frac{\varepsilon}{4}.\end{array}$
(2.8)

Plugging (2.8) into (2.7), rearranging terms and since that
$\left[\tau^{\varepsilon}_{2}<T\right]\subset\left[\tau^{\varepsilon}_{1}<T\right]$,
we see that

$Y^{i,N}_{t}\leq\mathbb{E}\Bigg{[}\sum^{1}_{j=0}\bigg{(}\int^{\tau^{\varepsilon}_{j+1}}_{\tau^{\varepsilon}_{j}}\psi_{\hat{\xi}_{j}}(r)dr-
g_{\hat{\xi}_{j}\hat{\xi}_{j+1}}\left(\tau^{\varepsilon}_{j+1}\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{j+1}<T\right]}\bigg{)}+Y_{\tau^{\varepsilon}_{2}}^{\hat{\xi}_{2},N-2}\mathbf{1}_{\left[\tau^{\varepsilon}_{2}<T\right]}\bigg{|}\mathcal{F}_{t}\Bigg{]}+\frac{\varepsilon}{2}+\frac{\varepsilon}{4}.$

Repeating this procedure $N$ times, we obtain

$Y^{i,N}_{t}\leq\mathbb{E}\Bigg{[}\sum^{N-1}_{j=0}\bigg{(}\int^{\tau^{\varepsilon}_{j+1}}_{\tau^{\varepsilon}_{j}}\psi_{\hat{\xi}_{j}}(s)ds-
g_{\hat{\xi}_{j}\hat{\xi}_{j+1}}\left(\tau^{\varepsilon}_{j+1}\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{j+1}<T\right]}\bigg{)}+Y_{\tau^{\varepsilon}_{N}}^{\hat{\xi}_{N},0}\mathbf{1}_{\left[\tau^{\varepsilon}_{N}<T\right]}\bigg{|}\mathcal{F}_{t}\Bigg{]}+\varepsilon\left(\sum^{N}_{i=1}\frac{1}{2^{i}}\right).$
(2.9)

But

$Y_{\tau^{\varepsilon}_{N}}^{\hat{\xi}_{N},0}=\mathbb{E}\bigg{[}\int^{T}_{\tau^{\varepsilon}_{N}}\psi_{\hat{\xi}_{N}}(s)ds\Big{|}\mathcal{F}_{\tau_{N}}\bigg{]}.$
(2.10)

Plugging (2.10) into (2.9), and noting that
$\left(\sum^{n}_{i=1}\frac{1}{2^{i}}\right)<1$, we deduce

$Y^{i,N}_{t}\leq\mathbb{E}\Bigg{[}\sum^{N}_{j=0}\int^{\tau^{\varepsilon}_{j+1}}_{\tau^{\varepsilon}_{j}}\psi_{\hat{\xi}_{j}}(s)ds-\sum^{N-1}_{j=0}g_{\hat{\xi}_{j}\hat{\xi}_{j+1}}\left(\tau^{\varepsilon}_{j+1}\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{j+1}<T\right]}\bigg{|}\mathcal{F}_{t}\Bigg{]}+\varepsilon\hskip
11.38092pt\text{for all}\hskip 5.69046pt\varepsilon>0.$

Since $(\tau^{\varepsilon}_{n},\hat{\xi}_{n})_{0\leq n\leq N+1}$ belongs to
$\mathcal{A}^{i,N}_{t}$, we can take essential supremum over
$\mathcal{S}\in\mathcal{A}^{i,N}_{t}$ and then sending $\varepsilon\rightarrow
0$ to obtain

$Y^{i,N}_{t}\leq\operatorname*{ess\,sup}_{\mathcal{S}\in\mathcal{A}^{i,N}_{t}}\mathbb{E}\Bigg{[}\sum^{N}_{j=0}\int^{\tau_{j+1}}_{\tau_{j}}\psi_{\hat{\xi}_{j}}(s)ds-\displaystyle\sum^{N-1}_{j=0}g_{\hat{\xi}_{j}\hat{\xi}_{j+1}}(\tau_{j+1})\mathbf{1}_{\left[\tau_{j+1}<T\right]}\bigg{|}\mathcal{F}_{t}\Bigg{]}.$
(2.11)

Now we derive the inverse inequality. Let
$\mathcal{S}=(\tau_{n},\xi_{n})\in\mathcal{A}^{i,N}_{t}$ be an arbitrary
strategy. Since $\tau_{1}\geq t$, $\mathbb{P}$-a.s., and $\xi_{0}=i$, then
from (2.5) we have

$\begin{array}[]{ll}Y^{i,N}_{t}&=\operatorname*{ess\,sup}\limits_{\tau\geq
t}\mathbb{E}\bigg{[}\displaystyle\int^{\tau}_{t}\psi_{i}\left(s\right)ds+\max_{k\in\mathcal{I}^{-i}}\left(Y^{k,N-1}_{\tau}-g_{ik}(\tau)\right)\mathbf{1}_{\left[\tau<T\right]}\Big{|}\mathcal{F}_{t}\bigg{]}\\\
\\\
&\geq\mathbb{E}\bigg{[}\displaystyle\int^{\tau_{1}}_{t}\psi_{i}\left(s\right)ds+\max_{k\in\mathcal{I}^{-i}}\left(Y^{k,N-1}_{\tau_{1}}-g_{ik}(\tau_{1})\right)\mathbf{1}_{\left[\tau_{1}<T\right]}\Big{|}\mathcal{F}_{t}\bigg{]}\\\
\\\
&\geq\mathbb{E}\bigg{[}\displaystyle\int^{\tau_{1}}_{t}\psi_{i}\left(s\right)ds+\left(Y^{\xi_{1},N-1}_{\tau_{1}}-g_{i\xi_{1}}(\tau_{1})\right)\mathbf{1}_{\left[\tau_{1}<T\right]}\Big{|}\mathcal{F}_{t}\bigg{]}.\end{array}$
(2.12)

In the same way, since $\tau_{2}\geq\tau_{1}$ and $\tau_{1}$ is also
$\mathcal{F}_{\tau_{2}}$\- measurable, then

$\begin{array}[]{ll}Y^{\xi_{1},N-1}_{\tau_{1}}&=\operatorname*{ess\,sup}\limits_{\tau\geq\tau_{1}}\mathbb{E}\bigg{[}\displaystyle\int^{\tau}_{\tau_{1}}\psi_{\xi_{1}}\left(s\right)ds+\max_{k\in\mathcal{I}^{-\xi_{1}}}\left(Y^{k,N-2}_{\tau}-g_{\xi_{1}k}(\tau)\right)\mathbf{1}_{\left[\tau<T\right]}\Big{|}\mathcal{F}_{\tau_{1}}\bigg{]}\\\
\\\
&\geq\mathbb{E}\bigg{[}\displaystyle\int^{\tau_{2}}_{\tau_{1}}\psi_{\xi_{1}}\left(s\right)ds+\max_{k\in\mathcal{I}^{-\xi_{1}}}\left(Y^{k,N-2}_{\tau_{2}}-g_{\xi_{1}k}(\tau_{2})\right)\mathbf{1}_{\left[\tau_{2}<T\right]}\Big{|}\mathcal{F}_{\tau_{1}}\bigg{]}\\\
\\\
&\geq\mathbb{E}\bigg{[}\displaystyle\int^{\tau_{2}}_{\tau_{1}}\psi_{\xi_{1}}\left(s\right)ds+\left(Y^{\xi_{2},N-2}_{\tau_{2}}-g_{\xi_{1}\xi_{2}}(\tau_{2})\right)\mathbf{1}_{\left[\tau_{2}<T\right]}\Big{|}\mathcal{F}_{\tau_{1}}\bigg{]}.\end{array}$

Plugging this last inequality into (2.12), rearranging terms and using that
$\left[\tau_{2}<T\right]\subset\left[\tau_{1}<T\right]\in\mathcal{F}_{\tau_{1}}$,
we see that

$Y^{i,N}_{t}\geq\mathbb{E}\Bigg{[}\displaystyle\sum^{1}_{j=0}\bigg{(}\int^{\tau_{j+1}}_{\tau_{j}}\psi_{\xi_{j}}\left(s\right)ds-
g_{\xi_{j}\xi_{j+1}}(\tau_{j})\mathbf{1}_{\left[\tau_{j+1}<T\right]}\bigg{)}+Y^{\xi_{2},N-2}_{\tau_{2}}\mathbf{1}_{\left[\tau_{2}<T\right]}\bigg{|}\mathcal{F}_{t}\Bigg{]}.$

Continuing this procedure, we have

$Y^{i,N}_{t}\geq\mathbb{E}\Bigg{[}\displaystyle\sum^{N-1}_{j=0}\bigg{(}\int^{\tau_{j+1}}_{\tau_{j}}\psi_{\xi_{j}}\left(s\right)ds-
g_{\xi_{j}\xi_{j+1}}(\tau_{j+1})\mathbf{1}_{\left[\tau_{j+1}<T\right]}\bigg{)}+Y^{\xi_{N},0}_{\tau_{N}}\mathbf{1}_{\left[\tau_{N}<T\right]}\bigg{|}\mathcal{F}_{t}\Bigg{]}.$

But again, since
$Y^{\xi_{N},0}_{\tau_{N}}=\mathbb{E}\big{[}\int^{T}_{\tau_{N}}\psi_{\xi_{n}}(s)ds\big{|}\mathcal{F}_{\tau_{N}}\big{]}$,
we get

$Y^{i,N}_{t}\geq\mathbb{E}\Bigg{[}\displaystyle\sum^{N}_{j=0}\int^{\tau_{j+1}}_{\tau_{j}}\psi_{\xi_{j}}\left(s\right)ds-\sum^{N-1}_{j=0}g_{\xi_{j}\xi_{j+1}}(\tau_{j+1})\mathbf{1}_{\left[\tau_{j+1}<T\right]}\bigg{|}\mathcal{F}_{t}\Bigg{]}\hskip
8.5359pt\text{for all}\hskip 5.69046pt\mathcal{S}\in\mathcal{A}^{i,N}_{t}.$

Thus, taking the essential supremum on $\mathcal{A}^{i,N}_{t}$, we get

$Y^{i,N}_{t}\geq\operatorname*{ess\,sup}_{\mathcal{S}\in\mathcal{A}^{i,N}_{t}}\mathbb{E}\Bigg{[}\sum^{N}_{j=0}\int^{\tau_{j+1}}_{\tau_{j}}\psi_{\xi_{j}}(s)ds-\sum^{N-1}_{j=0}g_{\xi_{j}\xi_{j+1}}(\tau_{j+1})\mathbf{1}_{\left[\tau_{j+1}<T\right]}\bigg{|}\mathcal{F}_{t}\Bigg{]}.$

This last inequality together with (2.11), yield the characterization (2.6).

Since $\mathcal{A}^{i,n}_{t}\subset\mathcal{A}^{i,n+1}_{t}$, we have
$Y^{i,n}_{t}\leq Y^{i,n+1}_{t}$, $\mathbb{P}$-a.s. for all
$t\in\left[0,T\right]$. On the other hand, by Assumption (A), we obtain for
each $i\in\mathcal{I}$,

$Y^{i,n}_{t}\leq\mathbb{E}\bigg{[}\int^{T}_{t}\max_{[i=1,\ldots,q]}\left|\psi_{i}(s)\right|ds\Big{|}\mathcal{F}_{t}\bigg{]}\hskip
19.91684pt\text{for all}\hskip 5.69046ptt\leq T\hskip
5.69046pt\text{and}\hskip 5.69046ptn\geq 0$

and hence the sequence $(Y^{i,n}_{t})_{n\geq 1}$ is convergent. We now let
$Y^{i}_{t}:=\lim_{n\rightarrow\infty}Y^{i,n}_{t}$ for $t\leq T$. Note that the
process $Y^{i}_{\cdot}$ satisfies

$Y^{i,0}_{t}\leq
Y^{i}_{t}\leq\mathbb{E}\bigg{[}\int^{T}_{t}\max_{[i=1,\ldots,q]}\left|\psi_{i}(s)\right|ds\Big{|}\mathcal{F}_{t}\bigg{]}\hskip
8.5359pt\text{for all}\hskip 5.69046ptt\leq T.$ (2.13)

Besides, $Y^{i}_{\cdot}$ is also càdlàg process. Indeed, from (2.5) the
process $(Y^{i,n}_{t}+\int^{t}_{0}\psi_{i}(s)ds)_{0\leq t\leq T}$ is a càdlàg
super-martingale for all $i\in\mathcal{I}$ and $n\geq 1$. Thus its limit
process $(Y^{i}_{t}+\int^{t}_{0}\psi_{i}(s)ds)_{0\leq t\leq T}$ is càdlàg as a
limit of increasing sequence of càdlàg super-martingales (see Dellacherie and
Meyer [[7], p. 86]), which gives the desired càdlàg property of $Y^{i}_{.}$.
Moreover, from (2.13), the $L^{2}$-properties of $\psi_{i}$ and by Doob’s
maximal inequality, for each $i\in\mathcal{I}$, we have

$\mathbb{E}\bigg{[}\sup_{0\leq t\leq
T}\left|Y^{i}_{t}\right|^{2}\bigg{]}<\infty$

and hence by the Lebesgue dominated convergence theorem the sequence
$(Y^{i,n}_{\cdot})_{n\geq 0}$ converges to $Y^{i}_{\cdot}$ in
$\mathcal{H}^{2}$. Thus, by Snell envelope properties (see Proposition 2-(iv)
in Djehiche, et al. [8]), the càdlàg processes
$Y^{1}_{\cdot},\ldots,Y^{q}_{\cdot}$ satisfy (2.4) since they are limits of
the increasing sequence of processes $Y^{i,n}_{\cdot}$, for $i\in\mathcal{I}$,
satisfying (2.6).$\hskip 28.45274pt\blacksquare$

Let us show now some properties of the $\varepsilon$-strategy introduced in
Theorem 2.2.

######  Proposition 2.3.

The $\varepsilon$-strategy
$\mathcal{S}^{\varepsilon}=\left(\tau^{\varepsilon}_{n},\xi^{\varepsilon}_{n}\right)_{n\geq
0}$ defined as follows:

  * •

$\tau^{\varepsilon}_{0}:=0$, $\tau^{\varepsilon}_{1}:=\inf\Big{\\{}s\geq
0:Y^{i}_{s}\leq\max\limits_{k\in\mathcal{I}^{-i}}\left(Y_{s}^{k}-g_{ik}\left(s\right)\right)+\frac{\varepsilon}{2}\Big{\\}}\wedge
T$

and, for $n\geq 2$,

  * •

$\tau^{\varepsilon}_{n}:=\inf\Big{\\{}s\geq\tau^{\varepsilon}_{n-1}:Y^{\xi^{\varepsilon}_{n-1}}_{s}\leq\max\limits_{k\in\mathcal{I}^{-\xi^{\varepsilon}_{n-1}}}\left(Y_{s}^{k}-g_{\xi^{\varepsilon}_{n-1}k}(s)\right)+\frac{\varepsilon}{2^{n}}\Big{\\}}\wedge
T$

and the sequence $(\xi^{\varepsilon}_{n})$ given by

  * •

$\xi^{\varepsilon}_{0}:=i$,
$\xi^{\varepsilon}_{1}=\arg\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{k}_{\tau^{\varepsilon}_{1}}-g_{ik}(\tau^{\varepsilon}_{1})\right\\}$

and for $n\geq 2$,

  * •

$\xi^{\varepsilon}_{n}=\arg\max\limits_{k\in\mathcal{I}^{-\xi_{n-1}}}\left\\{Y^{k}_{\tau^{\varepsilon}_{n}}-g_{\hat{\xi}_{n-1}k}(\tau^{\varepsilon}_{n})\right\\},$

is admissible.

Proof. Suppose for contradiction that $\mathcal{S}^{\varepsilon}$ is not
admissible, that is, $\mathbb{P}[\tau^{\varepsilon}_{n}<T,\text{for all}\hskip
5.69046ptn\geq 1]>0$. Then, by definition of $\tau^{\varepsilon}_{n}$ we have

$\mathbb{P}\left[Y^{\xi^{\varepsilon}_{n-1}}_{\tau^{\varepsilon}_{n}}\leq
Y^{\xi^{\varepsilon}_{n}}_{\tau^{\varepsilon}_{n}}-g_{\xi^{\varepsilon}_{n-1}\xi^{\varepsilon}_{n}}(\tau^{\varepsilon}_{n})+\frac{\varepsilon}{2^{n}},\hskip
5.69046pt\xi^{\varepsilon}_{n}\in\mathcal{I}^{-\xi^{\varepsilon}_{n-1}},\forall
n\geq 1\right]>0.$

If the event
$B=\left\\{\omega\in\Omega:\tau^{\varepsilon}_{n}(\omega)<T,\forall n\geq
1\right\\}$ has positive probability, then there is a state
$i_{1}\in\mathcal{I}$ and a loop $i_{1},i_{2}\ldots,i_{k}$ (with
$i_{1}=i_{k}$) of elements of $\mathcal{I}$ (recall that $\mathcal{I}$ is a
finite set), and subsequence
$\tau^{\varepsilon}_{n},\ldots,\tau^{\varepsilon}_{n+k}$ corresponding of this
configuration such that

$\mathbb{P}\left[Y^{i_{l}}_{\tau^{\varepsilon}_{n+l}}\leq
Y^{i_{l+1}}_{\tau^{\varepsilon}_{n+l}}-g_{i_{l},i_{l+1}}(\tau^{\varepsilon}_{n+l})+\frac{\varepsilon}{2^{n}},l=1,\ldots
k-1,(i_{k}=i_{1}),\forall n\geq 0\right]>0.$ (2.14)

Since $(\tau^{\varepsilon}_{n})_{n\geq 1}$ is monotone and bounded, then we
can define $\tau:=\lim_{n\rightarrow\infty}\tau^{\varepsilon}_{n}$. Taking the
limit with respect to $n$ in (2.14), we obtain

$\mathbb{P}\left[Y^{i_{l}}_{\tau-}\leq
Y^{i_{l+1}}_{\tau-}-g_{i_{l},i_{l+1}}(\tau-),l=1,\ldots
k-1,(i_{k}=i_{1})\right]>0.$ (2.15)

But it is easy to verify that

$\left\\{Y^{i_{l}}_{\tau-}\leq
Y^{i_{l+1}}_{\tau-}-g_{i_{l},i_{l+1}}(\tau-),l=1,\ldots
k-1,(i_{k}=i_{1})\right\\}\subseteq\Big{\\{}g_{i_{1},i_{2}}(\tau-)+\cdots+g_{i_{k-1},i_{1}}(\tau-)\leq
0\Big{\\}},$

then from (2.15) we have

$\mathbb{P}\left[g_{i_{1},i_{2}}(\tau-)+\cdots+g_{i_{k},i_{1}}(\tau-)\leq
0\right]>0.$

Since $g_{ij}\geq\gamma>0$ $\mathbb{P}$-a.s., we have a contradiction.
Therefore, $\mathcal{S}^{\varepsilon}$ is admissible.$\hskip
113.81102pt\blacksquare$

Our next result has to do with a so-called verification theorem for the
switching problem (2.3) in the context of càdlàg cost functions

######  Theorem 2.4.

The $q$ $\mathscr{S}^{2}$-processes
$(Y^{i}_{\cdot}:=\left(Y^{i}_{t}\right)_{t\leq T},i=1,\ldots,q)$ in Theorem
2.2 are unique and they have the following relation with the switching problem
(2.3):

  1. (i)

For each $i\in\mathcal{I}$,

$Y^{i}_{0}=\sup_{\mathcal{S}\in\mathcal{A}_{i}}J(\mathcal{S},i).$ (2.16)

  2. (ii)

The $\varepsilon$-strategy $\mathcal{S}^{\varepsilon}$ defined in Proposition
2.3 forms an $\varepsilon$-optimal strategy, i.e., for
$\mathcal{S}^{\varepsilon}=(\tau^{\varepsilon}_{n},\xi^{\varepsilon}_{n})_{n\geq
0}$,

$J\left(\mathcal{S}^{\varepsilon},i\right)\geq\sup\limits_{\mathcal{S}\in\mathcal{A}_{i}}J(\mathcal{S},i)-\varepsilon.$
(2.17)

_Proof._

  1. (i)

Assuming that at time $t=0$ the system is in mode $i$, it follows by (2.4)
that, for any $0\leq t\leq T$,

$Y^{i}_{t}+\int^{t}_{0}\psi_{i}(s)ds=\operatorname*{ess\,sup}_{\tau\geq
t}\mathbb{E}\bigg{[}\displaystyle\int^{\tau}_{0}\psi_{i}\left(s\right)ds+\max_{k\in\mathcal{I}^{-i}}\left(Y^{k}_{\tau}-g_{ik}\left(\tau\right)\right)\mathbf{1}_{\left[\tau<T\right]}\Big{|}\mathcal{F}_{t}\bigg{]}.$

Since $Y^{i}_{0}$ is $\mathcal{F}_{0}$-measurable, it is a $\mathbb{P}$-a.s.
constant, that is, $Y^{i}_{0}=\mathbb{E}\left[Y^{i}_{0}\right]$. Now take
$\mathcal{S}^{\varepsilon}$ defined in Proposition 2.3. Arguing similarly to
Theorem 2.2, we can deduce

$\begin{array}[]{ll}Y^{i}_{0}&\leq\mathbb{E}\bigg{[}\displaystyle\int^{\tau^{\varepsilon}_{1}}_{0}\psi_{i}\left(s\right)ds+\max_{k\in\mathcal{I}^{-i}}\left(Y^{k}_{\tau^{\varepsilon}_{1}}-g_{ik}\left(\tau^{\varepsilon}_{1}\right)\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{1}<T\right]}\bigg{]}+\frac{\varepsilon}{2}\\\
\\\
&=\mathbb{E}\bigg{[}\displaystyle\int^{\tau^{\varepsilon}_{1}}_{0}\psi_{i}\left(s\right)ds+\left(Y^{\xi^{\varepsilon}_{1}}_{\tau^{\varepsilon}_{1}}-g_{i\xi^{\varepsilon}_{1}}\left(\tau^{\varepsilon}_{1}\right)\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{1}<T\right]}\bigg{]}+\frac{\varepsilon}{2}.\end{array}$
(2.18)

The rest of the proof uses the same arguments as in the proof of Theorem 2.2.
Namely, for every $\tau^{\varepsilon}_{1}\leq t\leq T$, we can deduce

$Y^{\xi^{\varepsilon}_{1}}_{t}=\operatorname*{ess\,sup}_{\tau\geq
t}\mathbb{E}\bigg{[}\displaystyle\int^{\tau}_{t}\psi_{\xi^{\varepsilon}_{1}}\left(s\right)ds+\max_{j\in\mathcal{I}^{-\xi^{\varepsilon}_{1}}}\left(Y^{j}_{\tau}-g_{\xi^{\varepsilon}_{1}j}\left(\tau\right)\right)\mathbf{1}_{\left[\tau<T\right]}\Big{|}\mathcal{F}_{t}\bigg{]}.$

Then, from the definition of $\tau^{\varepsilon}_{2}$ and since
$(Y^{\xi^{\varepsilon}_{1}}_{t}+\int^{t}_{\tau^{\varepsilon}_{1}}\psi_{\xi^{\varepsilon}_{1}}\left(s\right)ds)_{\tau^{\varepsilon}_{1}\leq
t\leq\tau^{\varepsilon}_{2}}$ is a martingale, we get

$\begin{array}[]{ll}Y^{\xi^{\varepsilon}_{1}}_{\tau^{\varepsilon}_{1}}&\leq\mathbb{E}\bigg{[}\displaystyle\int^{\tau^{\varepsilon}_{2}}_{\tau^{\varepsilon}_{1}}\psi_{\xi^{\varepsilon}_{1}}\left(s\right)ds+\max_{j\in\mathcal{I}^{-\xi^{\varepsilon}_{1}}}\left(Y^{j}_{\tau^{\varepsilon}_{2}}-g_{\xi^{\varepsilon}_{1}j}\left(\tau^{\varepsilon}_{2}\right)\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{2}<T\right]}\Big{|}\mathcal{F}_{\tau^{\varepsilon}_{1}}\bigg{]}+\frac{\varepsilon}{4}\\\
\\\
&=\mathbb{E}\bigg{[}\displaystyle\int^{\tau^{\varepsilon}_{2}}_{\tau^{\varepsilon}_{1}}\psi_{\xi^{\varepsilon}_{1}}\left(s\right)ds+\left(Y^{\xi^{\varepsilon}_{2}}_{\tau^{\varepsilon}_{2}}-g_{\xi^{\varepsilon}_{1}\xi^{\varepsilon}_{2}}\left(\tau^{\varepsilon}_{2}\right)\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{2}<T\right]}\Big{|}\mathcal{F}_{\tau^{\varepsilon}_{1}}\bigg{]}+\frac{\varepsilon}{4}.\end{array}$
(2.19)

Plugging (2.19) into (2.18) and noting that
$\mathbf{1}_{\left[\tau^{\varepsilon}_{1}<T\right]}$ is
$\mathcal{F}_{\tau^{\varepsilon}_{1}}$-measurable, it follows that:

$\begin{array}[]{ll}Y^{i}_{0}&\leq\mathbb{E}\bigg{[}\displaystyle\int^{\tau^{\varepsilon}_{1}}_{0}\psi_{i}\left(s\right)ds-
g_{i\xi^{\varepsilon}_{1}}\left(\tau^{\varepsilon}_{1}\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{1}<T\right]}\bigg{]}\\\
\\\ &\hskip
28.45274pt+\mathbb{E}\bigg{[}\displaystyle\int^{\tau^{\varepsilon}_{2}}_{\tau^{\varepsilon}_{1}}\psi_{\xi^{\varepsilon}_{1}}\left(s\right)ds+\left(Y^{\xi^{\varepsilon}_{2}}_{\tau^{\varepsilon}_{2}}-g_{\xi^{\varepsilon}_{1}\xi^{\varepsilon}_{2}}\left(\tau^{\varepsilon}_{2}\right)\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{2}<T\right]}\bigg{]}+\varepsilon\left(\frac{1}{2}+\frac{1}{4}\right).\\\
\\\
&=\mathbb{E}\Bigg{[}\displaystyle\sum^{1}_{j=0}\bigg{(}\int^{\tau^{\varepsilon}_{j+1}}_{\tau^{\varepsilon}_{j}}\psi_{\xi^{\varepsilon}_{j}}(s)ds-
g_{\xi^{\varepsilon}_{j}\xi^{\varepsilon}_{j+1}}\left(\tau^{\varepsilon}_{j}\right)\mathbf{1}_{\left[\tau^{\varepsilon}_{j+1}<T\right]}\bigg{)}+Y^{\xi^{\varepsilon}_{2}}_{\tau^{\varepsilon}_{2}}\mathbf{1}_{\left[\tau^{\varepsilon}_{2}<T\right]}\Bigg{]}+\varepsilon\left(\frac{1}{2}+\frac{1}{4}\right)\end{array}$

since $[\tau^{\varepsilon}_{2}<T]\subset[\tau^{\varepsilon}_{1}<T]$. Repeating
this procedure $n$ times, we obtain

$Y^{i}_{0}\leq\mathbb{E}\Bigg{[}\displaystyle\sum^{n-1}_{j=0}\bigg{(}\int^{\tau^{\varepsilon}_{j+1}}_{\tau^{\varepsilon}_{j}}\psi_{\xi^{\varepsilon}_{j}}\left(s\right)ds-
g_{\xi^{\varepsilon}_{j}\xi^{\varepsilon}_{j+1}}(\tau^{\varepsilon}_{j+1})\mathbf{1}_{\left[\tau^{\varepsilon}_{j+1}<T\right]}\bigg{)}+Y^{\xi^{\varepsilon}_{n}}_{\tau^{\varepsilon}_{n}}\mathbf{1}_{\left[{\tau^{\varepsilon}_{n}<T}\right]}\Bigg{]}+\varepsilon\left(\frac{1}{2}+\cdots+\frac{1}{2^{n}}\right).$

Taking liminf as $n\rightarrow\infty$ we obtain

$Y^{i}_{0}\leq\mathbb{E}\Bigg{[}\displaystyle\sum^{\infty}_{j=0}\bigg{(}\int^{\tau^{\varepsilon}_{j+1}\wedge
T}_{\tau^{\varepsilon}_{j}\wedge
T}\psi_{\xi^{\varepsilon}_{j}}\left(s\right)ds-
g_{\xi^{\varepsilon}_{j}\xi^{\varepsilon}_{j+1}}(\tau^{\varepsilon}_{j+1})\mathbf{1}_{\left[\tau^{\varepsilon}_{j+1}<T\right]}\bigg{)}\bigg{]}+\varepsilon.$
(2.20)

By Proposition 2.3 we can take supremum over all admissible strategies
$\mathcal{A}_{i}$, to obtain

$\begin{array}[]{ll}Y^{i}_{0}&\leq\sup\limits_{\mathcal{S}\in\mathcal{A}_{i}}\mathbb{E}\Bigg{[}\displaystyle\sum^{\infty}_{j=0}\bigg{(}\int^{\tau_{j+1}\wedge
T}_{\tau_{j}\wedge T}\psi_{\xi_{j}}\left(s\right)ds-
g_{{\xi_{j}}\xi_{j+1}}(\tau_{j+1})\mathbf{1}_{\left[\tau_{j+1}<T\right]}\bigg{)}\Bigg{]}+\varepsilon\\\
\\\
&=\sup\limits_{\mathcal{S}\in\mathcal{A}_{i}}J(\mathcal{S},i)+\varepsilon.\end{array}$

Letting $\varepsilon\rightarrow 0$, it follows that
$Y^{i}_{0}\leq\sup_{\mathcal{S}\in\mathcal{A}_{i}}J(\mathcal{S},i)$. The
inverse inequality is analogous to the previous Theorem 2.2. Hence, the result
follows.

  2. (ii)

From part (i), specifically, (2.16) and inequality (2.20), we deduce

$\sup_{\mathcal{\mathcal{S}\in A}_{i}}J(\mathcal{S},i)-\varepsilon\leq
J(\mathcal{S}^{\varepsilon},i)\leq\sup_{\mathcal{S}\in\mathcal{A}_{i}}J(\mathcal{S},i),$

which proves (ii). $\hskip 398.33858pt\blacksquare$

##  3 Reflected Backward Stochastic Differential Systems

In this section we will provide the existence as well as uniqueness of the
solution of the system of reflected backward stochastic differential equations
(RBSDEs) of type

$\begin{cases}\forall i\in\mathcal{I},\hskip 5.69046pt\text{find}\hskip
5.69046pt\left(Y^{i}_{\cdot},Z^{i}_{\cdot},K^{i}_{\cdot}\right)\hskip
5.69046pt\text{such that}:&\\\
Y^{i}_{\cdot},K^{i}_{\cdot}\in\mathscr{S}^{2}\quad\text{and}\quad
Z^{i}_{\cdot}\in\mathcal{H}^{2,d};\hskip 2.84544ptK^{i}_{\cdot}\hskip
5.69046pt\text{is non-decreasing and}\quad K^{i}_{0}=0,&\\\
Y^{i}_{s}=h_{i}(X_{T})+\displaystyle\int^{T}_{s}f_{i}(r,X_{r},Y^{1}_{r},\ldots,Y^{q}_{r},Z^{i}_{r})dr+K^{i}_{T}-K^{i}_{s}-\int^{T}_{s}Z^{i}_{r}dB_{r}\hskip
11.38092pt\text{for all }0\leq s\leq T,&\\\
Y^{i}_{s}\geq\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{k}_{s}-\gamma_{ik}(s,X_{s})\right\\}\hskip
28.45274pt\text{for all }0\leq s\leq T,&\\\ \text{and if}\hskip
5.69046ptK^{i}_{\cdot}=K^{i,c}_{\cdot}+K^{i,d}_{\cdot},\hskip
2.84544pt\text{where}\hskip 5.69046ptK^{i,c}_{\cdot}\hskip
2.84544pt(\text{resp.}\hskip 5.69046ptK^{i,d}_{\cdot})\hskip 5.69046pt\text{is
the continuous}&\\\ \text{(resp. purely discontinuous) part of}\hskip
5.69046ptK^{i}_{\cdot},\hskip 5.69046pt\text{then:}\\\
\displaystyle\int^{T}_{0}\Big{(}Y^{i}_{r}-\max_{k\in\mathcal{I}^{-i}}\big{\\{}Y^{k}_{r}-\gamma_{ik}(r,X_{r})\big{\\}}\Big{)}dK^{i,c}_{r}=0.&\\\
\Delta_{s}Y_{\cdot}:=Y_{s}-Y_{s^{-}}=-\Big{(}\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{k}_{s}-\gamma_{ik}(s,X_{s})\right\\}-Y^{i}_{s}\Big{)}^{+}\hskip
28.45274pt\text{for all }0\leq s\leq T,\end{cases}$ (3.1)

in which the associated barriers are _càdlàg_ processes. This system is
connected with the previous switching problem. Actually when
$(f_{i})_{i\in{\cal I}}$ do not depend on $(Y^{i})_{i\in{\cal I}}$, the system
(3.1) is exactly the translation of the verification Theorem 2.2 in terms of
reflected BSDEs as it is well-known that the Snell envelope can be expressed
through reflected BSDEs (see e.g. El Karoui [12] or Hamadène [15]). On the
other hand, this form of system (3.1) allows to consider switching problems
when the cost functions are of risk sensitive type (utility functions) —see El
Karoui and Hamadène [13].

To begin with our analysis, we will first introduce the following assumptions
relate to the items involved in (3.1):

######  Assumption H.

  1. (H1)

: The stochastic process $(X_{t})_{t\geq 0}$ is in $\mathcal{H}^{2,r}$ for any
$r\in\mathbb{N}$.

  2. (H2)

: For any $i\in\mathcal{I}$, the function
$f_{i}:[0,T]\times\mathbb{R}^{r}\times\mathbb{R}^{q}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
satisfies:

    1. (i)

$\left(t,x\right)\mapsto f_{i}\left(t,x,y^{1},\ldots,y^{q},z\right)$ is
continuous uniformly with respect to $(y^{1},\ldots,y^{q},z)$;

    2. (ii)

$f_{i}$ is uniformly Lipschitz continuous with respect to
$(y^{1},\ldots,y^{q},z)$, i.e., for some $C\geq 0$,

$\left|f_{i}(t,x,y^{1},\ldots,y^{q},z)-f_{i}(t,x,\bar{y}^{1},\ldots,\bar{y}^{q},\bar{z})\right|\leq
C\left(\left|y^{1}-\bar{y}^{1}\right|+\cdots+\left|y^{q}-\bar{y}^{q}\right|+\left|z-\bar{z}\right|\right).$

    3. (iii)

the mapping $(t,x)\mapsto f_{i}(t,x,0,\ldots,0)$ is Borel measurable and of
polynomial growth.

    4. (iv)

_Monotonicity_ : For all $i\in{\cal I}$, for all $k\in\mathcal{I}^{-i}$, the
mapping $y_{k}\mapsto
f_{i}(t,x,y_{1},\ldots,y_{k-1},y_{k},y_{k+1},\ldots,y_{q},z)$ is non-
decreasing whenever the other components
$(t,x,y_{1},\ldots,y_{k-1},y_{k+1},\ldots,y_{q},z)$ are fixed.

  3. (H3)

: For each $i,k\in\mathcal{I}$, the function
$\gamma_{ik}:[0,T]\times\mathbb{R}^{r}\rightarrow\mathbb{R}$ is bounded from
below, i.e. there exists a real constant $\gamma>0$ such that,
$\gamma_{ik}\geq\gamma$. Furthermore it is càdlàg in $t$, continuous and of
polynomial growth in $x$.

  4. (H4)

: For each $i\in{\cal I}$, the function
$h_{i}:\mathbb{R}^{r}\rightarrow\mathbb{R}$ is continuous with polynomial
growth and satisfies

$\forall
x\in\mathbb{R}^{r},h_{i}(x)\geq\max_{k\in\mathcal{I}^{-i}}\left(h_{k}(x)-\gamma_{ik}(T,x)\right).\blacksquare$

Note that in the (3.1) the process $X$ does not play a specific role. We
consider this form of system (3.1) only in the perspective to deal with the
Hamilton-Jacobi-Bellman system associated with the switching problem.

######  Proposition 3.1.

Under Assumptions (H), the system of RBSDEs (3.1) has a solution
$(Y^{i}_{\cdot},Z^{i}_{\cdot},K^{i}_{\cdot})_{i=1,\ldots,q}$.

_Proof._ To begin with, we first consider the following standard BSDEs:

$\begin{cases}(\overline{Y}_{\cdot},\overline{Z}_{\cdot})\in\mathscr{S}^{2}\times\mathcal{H}^{2,d}\hskip
7.11317pt;&\\\
\overline{Y}_{s}=\max\limits_{i=1,\ldots,q}h_{i}(X_{T})+\displaystyle\int^{T}_{s}\big{[}\max_{i=1,\ldots,q}f_{i}\big{]}(r,X_{r},\overline{Y}_{r},\ldots,\overline{Y}_{r},\overline{Z}_{r})dr-\int^{T}_{s}\overline{Z}_{r}dB_{r},\hskip
7.11317pt\text{for all }s\leq T,\end{cases}$ (3.2)

and

$\begin{cases}(\underline{Y}_{\cdot},\underline{Z}_{\cdot})\in\mathscr{S}^{2}\times\mathcal{H}^{2,d}\hskip
7.11317pt;&\\\
\underline{Y}_{s}=\min\limits_{i=1,\ldots,q}h_{i}(X_{T})+\displaystyle\int^{T}_{s}\big{[}\min_{i=1,\ldots,q}f_{i}\big{]}(r,X_{r},\underline{Y}_{r},\ldots,\underline{Y}_{r},\underline{Z}_{r})dr-\int^{T}_{s}\underline{Z}_{r}dB_{r},\hskip
7.11317pt\text{for all }s\leq T.\end{cases}$ (3.3)

It is easy to verify that under (H) the data of (3.2) and (3.2) satisfy the
conditions of Pardoux and Peng’s result [23], pp. 59-60 and then in virtue of
Theorem 4.1 of this same reference, we claim the existence and uniqueness of
solutions of both (3.2) and (3.3). To solve the system (3.1), we shall use an
iterative method and regard (3.1) as a limit system. To this end, for any
$i\in\mathcal{I}$, we set $Y^{i,0}_{\cdot}:=\underline{Y}_{\cdot}$, and for
$n\geq 1$, we seek a triplet
$(Y^{i,n}_{\cdot},Z^{i,n}_{\cdot},K^{i,n}_{\cdot})$ such that, for $n\geq 1$

$\begin{cases}Y^{i,n}_{\cdot},K^{i,n}_{\cdot}\in\mathscr{S}^{2}\quad\text{and}\quad
Z^{i,n}_{\cdot}\in\mathcal{H}^{2,d};\hskip 2.84544ptK^{i,n}_{\cdot}\hskip
5.69046pt\text{is non-decreasing with}\quad K^{i}_{0}=0,&\\\
Y^{i,n}_{s}=h_{i}(X_{T})+\displaystyle\int^{T}_{s}f_{i}(r,X_{r},Y^{1,n-1}_{r},\ldots,Y^{i-1,n-1}_{r},Y^{i,n}_{r},Y^{i+1,n-1}_{r},\ldots
Y^{q,n-1}_{r},Z^{i,n}_{r})dr&\\\ \hskip
31.2982pt+K^{i,n}_{T}-K^{i,n}_{s}-\int^{T}_{s}Z^{i,n}_{r}dB_{r},\hskip
5.69046pt\text{for all }0\leq s\leq T;&\\\
Y^{i,n}_{s}\geq\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{k,n-1}_{s}-\gamma_{ik}(s,X_{s})\right\\},\hskip
7.11317pt\text{for all }0\leq s\leq T,&\\\ \text{and if}\hskip
5.69046ptK^{i,n}_{\cdot}=K^{i,n,c}_{\cdot}+K^{i,n,d}_{\cdot},\hskip
2.84544pt\text{where}\hskip 5.69046ptK^{i,n,c}_{\cdot}\hskip
2.84544pt(\text{resp.}\hskip 5.69046ptK^{i,n,d}_{\cdot})\hskip
5.69046pt\text{is the continuous}&\\\ \text{(resp. purely discontinuous) part
of}\hskip 5.69046ptK^{i,n}_{\cdot},\hskip 5.69046pt\text{then:}&\\\
\displaystyle\int^{T}_{0}\left(Y^{i,n}_{r}-\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{k,n-1}_{r}-\gamma_{ik}(r,X_{r})\right\\}\right)dK^{i,n,c}_{r}=0\,;&\\\
\Delta_{s}Y_{\cdot}:=Y^{i,n}_{s}-Y^{i,n}_{s^{-}}=-\Big{(}\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{k,n-1}_{s}-\gamma_{ik}(s,X_{s})\right\\}-Y^{i,n}_{s}\Big{)}^{+},\hskip
7.11317pt\text{for all}0\leq s\leq T.\end{cases}$

Note that for each $k\in\mathcal{I}$ the process $Y^{k,0}_{\cdot}$ is given.
Then, by letting

$\tilde{f}_{i}(s,Y^{i,1}_{s},Z^{i,1}_{s}):=f_{i}(s,X_{s},Y^{1,0}_{s},\ldots,Y^{i-1,0}_{s},Y^{i,1}_{s},Y^{i+1,0}_{s},\ldots
Y^{q,0}_{s},Z^{i,1}_{s})$

for $i\in\mathcal{I}$, the data of the RBSDE associated with
$(Y^{i,1}_{\cdot},Z^{i,1}_{\cdot},K^{i,1}_{\cdot})$ satisfy the assumptions in
Hamadène [15], Theorem 1.4, and hence the processes
$(Y^{i,1}_{\cdot},Z^{i,1}_{\cdot},K^{i,1}_{\cdot})$ do exist. Next, using the
comparison theorem of solutions of BSDEs (see e.g. Theorem 2.2 in El Karoui et
al. [14]) we deduce that for any $i\in\mathcal{I}$, $Y^{i,0}_{\cdot}\leq
Y^{i,1}_{\cdot}$. Besides, as $f_{i}$ satisfies the monotonicity property
(H2)-(H2)(iv) and using again the comparison of solutions of RBSDEs (see
Theorem 1.5 in Hamadène [15]) we obtain by induction that:

$\text{for all}\hskip 5.69046ptn\geq 0\hskip 5.69046pt\text{and}\hskip
5.69046pti\in\mathcal{I},\hskip 11.38092ptY^{i,n}_{\cdot}\leq
Y^{i,n+1}_{\cdot}.$ (3.4)

On the other hand, the process $(\overline{Y}_{\cdot},\overline{Z}_{\cdot})$
in (3.2), can be regarded as the triplet
$((\overline{Y}^{i}_{\cdot},\overline{Z}^{i}_{\cdot},0))_{i\in\mathcal{I}}$
(i.e. $K^{i}_{\cdot}=0$), which is solution for the system of RBSDEs with data

$([\max\limits_{i=1,\ldots,q}f_{i}](t,X_{t},y^{1},\ldots,y^{q},z),\gamma_{ik}(t,X_{t}),\max\limits_{i=1,\ldots,q}h_{i}(X_{T})),\quad
i,k\in\mathcal{I}.$

Note that

$\begin{array}[]{ll}f_{i}(t,\overline{Y}_{t},\overline{Z}_{t})&:=f_{i}(t,X_{t},Y^{1,0}_{t},\ldots,\overline{Y}^{i}_{t},\ldots,Y^{q,0}_{t},\overline{Z}_{t})\\\
&\leq\max_{i=1,\ldots,q}f_{i}(t,X_{t},\overline{Y}^{1}_{t},\ldots,\overline{Y}^{i}_{t},\ldots,\overline{Y}^{q}_{t},\overline{Z}_{t})\\\
&:=\max_{i=1,\ldots,q}f_{i}(t,\overline{Y}_{t},\overline{Z}_{t}),\end{array}$

since $f_{i}$ satisfies the monotonicity property (H2)-(H2)(iv) and due that,
for each $k\in\mathcal{I}^{-i}$ (the fixed processes),
$Y^{k,0}_{\cdot}=\underline{Y}_{\cdot}\leq\overline{Y}_{\cdot}$. Therefore, by
comparison Theorem 1.5 in Hamadène [15], we get that
$Y^{i,1}_{\cdot}\leq\overline{Y}_{\cdot}$. In general, through an induction
procedure, we can obtain for all $n\geq 0$ and $i\in\mathcal{I}$,
$Y^{i,n}_{\cdot}\leq\overline{Y}_{\cdot}$ and hence

$\underline{Y}_{\cdot}=Y^{i,0}_{\cdot}\leq Y^{i,n}_{\cdot}\leq
Y^{i,n+1}_{\cdot}\leq\overline{Y}_{\cdot}.$ (3.5)

Arguing as in Theorem 2.2, we can see that there exists $Y^{i}_{\cdot}$ such
that $Y^{i,n}_{\cdot}\nearrow Y^{i}_{\cdot}$ and $\mathbb{E}\big{[}\sup_{0\leq
t\leq T}\left|Y^{i}_{t}\right|^{2}\big{]}<\infty$. Therefore, using Peng’s
monotonic limit theorem (see Theorem 2.1 and Theorem 3.6 in Peng [24]), we
deduce that for any $i\in\mathcal{I}$, the limit process $Y^{i}_{\cdot}$ is a
càdlàg process and there exists
$(Z^{i}_{\cdot},K^{i}_{\cdot})\in\mathcal{H}^{2,d}\times\mathscr{S}^{2}$ with
$K^{i}$ non-decreasing process and $K^{i}_{0}=0$ such that: $\forall s\leq T$,

$\begin{cases}Y^{i}_{s}=h_{i}(X_{T})+\displaystyle\int^{T}_{s}f_{i}(r,X_{r},Y^{1}_{r},\ldots,Y^{i}_{r},\ldots,Y^{q}_{r},Z^{i}_{r})dr+K^{i}_{T}-K^{i}_{s}-\int^{T}_{s}Z^{i}_{r}dB_{r}\,;&\\\
Y^{i}_{s}\geq\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{k}_{s}-\gamma_{ik}(s,X_{s})\right\\}.\end{cases}$

Now we claim that $\left(Y^{i},Z^{i},K^{i}\right)_{i=1,\ldots,q}$ is, in fact,
the desired solution of (3.1). Indeed, consider the RBSDEs at the $i$-th
variable and the other variables $Y^{1},\dots,Y^{i-1},Y^{i+1},\dots,Y^{q}$
fixed, that is to say

$\begin{cases}\forall i\in\mathcal{I},\hskip 5.69046pt\text{find}\hskip
5.69046pt\left(\tilde{Y}^{i}_{\cdot},\tilde{Z}^{i}_{\cdot},\tilde{K}^{i}_{\cdot}\right)\hskip
5.69046pt\text{such that}:\\\
\tilde{Y}^{i}_{\cdot},\tilde{K}^{i}_{\cdot}\in\mathscr{S}^{2}\quad\text{and}\quad\tilde{Z}^{i}_{\cdot}\in\mathcal{H}^{2,d};\tilde{K}^{i}_{\cdot}\hskip
5.69046pt\text{is non-decreasing and}\hskip 5.69046pt\tilde{K}^{i}_{0}=0&\\\
\tilde{Y}^{i}_{s}=h_{i}(X_{T})+\displaystyle\int^{T}_{s}f_{i}(r,X_{r},Y^{1}_{r},\ldots,Y^{i-1}_{r},\tilde{Y}^{i}_{r},Y^{i+1}_{r},\ldots
Y^{q}_{r},Z^{i}_{r})dr+\tilde{K}^{i}_{T}-\tilde{K}^{i}_{s}-\int^{T}_{s}\tilde{Z}^{i}_{r}dB_{r},\,\,\text{for
all }0\leq s\leq T;&\\\
\tilde{Y}^{i}_{s}\geq\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{k}_{s}-\gamma_{ik}(s,X_{s})\right\\}\hskip
14.22636pt\text{for all }0\leq s\leq T\\\ \text{and if}\hskip
5.69046pt\tilde{K}^{i}_{\cdot}=\tilde{K}^{i,c}_{\cdot}+K^{i,d}_{\cdot},\hskip
2.84544pt\text{where}\hskip 5.69046pt\tilde{K}^{i,c}_{\cdot}\hskip
2.84544pt(\text{resp.}\tilde{K}^{i,d}_{\cdot})\hskip 5.69046pt\text{is the
continuous}&\\\ \text{(resp. purely discontinuous) part of}\hskip
5.69046pt\tilde{K}^{i},\hskip 5.69046pt\text{then:}\\\
\displaystyle\int^{T}_{0}\Big{(}\tilde{Y}^{i}_{r}-\max\limits_{k\in\mathcal{I}^{-i}}\big{\\{}Y^{k}_{r}-\gamma_{ik}(r,X_{r})\big{\\}}\Big{)}d\tilde{K}^{i,c}_{r}=0.\\\
\Delta_{s}\tilde{Y}_{\cdot}:=\tilde{Y}^{i}_{s}-\tilde{Y}^{i}_{s^{-}}=-\Big{(}\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{k}_{s}-\gamma_{ik}(s,X_{s})\right\\}-\tilde{Y}^{i}_{s}\Big{)}^{+}\hskip
14.22636pt\text{for all }0\leq s\leq T.\end{cases}$ (3.6)

The solution of $(\ref{rbsdefija})$ do exist by using again Theorem 1.4 in
Hamadène [15]. Such a solution $\tilde{Y}^{i}_{\cdot}$ becomes the smallest
$f_{i}$-supermartingale that dominates
$\max_{k\in\mathcal{I}^{-i}}\left\\{Y^{k}_{s}-\gamma_{ik}(s,X_{s})\right\\}$
(for more details on this last assertion, see Peng and Xu [25]). Whence
$\tilde{Y}^{i}_{t}\leq Y^{i}_{t}$. On the other hand, since $Y^{i,n-1}_{t}\leq
Y^{i}_{t}$ for any $i\in\mathcal{I}$ and $n\geq 1$, we get

$\max\limits_{k\in\mathcal{I}^{-i}}\big{\\{}Y^{k,n-1}_{s}-\gamma_{ik}(s,X_{s})\big{\\}}\leq\max\limits_{k\in\mathcal{I}^{-i}}\big{\\{}Y^{k}_{s}-\gamma_{ik}(s,X_{s})\big{\\}}.$

Also observe that assumptions ((H2))-((H2)(iv)) yields that

$f_{i}(t,x,Y^{1,n-1}_{t},\ldots,\tilde{Y}^{i}_{t},\ldots,Y^{q,n-1}_{t},Z^{i}_{t})\leq
f_{i}(t,x,Y^{1}_{t},\ldots,\tilde{Y}^{i}_{t},\ldots,Y^{q}_{t},Z^{i}_{t}).$

Then using again the comparison theorem for RBSDEs given in Theorem 1.5 in
Hamadène [15], we have $Y^{i,n}_{t}\leq\tilde{Y}^{i}_{t}$. This implies that
$Y^{i}_{t}\leq\tilde{Y}^{i}_{t}$, and hence $\tilde{Y}^{i}_{t}=Y^{i}_{t}$.
Moreover, this also implies that $\tilde{Z}^{i}_{t}=Z^{i}_{t}$ and
$\tilde{K}^{i}_{t}=K^{i}_{t}$ for any $0\leq t\leq T$, $\mathbb{P}$-a.s. This
proves the existence of solution for (3.1). $\blacksquare$

We now provide a representation result for the solutions of system (3.1) and,
as a by-product, we obtain the uniqueness. For later use, let us fix
$\textbf{u}_{.}:=\left(u^{1}_{.},\ldots,u^{q}_{.}\right)$ in
$\mathcal{H}^{2,q}$ and let us consider the following system of RBSDEs:

$\begin{cases}\forall i\in\mathcal{I},\hskip 5.69046pt\text{find}\hskip
5.69046pt\left(Y^{\textbf{u},i}_{\cdot},Z^{\textbf{u},i}_{\cdot},K^{\textbf{u},i}_{\cdot}\right)\in\mathscr{S}^{2}\times\mathscr{S}^{2}\times\mathcal{H}^{2,d}\hskip
5.69046pt\text{such that}:&\\\
Y^{\textbf{u},i}_{s}=h_{i}(X_{T})+\displaystyle\int^{T}_{s}f_{i}(r,X_{r},\textbf{u}_{r},Z^{\textbf{u},i}_{r})dr+K^{\textbf{u},i}_{T}-K^{\textbf{u},i}_{s}-\int^{T}_{s}Z^{\textbf{u},i}_{r}dB_{r}\hskip
14.22636pt\text{for all }0\leq s\leq T;&\\\
Y^{\textbf{u},i}_{s}\geq\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{\textbf{u},k}_{s}-\gamma_{ik}(s,X_{s})\right\\}\hskip
14.22636pt\text{for all }0\leq s\leq T.&\\\ \text{and if}\hskip
5.69046ptK^{\textbf{u},i}_{\cdot}=K^{\textbf{u},i,c}_{\cdot}+K^{\textbf{u},i,d}_{\cdot},\hskip
2.84544pt\text{where}\hskip 5.69046ptK^{i,\textbf{u},c}_{\cdot}\hskip
2.84544pt(\text{resp.}K^{i,\textbf{u},d}_{\cdot})\hskip 5.69046pt\text{is the
continuous}&\\\ \text{(resp. purely discontinuous) part of}\hskip
5.69046ptK^{\textbf{u},i}_{\cdot},\hskip 5.69046pt\text{then:}&\\\
\displaystyle\int^{T}_{0}\Big{(}Y^{\textbf{u},i}_{r}-\max\limits_{k\in\mathcal{I}^{-i}}\big{\\{}Y^{\textbf{u},k}_{r}-\gamma_{ik}(r,X_{r})\big{\\}}\Big{)}dK^{i,\textbf{u},c}_{r}=0.&\\\
\Delta_{s}\tilde{Y}^{\textbf{u}}_{\cdot}:=\tilde{Y}^{\textbf{u},i}_{s}-\tilde{Y}^{\textbf{u},i}_{s^{-}}=-\Big{(}\max\limits_{k\in\mathcal{I}^{-i}}\left\\{Y^{\textbf{u},k}_{s}-\gamma_{ik}(s,X_{s})\right\\}-\tilde{Y}^{\textbf{u},i}_{s}\Big{)}^{+}\hskip
14.22636pt\text{for all }0\leq s\leq T.\end{cases}$ (3.7)

Observe that $f_{i}$ does not depend on $Y^{1},\ldots,Y^{q}$. Let $s\leq T$ be
fixed, $i\in\mathcal{I}$ and let $\mathcal{D}^{i}_{s}$ be the following set of
strategies as in (2.1), such that:

$\mathcal{D}^{i}_{s}:=\left\\{\alpha=\left(\theta_{n},\kappa_{n}\right)_{n\geq
0}:\theta_{0}=s,\kappa_{0}=i\hskip 5.69046pt\text{and}\hskip
5.69046pt\mathbb{E}[(\mathtt{C}^{\alpha}_{T})^{2}]<\infty\right\\}$

where $\mathtt{C}^{\alpha}_{r}$, $r\leq T$, is the following cumulative costs
up to time $r$, i.e.,

$\mathtt{C}^{\alpha}_{r}:=\sum^{\infty}_{n=1}\gamma_{\kappa_{n-1},\kappa_{n}}(\theta_{n},X_{\theta_{n}})\mathbf{1}_{[\theta_{n}\leq
r]}\hskip 11.38092pt\text{for}\hskip 5.69046ptr<T\hskip
5.69046pt\text{and}\hskip 5.69046pt\mathtt{C}^{\alpha}_{T}=\lim_{r\rightarrow
T}\mathtt{C}^{\alpha}_{r},\hskip 5.69046pt\mathbb{P}\text{-a.s.}$

Therefore and for any admissible strategy $\alpha\in\mathcal{D}^{i}_{s}$ we
have:

$\mathtt{C}^{\alpha}_{T}=\sum^{\infty}_{n=1}\gamma_{\kappa_{n-1},\kappa_{n}}(\theta_{n},X_{\theta_{n}})\mathbf{1}_{[\theta_{n}<T]}.$

Consider a strategy $\alpha=(\theta_{n},\kappa_{n})_{n\geq
0}\in\mathcal{D}^{i}_{s}$ and let
$(P^{\alpha}_{\cdot},Q^{\alpha}_{\cdot}):=(P^{\alpha}_{s},Q^{\alpha}_{s})_{s\leq
T}$ be the solution of the following BSDE

$\begin{cases}P^{\alpha}_{\cdot}\hskip 2.84544pt\text{is c\\`{a}dl\\`{a}g
and}\hskip 2.84544pt\mathbb{E}\big{[}\sup_{s\leq
T}\left|P^{\alpha}_{s}\right|^{2}\big{]}<\infty,\hskip
5.69046ptQ^{\alpha}_{\cdot}\in\mathcal{H}^{2,d};&\\\
P^{\alpha}_{s}=h_{\alpha}(X_{T})+\displaystyle\int^{T}_{s}f_{\alpha}(r,X_{r},\textbf{u}_{r},Q^{\alpha}_{r})dr-(\mathtt{C}^{\alpha}_{T}-\mathtt{C}^{\alpha}_{s})-\int^{T}_{s}Q^{\alpha}_{r}dB_{r},\,s\leq
T,\end{cases}$ (3.8)

with

$\begin{array}[]{ll}h_{\alpha}(x)=h_{\kappa_{n}}(x)\mathbf{1}_{[\theta_{n}<T\leq\theta_{n+1}]}\hskip
2.84544pt\text{and}&\\\ &\\\
f_{\alpha}(r,x,v_{1},\ldots,v_{q},z):=\displaystyle\sum\limits^{\infty}_{n=0}f_{\kappa_{n}}(r,x,v_{1},\ldots,v_{q},z)\mathbf{1}_{[\theta_{n}\leq
r<\theta_{n+1})}.\end{array}$ (3.9)

Making the change of variable
$\bar{P}^{\alpha}_{\cdot}:=P^{\alpha}_{\cdot}-\mathtt{C}^{\alpha}_{\cdot}$,
the equation in (3.8) is transformed in a standard BSDE. Since
$\mathtt{C}^{\alpha}$ is adapted and
$\mathbb{E}[(\mathtt{C}^{\alpha}_{T})^{2}]<\infty$, we easily deduce the
existence and uniqueness of the process
$(P^{\alpha}_{\cdot},Q^{\alpha}_{\cdot})$. We then have the following
representation for the solution of (3.7).

######  Proposition 3.2.

Assume that for any $i,k\in\mathcal{I}$:

  1. (i)

$f_{i}$ satisfies (H2)-(H2)(ii),(H2)(iii);

  2. (ii)

$\gamma_{ik}$ (resp. $h_{i}$) satisfies (H3) (resp. (H4)).

Then the solution of system of RBSDEs (3.7) exists, it is unique and
satisfies:

$Y^{\textbf{u},i}_{s}=\operatorname*{ess\,sup}_{\alpha\in\mathcal{D}^{i}_{s}}\left\\{P^{\alpha}_{s}-\mathtt{C}^{\alpha}_{s}\right\\}\quad\forall
s\leq T,\quad\forall i\in\mathcal{I}.$ (3.10)

_Proof._ Since $f_{i}$ does not depend on variables
$Y^{1}_{\cdot},\ldots,Y^{q}_{\cdot}$, then, it trivially satisfies
(H2)-(H2)(iv). Then, by hypothesis (i) and (ii), and Proposition 3.1, the
solution
($Y^{\textbf{u},i}_{\cdot},Z^{\textbf{u},i}_{\cdot},K^{\textbf{u},i}_{\cdot})$
of the system (3.7) exists. Therefore, plugging an arbitrary strategy
$\alpha\in\mathcal{D}^{i}_{s}$ in (3.7), we obtain:

$Y^{\textbf{u},i}_{s}\geq
h_{\alpha}(X_{T})+\displaystyle\int^{T}_{s}f_{\alpha}(r,X_{r},\textbf{u}_{r},Z^{\alpha}_{r})dr+\tilde{K}^{\alpha}_{T}-\mathtt{C}^{\alpha}_{T}-\int^{T}_{s}Z^{\alpha}_{r}dB_{r}.$
(3.11)

with $h_{\alpha}$ and $f_{\alpha}$ as in (3.9), and,

$\tilde{K}^{\alpha}_{T}=(K^{\textbf{u},i}_{\theta_{1}}-K^{\textbf{u},i}_{s})+\sum\limits^{\infty}_{n=1}(K^{\textbf{u},\kappa_{n}}_{\theta_{n+1}}-K^{\textbf{u},\kappa_{n}}_{\theta_{n}})\text{
and
}Z^{\alpha}_{r}=\sum\limits^{\infty}_{n=0}Z^{\textbf{u},\kappa_{n}}_{r}\mathbf{1}_{[\theta_{n}\leq
r<\theta_{n+1})},\forall r\leq T.$ (3.12)

Adding $\mathtt{C}^{\alpha}_{s}$ from both sides of (3.11) and taking into
account that $\tilde{K}^{\alpha}_{T}\geq 0$, we have

$\begin{array}[]{ll}Y^{\textbf{u},i}_{s}+\mathtt{C}^{\alpha}_{s}\geq
h_{\alpha}(X_{T})+\displaystyle\int^{T}_{s}f_{\alpha}(r,X_{r},\textbf{u}_{r},Z^{\alpha}_{r})dr-(\mathtt{C}^{\alpha}_{T}-\mathtt{C}^{\alpha}_{s})-\int^{T}_{s}Z^{\alpha}_{r}dB_{r}&\\\
\hskip 48.36958pt=P^{\alpha}_{s}.\end{array}$

Therefore, we have

$Y^{\textbf{u},i}_{s}\geq\operatorname*{ess\,sup}\limits_{\alpha\in\mathcal{D}^{i}_{s}}\left\\{P^{\alpha}_{s}-\mathtt{C}^{\alpha}_{s}\right\\},\hskip
8.5359pt\forall\alpha\in\mathcal{D}^{i}_{s}.$ (3.13)

Next let
$\alpha^{\varepsilon}=(\theta^{\varepsilon}_{n},\kappa^{\varepsilon}_{n})_{n\geq
0}$ be the strategy defined recursively as follows (compare to the
$\varepsilon$-strategy $\mathcal{S}^{\varepsilon}$ in Proposition 2.3):
$\theta^{\varepsilon}_{0}=0,\kappa^{\varepsilon}_{0}=i$ and for $n\geq 0$,

$\theta^{\varepsilon}_{n+1}=\inf\left\\{s\geq\theta^{\varepsilon}_{n}:Y^{\textbf{u},\kappa^{\varepsilon}_{n}}_{s}\leq\max\limits_{k\in\mathcal{I}^{-\kappa^{\varepsilon}_{n}}}\left(Y^{\textbf{u},k}_{s}-\gamma_{\kappa^{\varepsilon}_{n},k}(s,X_{s})\right)+\frac{\varepsilon}{2^{n+1}}\right\\}\wedge
T$

and

$\kappa^{\varepsilon}_{n+1}=\arg\max_{k\in\mathcal{I}^{-\kappa^{\varepsilon}_{n}}}\left\\{Y^{\textbf{u},k}_{\theta^{\varepsilon}_{n+1}}-\gamma_{\kappa^{\varepsilon}_{n},k}(\theta^{\varepsilon}_{n+1},X_{\theta^{\varepsilon}_{n+1}})\right\\}.$

In a similar manner as in the proof of Proposition 2.3, we can ensure that
$\alpha^{\varepsilon}\in\mathcal{D}^{i}_{s}$ satisfies that
$P[\theta^{\varepsilon}_{n}<T,\forall n\geq 0]=0$. Let us prove now that
$\mathbb{E}\big{[}(\mathtt{C}^{\varepsilon}_{T})^{2}\big{]}<\infty$ and that
$\alpha^{\varepsilon}$ is $\varepsilon$-optimal in $\mathcal{D}^{i}_{s}$ for
the problem (3.10). Following the strategy $\alpha^{\varepsilon}$ and since
$(Y^{\textbf{u},i})_{i\in\mathcal{I}}$ solves the RBSDE (3.7), it turns out
that,

$Y^{\textbf{u},i}_{s}\leq
Y^{\textbf{u},\kappa^{\varepsilon}_{n}}_{\theta^{\varepsilon}_{n}}+\displaystyle\int^{\theta^{\varepsilon}_{n}}_{s}f_{\alpha^{\varepsilon}}(r,X_{r},\textbf{u}_{r},Z^{\alpha^{\varepsilon}}_{r})dr-\mathtt{C}^{\alpha^{\varepsilon}}_{\theta^{\varepsilon}_{n}}-\int^{\theta^{\varepsilon}_{n}}_{s}Z^{\alpha^{\varepsilon}}_{r}dB_{r}+\varepsilon\sum^{n}_{i=1}\frac{1}{2^{i}},\hskip
5.69046pt\forall n\geq 1$ (3.14)

since
$K^{\textbf{u},\kappa^{\varepsilon}_{n}}_{r}-K^{\textbf{u},\kappa^{\varepsilon}_{n}}_{\theta^{\varepsilon}_{n}}=0$
for $\theta^{\varepsilon}_{n}\leq r<\theta^{\varepsilon}_{n+1}$. Taking now
the limit with respect to $n$ in (3.14) we get:

$\begin{array}[]{ll}Y^{\textbf{u},i}_{s}&\leq
h_{\alpha^{\varepsilon}}(X_{T})+\displaystyle\int^{T}_{s}f_{\alpha^{\varepsilon}}(r,X_{r},\textbf{u},Z^{\alpha^{\varepsilon}}_{r})dr-\mathtt{C}^{\alpha^{\varepsilon}}_{T}-\int^{T}_{s}Z^{\alpha^{\varepsilon}}_{r}dB_{r}+\varepsilon\\\
\\\
&=P^{\alpha^{\varepsilon}}_{s}-\mathtt{C}^{\alpha^{\varepsilon}}_{s}+\varepsilon.\end{array}$
(3.15)

Taking supremum over all $\alpha\in\mathcal{D}^{i}_{s}$, and next letting
$\varepsilon\rightarrow 0$ and using the assumptions ((H4)) and
((H2))-((H2)(ii)),((H2)(iii)) satisfied for $h_{i}$ and $f_{i}$ respectively
and since $\textbf{u}\in\mathcal{H}^{2,q}$,
$Z^{\alpha^{\varepsilon}}\in\mathcal{H}^{2,d}$ and
$(Y^{1}_{\cdot},\ldots,Y^{q}_{\cdot})\in(\mathscr{S}^{2})^{q}$, we deduce from
(3.15) that $\mathbb{E}[(\mathtt{C}^{\alpha^{\varepsilon}}_{T})^{2}]<\infty$.
It follows that
$Y^{\textbf{u},i}_{s}\leq\operatorname*{ess\,sup}_{\alpha\in\mathcal{D}^{i}_{s}}\left\\{P^{\alpha}_{s}-\mathtt{C}^{\alpha}_{s}\right\\}$.
This last fact together with (3.13) yield (3.10). As a by-product, we obtain
that the solution of (3.7) is unique. $\hskip 28.45274pt\blacksquare$

Next for $\textbf{u}:=(u^{1},\ldots,u^{q})\in\mathcal{H}^{2,q}$ let us define

$\Phi(\textbf{u}):=(Y^{\textbf{u},1}_{\cdot},\ldots,Y^{\textbf{u},q}_{\cdot}),$

where
$(Y^{\textbf{u},i}_{\cdot},Z^{\textbf{u},i}_{\cdot},K^{\textbf{u},i}_{\cdot})_{i=1,\ldots,q}$
is the solution of system (3.7) which exists and is unique under the
assumptions of Proposition 3.2. Note that the processes
$(Y^{\textbf{u},i}_{\cdot},\ldots,Y^{\textbf{u},q}_{\cdot})$ belong to
$(\mathscr{S}^{2})^{q}\subseteq\mathcal{H}^{2,q}$. Hence, $\Phi$ is a mapping
from $\mathcal{H}^{2,q}$ to $\mathcal{H}^{2,q}$.

We introduce the norm defined on $\mathcal{H}^{2,q}$ by

$\left\|(u^{1},\ldots,u^{q})\right\|^{2}_{\beta}:=\mathbb{E}\Bigg{[}\displaystyle\int^{T}_{0}e^{\beta
s}\left(\sum^{q}_{i=1}\left|u^{i}_{s}\right|^{2}\right)ds\Bigg{]}.$

Note that
$\left\|w\right\|_{\mathcal{H}^{2,q}}\leq\left\|w\right\|_{\beta}\leq e^{\beta
T}\left\|w\right\|_{\mathcal{H}^{2,q}}$, for all $w\in\mathcal{H}^{2,q}$,
implies that these norms are equivalent. For sake of completeness, we present
the following result, established in Chassagneux et al. [6], which ensures
that $\Phi$ is a contraction on the Banach space
$(\mathcal{H}^{2,q},\left\|\cdot\right\|_{\beta})$.

######  Proposition 3.3.

Assume that for any $i,j\in\mathcal{I}$ the following hypotheses are in force:

  1. (i)

$f_{i}$ verifies (H2)-(H2)(ii),(H2)(iii);

  2. (ii)

$\gamma_{ij}$ (resp. $h_{i}$) verifies (H3) (resp. (H4)).

Then, there exists $\beta_{0}\in\mathbb{R}$ such that the mapping $\Phi$ is a
contraction operator on
$(\mathcal{H}^{2,q},\left\|\cdot\right\|_{\beta_{0}})$. Therefore $\Phi$ has a
fixed point $(Y^{1}_{\cdot},\ldots,Y^{q}_{\cdot})$ which belongs to
$(\mathscr{S}^{2})^{q}$ and which provides a unique solution for system (3.1).

Proof. Let $\textbf{u},\textbf{v}\in\mathcal{H}^{2,q}$ and consider the
respective images under $\Phi$, $Y^{\textbf{u},i}_{\cdot}:=\Phi(\textbf{u})$
and $Y^{\textbf{v},i}_{\cdot}:=\Phi(\textbf{v})$. Besides, let us introduce
the following “auxiliary dominating” RBSDE, for $i\in\mathcal{I}$:

$\begin{cases}\check{Y}^{i}_{s}=h_{i}(X_{T})+\int^{T}_{s}\check{f}_{i}(r,X_{r},\check{Z}^{i}_{r})dr+\check{K}^{i}_{T}-\check{K}^{i}_{s}-\int^{T}_{s}\check{Z}^{i}_{r}dB_{r}\hskip
14.22636pt\text{for all }0\leq s\leq T,&\\\
\check{Y}^{i}_{s}\geq\max\limits_{k\in\mathcal{I}^{-i}}\left\\{\check{Y}^{k}_{s}-\gamma_{ik}(s,X_{s})\right\\}\hskip
14.22636pt\text{for all }0\leq s\leq T,&\\\
\int^{T}_{0}\Big{(}\check{Y}^{i}_{s}-\max\limits_{k\in\mathcal{I}^{-i}}\left\\{\check{Y}^{k}_{s}-\gamma_{ik}(s,X_{s})\right\\}\Big{)}d\check{K}^{i,c}_{s}=0.&\\\
\Delta_{s}\check{Y}^{i}_{\cdot}:=\check{Y}^{i}_{s}-\check{Y}^{i}_{s^{-}}=-\Big{(}\max\limits_{k\in\mathcal{I}^{-i}}\left\\{\check{Y}^{k}_{s}-\gamma_{ik}(s,X_{s})\right\\}-\check{Y}^{i}_{s}\Big{)}^{+}\hskip
14.22636pt\text{for all }0\leq s\leq T\end{cases}$ (3.16)

where
$\check{f}(s,X_{s},z^{i})=\max\\{f(s,X_{s},\textbf{u}_{r},z^{i}),f(s,X_{s},\textbf{v}_{r},z^{i})\\}$,
and $\check{K}^{i,c}_{\cdot}$ and $\check{K}^{i,d}_{\cdot}$ are the continuous
and discontinuous parts of $\check{K}^{i}$. Note that by Proposition 3.2 a
unique solution
$(\check{Y}^{i}_{\cdot},\check{Z}^{i}_{\cdot},\check{K}^{i}_{\cdot})$ exists
for (3.16).
For $s\in[0,T]$ fixed, and for any $\alpha\in\mathcal{D}^{i}_{s}$, denote by
$(U^{\alpha}_{\cdot},Z^{\alpha}_{\cdot})$,$(\bar{U}^{\alpha}_{\cdot},\bar{Z}^{\alpha}_{\cdot})$
and $(\check{U}^{\alpha}_{\cdot},\check{Z}^{\alpha}_{\cdot})$ the respective
solutions of the following one-dimensional BSDEs: $\forall s\leq T$,

$\begin{array}[]{ll}U^{\alpha}_{s}=h_{\alpha}(X_{T})+\int^{T}_{s}f_{\alpha}(r,X_{r},\textbf{u}_{r},Z^{\alpha}_{r})dr-(\mathtt{C}^{\alpha}_{T}-\mathtt{C}^{\alpha}_{s})-\int^{T}_{s}Z^{\alpha}_{r}dB_{r},&\\\
\bar{U}^{\alpha}_{s}=h_{\alpha}(X_{T})+\int^{T}_{s}f_{\alpha}(r,X_{r},\textbf{v}_{r},\bar{Z}^{\alpha}_{r})dr-(\mathtt{C}^{\alpha}_{T}-\mathtt{C}^{\alpha}_{s})-\int^{T}_{s}\bar{Z}^{\alpha}_{r}dB_{r},&\\\
\check{U}^{\alpha}_{s}=h_{\alpha}(X_{T})+\int^{T}_{s}\check{f}_{\alpha}(r,X_{r},\check{Z}^{\alpha}_{r})dr-(\mathtt{C}^{\alpha}_{T}-\mathtt{C}^{\alpha}_{s})-\int^{T}_{s}\check{Z}^{\alpha}_{r}dB_{r}.\end{array}$

We deduce from Proposition 3.2 that

$Y^{\textbf{u},i}_{s}=\operatorname*{ess\,sup}_{\alpha\in\mathcal{D}^{i}_{s}}\left\\{U^{\alpha}_{s}-\mathtt{C}^{\alpha}_{s}\right\\},\hskip
5.69046pt\
Y^{\textbf{v},i}_{s}=\operatorname*{ess\,sup}_{\alpha\in\mathcal{D}^{i}_{s}}\left\\{\bar{U}^{\alpha}_{s}-\mathtt{C}^{\alpha}_{s}\right\\},\hskip
8.5359pt\check{Y}^{i}_{s}=\operatorname*{ess\,sup}_{\alpha\in\mathcal{D}^{i}_{s}}\left\\{\check{U}^{\alpha}_{s}-\mathtt{C}^{\alpha}_{s}\right\\}.$
(3.17)

Besides, note that for an $\varepsilon$-optimal strategy
$\alpha^{\varepsilon}\in\mathcal{D}^{i}_{s}$, we have

$\check{Y}^{i}_{s}\leq\check{U}^{\alpha^{\varepsilon}}_{s}-\mathtt{C}^{\alpha^{\varepsilon}}_{s}+\varepsilon.$
(3.18)

Using a comparison argument, we easily check that
$\check{U}^{\alpha}_{\cdot}\geq
U^{\alpha}_{\cdot}\vee\bar{U}^{\alpha}_{\cdot}$ for any strategy
$\alpha\in\mathcal{D}^{i}_{s}$, and hence, by (3.17) we get that
$\check{Y}^{i}_{s}\geq Y^{\textbf{u},i}_{s}\vee Y^{\textbf{v},i}_{s}$.
Therefore, taking into account the last two inequalities and (3.18), we get
that

$U^{\alpha^{\varepsilon}}_{s}-\mathtt{C}^{\alpha^{\varepsilon}}_{s}\leq
Y^{\textbf{u},i}_{s}\leq\check{Y}^{i}_{s}\leq\check{U}^{\alpha^{\varepsilon}}_{s}-\mathtt{C}^{\alpha^{\varepsilon}}_{s}+\varepsilon\hskip
11.38092pt\text{and}\hskip
11.38092pt\bar{U}^{\alpha^{\varepsilon}}_{s}-\mathtt{C}^{\alpha^{\varepsilon}}_{s}\leq
Y^{\textbf{v},i}_{s}\leq\check{Y}^{i}_{s}\leq\check{U}^{\alpha^{\varepsilon}}_{s}-\mathtt{C}^{\alpha^{\varepsilon}}_{s}+\varepsilon.$

This implies

$\left|Y^{\textbf{u},i}_{s}-Y^{\textbf{v},i}_{s}\right|\leq\left|\check{U}^{\alpha^{\varepsilon}}_{s}-U^{\alpha^{\varepsilon}}_{s}\right|+\left|\check{U}^{\alpha^{\varepsilon}}_{s}-\bar{U}^{\alpha^{\varepsilon}}_{s}\right|+2\varepsilon,$

and by using the inequality $(a+b+c)^{2}\leq 4a^{2}+4b^{2}+2c^{2}$, we have

$\left|Y^{\textbf{u},i}_{s}-Y^{\textbf{v},i}_{s}\right|^{2}\leq
4\left|\check{U}^{\alpha^{\varepsilon}}_{s}-U^{\alpha^{\varepsilon}}_{s}\right|^{2}+4\left|\check{U}^{\alpha^{\varepsilon}}_{s}-\bar{U}^{\alpha^{\varepsilon}}_{s}\right|^{2}+4\varepsilon^{2}.$
(3.19)

Now, applying Ito’s formula to $e^{\beta
s}\left|\check{U}^{\alpha^{\varepsilon}}_{s}-U^{\alpha^{\varepsilon}}_{s}\right|^{2}$,
using the inequality $\left|x\vee y-y\right|\leq\left|x-y\right|$ and the fact
that $f_{\alpha_{\varepsilon}}$ is Lipschitz, taking expectation, to obtain:
$\forall s\leq T$,

$\begin{array}[]{ll}\mathbb{E}\Big{[}e^{\beta
s}\left|\check{U}^{\alpha^{\varepsilon}}_{s}-U^{\alpha^{\varepsilon}}_{s}\right|^{2}+\int^{T}_{s}e^{\beta
r}\left|\check{Z}^{\alpha^{\varepsilon}}_{r}-Z^{\alpha^{\varepsilon}}_{r}\right|^{2}dr\Big{]}\leq-\mathbb{E}\Big{[}\int^{T}_{s}\beta
e^{\beta
r}\left|\check{U}^{\alpha^{\varepsilon}}_{r}-U^{\alpha^{\varepsilon}}_{r}\right|^{2}dr\Big{]}\\\
\\\ \hskip 56.9055pt+2C\mathbb{E}\Big{[}\int^{T}_{s}e^{\beta
r}\left|\check{U}^{\alpha^{\varepsilon}}_{r}-U^{\alpha^{\varepsilon}}_{r}\right|\left(\left|\textbf{v}_{r}-\textbf{u}_{r}\right|+\left|\check{Z}^{\alpha^{\varepsilon}}_{r}-Z^{\alpha^{\varepsilon}}_{r}\right|\right)dr\Big{]}.\end{array}$

The inequalities $2ab\leq\beta a^{2}+\frac{1}{\beta}b^{2}$ and $(a+b)^{2}\leq
2a^{2}+2b^{2}$ also imply

$\begin{array}[]{ll}\mathbb{E}\Big{[}e^{\beta
s}\left|\check{U}^{\alpha^{\varepsilon}}_{s}-U^{\alpha^{\varepsilon}}_{s}\right|^{2}\Big{]}+\mathbb{E}\Big{[}\int^{T}_{s}e^{\beta
r}\left|\check{Z}^{\alpha^{\varepsilon}}_{r}-Z^{\alpha^{\varepsilon}}_{r}\right|^{2}dr\Big{]}\leq-\mathbb{E}\Big{[}\int^{T}_{s}\beta
e^{\beta
r}\left|\check{U}^{\alpha^{\varepsilon}}_{r}-U^{\alpha^{\varepsilon}}_{r}\right|^{2}dr\Big{]}\\\
\\\ \hskip 56.9055pt+\mathbb{E}\Big{[}\int^{T}_{s}\big{\\{}\beta e^{\beta
r}\left|\check{U}^{\alpha^{\varepsilon}}_{r}-U^{\alpha^{\varepsilon}}_{r}\right|^{2}+\frac{2C^{2}}{\beta}e^{\beta
r}\left|\textbf{v}_{r}-\textbf{u}_{r}\right|^{2}+\frac{2C^{2}}{\beta}e^{\beta
r}\left|\check{Z}^{\alpha^{\varepsilon}}_{r}-Z^{\alpha^{\varepsilon}}_{r}\right|^{2}\big{\\}}dr\Big{]}.\end{array}$

Rearranging terms, we obtain:

$\mathbb{E}\Big{[}e^{\beta
s}\left|\check{U}^{\alpha^{\varepsilon}}_{s}-U^{\alpha^{\varepsilon}}_{s}\right|^{2}\Big{]}+\Big{(}1-\frac{2C^{2}}{\beta}\Big{)}\mathbb{E}\bigg{[}\int^{T}_{s}e^{\beta
r}\left|\check{Z}^{\alpha^{\varepsilon}}_{r}-Z^{\alpha^{\varepsilon}}_{r}\right|^{2}dr\bigg{]}\leq\frac{2C^{2}}{\beta}\mathbb{E}\Big{[}\int^{T}_{s}e^{\beta
r}\left|\textbf{v}_{r}-\textbf{u}_{r}\right|^{2}dr\Big{]}.$

Taking $\beta\geq 2C^{2}$, we get

$\mathbb{E}\Big{[}e^{\beta
s}\left|\check{U}^{\alpha^{\varepsilon}}_{s}-U^{\alpha^{\varepsilon}}_{s}\right|^{2}\Big{]}\leq\frac{2C^{2}}{\beta}\mathbb{E}\bigg{[}\int^{T}_{0}e^{\beta
r}\left|\textbf{v}_{r}-\textbf{u}_{r}\right|^{2}dr\bigg{]}.$

Now, an analogous procedure to $e^{\beta
s}\left|\check{U}^{\alpha_{\varepsilon}}_{s}-\bar{U}^{\alpha_{\varepsilon}}_{s}\right|^{2}$
lead to similar result, namely

$\mathbb{E}\Big{[}e^{\beta
s}\left|\check{U}^{\alpha^{\varepsilon}}_{s}-\bar{U}^{\alpha^{\varepsilon}}_{s}\right|^{2}\Big{]}\leq\frac{2C^{2}}{\beta}\mathbb{E}\bigg{[}\int^{T}_{0}e^{\beta
r}\left|\textbf{v}_{r}-\textbf{u}_{r}\right|^{2}dr\bigg{]}.$

Combining these two inequalities with (3.19), we deduce

$\mathbb{E}\Big{[}e^{\beta
s}\left|Y^{\textbf{u},i}_{s}-Y^{\textbf{v},i}_{s}\right|^{2}\Big{]}\leq\frac{16C^{2}}{\beta}\mathbb{E}\bigg{[}\int^{T}_{0}e^{\beta
r}\left|\textbf{v}_{r}-\textbf{u}_{r}\right|^{2}dr\bigg{]}+4\varepsilon^{2}e^{\beta
T}.$

By integrating with respect to $s$ on both sides of the last inequality and
taking into account the fact that such inequality holds true for any
$i=1,\ldots,q$ and for all $s\in[0,T]$, we get

$\left\|\Phi(Y^{\textbf{u}})-\Phi(Y^{\textbf{v}})\right\|_{\beta}\leq
4C\sqrt{Tq\beta^{-1}}\left\|\textbf{u}-\textbf{v}\right\|_{\beta}+2\varepsilon\sqrt{Tqe^{\beta
T}}.$

Finally, choosing $\beta_{0}>\max\left(16C^{2}Tq,2C^{2}\right)$ and taking
$\varepsilon\rightarrow 0$, we see that this mapping is a contraction. This
gives the existence and uniqueness of the system of RBSDE (3.1).
$\blacksquare$

##  4 The Markovian Framework

In this section we will provide more specifications to the process $X_{\cdot}$
treated in previous sections. Namely, we will assume now that this process has
a Markovian evolution described by means of a stochastic differential equation
(diffusion process) as in (4.3) below. Under this framework our previous
analysis can be reduced to study a system of partial differential equations
with obstacles (quasi-variational system). Among the main result of this
section we can highlight the characterization of both the optimal function
(2.3) and the solution of the system of RBSDEs (3.1) as a viscosity solution
in a weak sense (see Theorem 4.9). We will start to introduce the following
functions:

$\begin{cases}b:(t,x)\in\left[0,T\right]\times\mathbb{R}^{r}\mapsto
b(t,x)\in\mathbb{R}^{r};&\\\
\sigma:(t,x)\in\left[0,T\right]\times\mathbb{R}^{r}\mapsto\sigma(t,x)\in\mathbb{R}^{r\times
d},\end{cases}$

satisfying the following hypotheses:

The functions $b$ and $\sigma$ are jointly continuous and Lipschitz continuous
with respect to $x$ uniformly in $t$, that is, there exists a constant $C\geq
0$ such that for any $t\in[0,T]$ and $x$, $x^{\prime}\in\mathbb{R}^{r}$

$\left|b(t,x)-b(t,x^{\prime})\right|+\left|\sigma(t,x)-\sigma(t,x^{\prime})\right|\leq
C\left|x-x^{\prime}\right|.$ (4.1)

Note that continuity and (4.1) imply that $b$ and $\sigma$ are of linear
growth, i.e., there exists a constant $C$ such that:

$\left|b(t,x)\right|+\left|\sigma(t,x)\right|\leq
C(1+\left|x\right|),\,\,\forall(t,x).$ (4.2)

It is well known that under (4.1)-(4.2), there exists a unique Markov process
$(X^{t,x}_{s})_{s\leq T}$, for
$\left(t,x\right)\in\left[0,T\right]\times\mathbb{R}^{r}$, that is a (strong)
solution of the following standard stochastic differential equation:

$\begin{cases}dX^{t,x}_{s}=b(s,X^{t,x}_{s})ds+\sigma(s,X^{t,x}_{s})dB_{s}\hskip
14.22636pt\text{for all }t\leq s\leq T;&\\\ X^{t,x}_{s}=x\hskip
14.22636pt\text{for all }0\leq s\leq t,\end{cases}$ (4.3)

satisfying the following estimates: For any $p\geq 2$,
$x,x^{\prime}\in\mathbb{R}^{r}$ and $s\geq t$

$\displaystyle\mathbb{E}\big{[}\sup_{s\leq
T}\left|X^{t,x}_{s}\right|^{p}\big{]}\leq
C\left(1+\left|x\right|^{p}\right),\quad\mathbb{E}[\sup_{r\in[t,s]}|X^{t,x}_{r}-x|^{p}]\leq
M_{p}(s-t)(1+|x|^{p})]\mbox{ and }$ (4.4)
$\displaystyle\quad\qquad\mathbb{E}[\sup_{r\in[t,s]}|X^{t,x}_{r}-X^{t,x^{\prime}}_{r}-(x-x^{\prime})|^{p}]\leq
M_{p}(s-t)|x-x^{\prime}|^{p}$

for some constant $M_{p}$ (one can see Karatzas and Shreve [22] or Revuz and
Yor [26], for more details).

Recall that the associated infinitesimal generator to $(X^{t,x}_{s})_{s\leq
T}$ is given by :

$\mathcal{L}\phi(t,x)=\frac{1}{2}Tr\left[\left(\sigma.\sigma^{\mathtt{T}}\right)(t,x)D^{2}_{xx}\phi(t,x)\right]+b(t,x)^{\mathtt{T}}D_{x}\phi(t,x)$

for $\phi$ in $C^{1,2}([0,T]\times\mathbb{R}^{r};\mathbb{R})$ ($Tr(.)$ is the
trace of a square matrix and, $\mathtt{A}^{\mathtt{T}}$ is the transpose of a
matrix $\mathtt{A}$).

Now let $(t,x)\in\left[0,T\right]\times\mathbb{R}^{r}$ be fixed and let
$((Y^{t,x,i}_{s},Z^{t,x,i}_{s},K^{t,x,i}_{s})_{t\leq s\leq T})_{i=1,\ldots,q}$
be the unique solution of system (3.1) when the process $X$ is taken to be
equal to $X^{t,x}$ of (4.3), i.e., the solution associated with
$(f_{i}(s,X^{t,x}_{s},y^{1},\ldots,y^{i},\ldots
y^{q},z^{i}),h_{i}(X^{t,x}_{T}),g_{ik}(s,X^{t,x}_{s}))$ ($g_{ik}$ are the
switching costs and they satisfy the same assumptions as $\gamma_{ik}$ in
Assumption (H)) with $y^{i}\in\mathbb{R}$ and $z^{i}\in\mathbb{R}^{d}$.

Assume now that Assumptions (H) are satisfied. Since we are in the Markovian
framework then there exist deterministic functions $u^{i}$, $i\in{\cal I}$,
with polynomial growth such that for any $(t,x)$

$Y^{t,x,i}_{s}=u^{i}(s,X^{t,x}_{s}),\,\,i\in{\cal I},\
\mathbb{P}-a.s.,\,\,\forall s\in[t,T].$

Note that we also have

$u^{i}(t,x)=Y^{t,x,i}_{t},\hskip 5.69046pt\text{for}\hskip
5.69046pt(t,x)\in[0,T]\times\mathbb{R}^{r}\hskip 5.69046pt\text{and}\hskip
5.69046pti\in{\cal I}.$ (4.5)

On the other hand the polynomial growth of $u^{i}$ stems from the polynomial
growths of the data assumed in Assumption (H) and the BSDEs (3.2), (3.3) as
well.

_Notation:_ For a sake of simplicity of notation, hereafter we sometimes
denote by $(\psi)_{k=1,\dots,q}:=(\psi_{1},\dots,\psi_{q})$, for some generic
function or vector $\psi$.

######  Remark 4.1.

From now on we will assume that $f_{i}$ is non-decreasing w.r.t $y^{k}$ for
any $k=1,...,q$ and not only w.r.t $y^{1},...,y^{i-1},y^{i+1},...,y^{q}$ (as
precised in (H2)-(H2)(iv)). This assumption is not really restrictive since by
considering the system of RBSDEs verified by $(e^{\alpha t}Y^{i}_{t})_{t\leq
T}$ , we obtain new generators $F_{i}$ given by

$F_{i}(t,y^{1},...,y^{m},z^{i}):=e^{\alpha t}f_{i}(t,x,e^{-\alpha
t}y^{1},...,e^{-\alpha t}y^{m},e^{-\alpha t}z^{i})-\alpha y^{i}$

which have the same properties as $(f_{i})_{i=1,\dots,q}$. Moreover, with an
appropriate choice of $\alpha$, those new generators are non-decreasing w.r.t
$y^{k}$ for any $k=1,\dots,q$, i.e., they fulfill the property we are
requiring for $(f_{i})_{i=1,\dots,q}$ (one can see Hamadène and Morlais [17],
for more details on this transform).

Our main interest will be to show that the function
$(u^{i})_{i=1,\dots,q}:(t,x)\in\left[0,T\right]\times\mathbb{R}^{r}\mapsto(u^{i}(t,x))_{i=1,\dots,q}\in\mathbb{R}^{q}$
is a solution in a weak viscosity sense for the Hamilton-Jacobi-Bellman system
of PDEs associated with the switching problem. In the case when the functions
$g_{ij}$ and $h_{i}$, $i,j\in\mathcal{I}$, are continuous, this system reads
as: for all $i\in\mathcal{I}$,

$\begin{cases}\min\\{v^{i}(t,x)-\max\limits_{k\in\mathcal{I}^{-i}}\left(v^{k}(t,x)-g_{ik}(t,x)\right);\hskip
2.84544pt-\partial_{t}v^{i}(t,x)-\mathcal{L}v^{i}(t,x)-&\\\ \hskip 56.9055pt-
f_{i}(t,x,\left(v^{1},\ldots,v^{i},\ldots,v^{q}\right)(t,x),\sigma^{\mathtt{T}}(t,x)D_{x}v^{i}(t,x))\\}=0;&\\\
v^{i}(T,x)=h_{i}(x).\end{cases}$ (4.6)

and it is shown that $(u^{i})_{i=1,\dots,q}$ is the unique viscosity solution
of system (4.6). But in our framework the functions $g_{ij}$,
$i,j\in\mathcal{I}$, are no longer continuous w.r.t $t$, therefore the
definition should be adapted. We are going to show that
$(u^{i})_{i=1,\dots,q}$ is a viscosity solution in a weak sense for the HJB
system of PDEs (4.6), associated with the swiching problem, and which we are
going to define in what follows. This definition is inspired by Ishii’s works
[21, 20], and also by the paper of Barles and Perthame [1].

To proceed for a locally bounded $\mathbb{R}$-valued function $v(t,x)$,
$(t,x)\in[0,T]\times\mathbb{R}^{\ell}$ ($\ell\geq 1$), we define its lower
(resp. upper) semi-continuous envelope $v_{*}$ (resp. $v^{*}$) as follows: For
any $(t,x)\in[0,T]\times\mathbb{R}^{\ell}$,

$v^{\ast}(t,x):=\limsup_{(t^{\prime},x^{\prime})\rightarrow(t,x),\hskip
2.84544ptt^{\prime}<T}v(t^{\prime},x^{\prime})\hskip 8.5359pt\text{(resp.
}\hskip
8.5359ptv_{\ast}(t,x):=\liminf_{(t^{\prime},x^{\prime})\rightarrow(t,x),\hskip
2.84544ptt^{\prime}<T}v(t^{\prime},x^{\prime})).$

Note that the function $v^{*}$ (resp. $v_{*}$) can also be seen as the
smallest usc (resp. lsc) function which is greater (resp. smaller) than $v$.
On the other hand, the following properties of the semi-continuous envelopes
of functions will be useful later.

######  Lemma 4.2.

Let $(t,x)\in[0,T]\times\mathbb{R}^{\ell}$ and $\varphi_{i}(t,x)$, $i=1,2$, be
two locally bounded $\mathbb{R}$-valued functions. We then have:

  1. (i)

If $\varphi_{1}$ is continuous then
$(\varphi_{1}+\varphi_{2})_{*}=\varphi_{1}+(\varphi_{2})_{*}$ and
$(\varphi_{1}+\varphi_{2})^{*}=\varphi_{1}+(\varphi_{2})^{*}$.

  2. (ii)

$(-\varphi_{1})_{*}=-(\varphi_{1})^{*}$.

  3. (iii)

$(\varphi_{1}\wedge\varphi_{2})_{*}=(\varphi_{1})_{*}\wedge(\varphi_{2})_{*}$
and $(\varphi_{1}\vee\varphi_{2})^{*}=(\varphi_{1})^{*}\vee(\varphi_{2})^{*}$.

  4. (iv)

If $\varphi_{1}$ is continuous then
$(\varphi_{1}\wedge\varphi_{2})^{*}=\varphi_{1}\wedge(\varphi_{2})^{*}$ and
$(\varphi_{1}\vee\varphi_{2})_{*}=\varphi_{1}\vee(\varphi_{2})_{*}$.

_Proof._ (i) Obviously we have
$\varphi_{1}+\varphi_{2}\geq\varphi_{1}+(\varphi_{2})_{*}$ and then
$(\varphi_{1}+\varphi_{2})_{*}\geq\varphi_{1}+(\varphi_{2})_{*}$ since this
latter is lsc. On the other hand
$(\varphi_{1}+\varphi_{2})_{*}-\varphi_{1}\leq\varphi_{2}$ and then
$(\varphi_{1}+\varphi_{2})_{*}-\varphi_{1}\leq(\varphi_{2})_{*}$ since
$(\varphi_{1}+\varphi_{2})_{*}-\varphi_{1}$ is lsc. This completes the proof
of the claim as the other property can be obtained similarly.

Points (ii) and (iii) are rather obvious, we then leave their proofs to the
care of the reader.

(iv) First note that
$(\varphi_{1}\wedge\varphi_{2})^{*}\leq\varphi_{1}\wedge(\varphi_{2})^{*}$.
Next let $((t_{n},x_{n}))_{n}$ be a sequence such that
$(\varphi_{2}(t_{n},x_{n}))_{n}\rightarrow(\varphi_{2})^{*}(t,x)$ as
$n\to\infty$. As $\varphi_{1}$ is continuous then
$\varphi_{1}(t_{n},x_{n})\wedge\varphi_{2}(t_{n},x_{n})\rightarrow\varphi_{1}(t,x)\wedge(\varphi_{2})^{*}(t,x)$
as $n\to\infty$. Therefore, by definition of the usc envelope,
$(\varphi_{1}\wedge\varphi_{2})^{*}(t,x)\geq\varphi_{1}(t,x)\wedge(\varphi_{2})^{*}(t,x)$
which completes the proof of the first claim. The proof of the other one is
similar. $\blacksquare$

Next for $i=1,...,q$, let us denote by $F_{i}$ the non-linearity which defines
the $i-th$ equation in (4.6), i.e.,

$F_{i}(t,x,(y^{j})_{j=1,\ldots,q},r,p,X)=\min\big{\\{}y^{i}-\max\limits_{k\in\mathcal{I}^{-i}}(y^{k}-g_{ik}(t,x));\hskip
2.84544ptG_{i}(t,x,(y^{j})_{j=1,\ldots,q},r,p,X)\big{\\}}$ (4.7)

where

$G_{i}(t,x,(y^{j})_{j=1,\ldots,q},r_{i},p_{i},X_{i})=-r_{i}-\frac{1}{2}Tr(\sigma\sigma^{\mathtt{T}}X_{i})-b^{\mathtt{T}}p_{i}-f_{i}(t,x,(y^{j})_{j=1,\ldots,q},\sigma^{\mathtt{T}}p_{i}).$
(4.8)

Note that by Assumption (H) on $f_{i}$ and (4.1), the function $G_{i}$ is
jointly continuous in its arguments. Therefore, taking into account the
results of Lemma 4.2, for any $i=1,\dots,q$, the semi-continuous envelopes of
$F_{i}$ (in all arguments) are given by:

$F_{i}^{*}(t,x,(y^{j})_{j=1,\ldots,q},r,p,X)=\min\big{\\{}y^{i}-(\max\limits_{k\in\mathcal{I}^{-i}}(y^{k}-g_{ik}(t,x)))_{*};\hskip
2.84544ptG_{i}(t,x,(y^{j})_{j=1,\ldots,q},r,p,X)\big{\\}}$

and

$(F_{i})_{*}(t,x,(y^{j})_{j=1,\ldots,q},r,p,X)=\min\big{\\{}y^{i}-(\max\limits_{k\in\mathcal{I}^{-i}}(y^{k}-g_{ik}(t,x)))^{*};\hskip
2.84544ptG_{i}(t,x,(y^{j})_{j=1,\ldots,q},r,p,X)\big{\\}}.$

We are now ready to precise the definition of the viscosity solution of HJB
system associated with the switching problem. As noticed previously it is
inspired by the papers [1, 21, 20]. On the other hand, the discontinuities of
the functions $(u^{i})_{i=1,\dots,q}$ generated by the ones of
$(g_{ij})_{i,j\in\mathcal{I}}$ make that the terminal condition at time $t=T$
is not the same as in (4.6), but should be adapted as well to this weak sense
(see e.g. [2]).

######  Definition 4.3.

Let $\mathbf{v}:=(v^{1},\ldots,v^{q})$ be a locally bounded function from
$[0,T]\times\mathbb{R}^{r}$ into $\mathbb{R}^{q}$.

(1)

We say that $\mathbf{v}$ is a viscosity subsolution of (4.6) if for any
$i\in\mathcal{I}$, and $x_{0}\in\mathbb{R}$,

(a)

$v^{i*}$ verifies the following inequality at point $(T,x_{0})$:

$\min\big{\\{}v^{i\ast}(T,x_{0})-h_{i}(x_{0});\hskip
5.69046ptu^{i\ast}(T,x_{0})-\max\limits_{j\in\mathcal{I}^{-i}}\left(v^{j\ast}-g_{ij}\right)^{\ast}(T,x_{0})\big{\\}}\leq
0.$ (4.9)

(b)

Moreover, at $(t_{0},x_{0})\in[0,T)\times\mathbb{R}^{r}$, the function $v^{i}$
is such that, for and any $\phi\in C^{1,2}([0,T]\times\mathbb{R}^{r})$ with
$\phi(t_{0},x_{0})=v^{i\ast}(t_{0},x_{0})$ and $\phi-v^{i\ast}$ attaining its
minimum at $(t_{0},x_{0})$, we have

$\begin{array}[]{l}(F_{i})_{*}(t_{0},x_{0},(v^{j\ast}(t_{0},x_{0}))_{j=1,\ldots,q},\partial_{t}\phi(t_{0},x_{0}),D_{x}\phi(t_{0},x_{0}),D^{2}_{xx}\phi(t_{0},x_{0}))\\\
\qquad\qquad=\min\bigg{\\{}v^{i\ast}(t_{0},x_{0})-\max\limits_{k\in\mathcal{I}^{-i}}\left(v^{k*}-g_{ik}\right)^{\ast}(t_{0},x_{0});\\\
\qquad\qquad-\left(\partial_{t}+\mathcal{L}\right)\phi(t_{0},x_{0})-f_{i}(t_{0},x_{0},v^{j*}(t_{0},x_{0}))_{j=1,\ldots,q}(t_{0},x_{0}),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t_{0},x_{0}))\bigg{\\}}\leq
0.\end{array}$

(2)

In the same manner, $\mathbf{v}$ is said to be a viscosity supersolution of
(4.6) if for any $i\in\mathcal{I}$, and $x_{0}\in\mathbb{R}$,

(a)

$v^{i}_{*}$ verifies at $(T,x_{0})$ the following:

$\min\Big{\\{}v^{i}_{\ast}(T,x_{0})-h_{i}(x_{0});\hskip
5.69046ptv^{i}_{\ast}(T,x_{0})-\big{(}\max\limits_{j\in\mathcal{I}^{-i}}\left(v^{j}_{\ast}-g_{ij}\right)\big{)}_{\ast}(T,x_{0})\Big{\\}}\geq
0.$ (4.10)

(b)

Similarly, at $(t_{0},x_{0})\in[0,T)\times\mathbb{R}^{r}$, $v^{i}$ satisfies
the next: for and any $\phi\in C^{1,2}([0,T]\times\mathbb{R}^{r})$ with
$\phi(t_{0},x_{0})=v^{i}_{*}(t_{0},x_{0})$ and $\phi-v^{i}_{*}$ attaining its
maximum at $(t_{0},x_{0})$, we have

$\begin{array}[]{l}(F_{i})^{*}(t_{0},x_{0},(v^{j}_{*}(t_{0},x_{0}))_{j=1,\ldots,q},\partial_{t}\phi(t_{0},x_{0}),D_{x}\phi(t_{0},x_{0}),D^{2}_{xx}\phi(t_{0},x_{0}))\\\
\qquad=\min\bigg{\\{}v^{i}_{*}(t_{0},x_{0})-(\max\limits_{k\in\mathcal{I}^{-i}}\left(v^{k}_{*}-g_{ik}\right))_{\ast}(t_{0},x_{0});\\\
\qquad\qquad-\left(\partial_{t}+\mathcal{L}\right)\phi(t_{0},x_{0})-f_{i}(t_{0},x_{0},(v^{j}_{*}(t_{0},x_{0}))_{j=1,\dots,q},\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t_{0},x_{0}))\bigg{\\}}\geq
0.\end{array}$

(3)

We say that $\mathbf{v}$ is viscosity solution of (4.6) if it is both a
viscosity sub. and supersolution.

To proceed we are going to show that the functions $(u^{i})_{i=1,...,q}$ is a
viscosity solution of the system (4.6) in a weak sense, i.e., according to
Definition 4.3. However we need some preliminary results which we give as
lemmas hereafter. From now $B_{\eta}(t_{0},x_{0})$ is the open ball of radius
$\eta$ and center $(t_{0},x_{0})$.

######  Lemma 4.4.

Under the Assumption ((H2)), the mapping

$(t,x)\longmapsto
f_{i}\big{(}t,x,\big{(}v^{1\ast},\ldots,v^{q\ast}\big{)}(t,x),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t,x)\big{)}$

is u.s.c. for any $\phi\in C^{1,2}([0,T]\times\mathbb{R}^{r})$.

Proof. Let $(t_{0},x_{0})\in[0,T]\times\mathbb{R}^{r}$. Since $v^{k\ast}$ is
u.s.c for $k=1,\dots,q$, then for all $\varepsilon>0$ there exists
$\eta_{\varepsilon}>0$ such that for all $(t,x)$, satisfying
$\left\|(t,x)-(t_{0},x_{0})\right\|<\eta_{\varepsilon}$, we have

$v^{k\ast}(t_{0},x_{0})\geq v^{k\ast}(t,x)-\varepsilon\hskip
19.91684pt\text{for all}\hskip 8.5359ptk=1,\ldots,q.$

Next, by monotonicity and Lipschitz properties of $f_{i}$, for all $(t,x)\in
B_{\eta_{\varepsilon}}(t_{0},x_{0})$ we get

$\begin{split}&\
f_{i}\big{(}t_{0},x_{0},\big{(}v^{k\ast}\big{)}_{k=1,\ldots,q}(t_{0},x_{0}),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t_{0},x_{0})\big{)}\geq
f_{i}\big{(}t_{0},x_{0},\big{(}v^{k\ast}(t,x)-\varepsilon\big{)}_{k=1,\ldots,q},\big{(}\sigma^{\mathtt{T}}D_{x}\big{)}\phi(t_{0},x_{0})\big{)}\\\
&\ \geq
f_{i}\big{(}t_{0},x_{0},\big{(}v^{k\ast}(t,x)\big{)}_{k=1,\dots,q},\big{(}\sigma^{\mathtt{T}}D_{x}\big{)}\phi(t_{0},x_{0})\big{)}-C\varepsilon=f_{i}(t,x,\big{(}v^{k\ast}(t,x)\big{)}_{k=1,\ldots,q},\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t,x))-C\varepsilon+\\\
&\
+\Big{\\{}f_{i}\big{(}t_{0},x_{0},\big{(}v^{k\ast}(t,x)\big{)}_{k=1,\ldots,q},\big{(}\sigma^{\mathtt{T}}D_{x}\big{)}\phi(t_{0},x_{0})\big{)}-f_{i}(t,x,\big{(}v^{k\ast}(t,x)\big{)}_{k=1,\ldots,q},\big{(}\sigma^{\mathtt{T}}D_{x}\big{)}\phi(t,x))\Big{\\}},\end{split}$

where $C$ is the Lipschitz constant of $f_{i}$. By continuity of $f_{i}$ with
respect to $(t,x)$ and Lipschitz in $z^{i}$ (Assumptions ((H2))-((H2)(i)) and
((H2))-((H2)(ii))), the quantity inside the brackets goes to zero as
$(t,x)\rightarrow(t_{0},x_{0})$. Therefore, taking a suitable
$\eta_{\varepsilon}>0$, we obtain

$f_{i}\big{(}t_{0},x_{0},\big{(}v^{k\ast}(t_{0},x_{0})\big{)}_{k=1,\ldots,q},\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t_{0},x_{0})\big{)}\geq
f_{i}(t,x,\big{(}v^{k\ast}(t,x)\big{)}_{k=1,\ldots,q},\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t,x))-C^{\prime}\varepsilon$

for all $(t,x)\in B_{\eta_{\varepsilon}}(t_{0},x_{0})$ and for some other
constant $C^{\prime}$ and the claim follows. $\blacksquare$

######  Lemma 4.5.

Let $\phi\in C^{1,2}([0,T]\times\mathbb{R}^{r})$,
$(t_{0},x_{0})\in[0,T)\times\mathbb{R}^{r}$ and
$\phi(t_{0},x_{0})=v^{i\ast}(t_{0},x_{0})$. If

$\phi(t_{0},x_{0})=v^{i\ast}(t_{0},x_{0})>\max\limits_{k\in\mathcal{I}^{-i}}\big{(}v^{k\ast}-g_{ik}\big{)}^{\ast}(t_{0},x_{0})$
(4.11)

and

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t_{0},x_{0})>f_{i}(t_{0},x_{0},(v^{k\ast})_{k=1,\dots,q}(t_{0},x_{0}),(\sigma^{\mathtt{T}}D_{x})\phi(t_{0},x_{0})),$
(4.12)

then there exist $\varepsilon$ and a ball
$B_{\eta_{\varepsilon}}(t_{0},x_{0})$ such that for all $(t,x)\in
B_{\eta_{\varepsilon}}(t_{0},x_{0})$ we have:

$\phi(t,x)\geq\max\limits_{k\in\mathcal{I}^{-i}}\big{(}v^{k\ast}(t,x)-g_{ik}(t,x)\big{)}+\varepsilon$
(4.13)

and

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t,x)\geq
f_{i}\big{(}t,x,(v^{k\ast})_{k=1,\dots,q}(t,x),(\sigma^{\mathtt{T}}D_{x})\phi(t,x)\big{)}+\varepsilon.$
(4.14)

Proof: By (4.11) and the continuity of $\phi$ there exist $\varepsilon$ and a
ball $B_{\eta_{\varepsilon}}(t_{0},x_{0})$ such that

$\phi(t,x)\geq\max\limits_{k\in\mathcal{I}^{-i}}\big{(}v^{k\ast}(t_{0},x_{0})-g_{ik}(t_{0},x_{0})\big{)}^{\ast}+2\varepsilon$
(4.15)

for all $(t,x)\in B_{\eta_{\varepsilon}}(t_{0},x_{0})$. Next, by the u.s.c
property, there exists $\eta_{\varepsilon}^{{}^{\prime}}$ such that for all
$(t,x)\in B_{\eta^{{}^{\prime}}_{\varepsilon}}(t_{0},x_{0})$ we have

$\begin{array}[]{ll}\max\limits_{k\in\mathcal{I}^{-i}}\left(v^{k\ast}(t_{0},x_{0})-g_{ik}(t_{0},x_{0})\right)^{\ast}\geq\max\limits_{k\in\mathcal{I}^{-i}}\left(v^{k\ast}(t,x)-g_{ik}(t,x)\right)^{\ast}-\varepsilon&\\\
&\\\ \hskip
153.6447pt\geq\max\limits_{k\in\mathcal{I}^{-i}}\left(v^{k\ast}(t,x)-g_{ik}(t,x)\right)-\varepsilon\end{array}$
(4.16)

where in the last inequality we use that the usc envelope of a function is
greater or equal to the function itself. Therefore, from (4.15), (4.16) and
assuming , without loss of generality, that
$\eta_{\varepsilon}\leq\eta_{\varepsilon}^{{}^{\prime}}$ we have

$\phi(t,x)\geq\max\limits_{k\in\mathcal{I}^{-i}}\big{(}v^{k\ast}(t,x)-g_{ik}(t,x)\big{)}+\varepsilon$
(4.17)

for all $(t,x)\in B_{\eta_{\varepsilon}}(t_{0},x_{0})$.

As for the second inequality we can do a similar procedure since
$\left(\partial_{t}+\mathcal{L}\right)\phi$ is continuous and $(t,x)\mapsto
f_{i}(t,x,(v^{k\ast})_{k=1,\dots,q}(t,x),(\sigma^{\mathtt{T}}D_{x})\phi(t,x))$
is u.s.c. Namely, there exist $\varepsilon^{{}^{\prime}}$ and
$\eta_{\varepsilon}^{{}^{\prime\prime}}$ such that for each $(t,x)\in
B_{\eta_{\varepsilon}^{{}^{\prime\prime}}}(t_{0},x_{0})$ we have

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t,x)\geq
f_{i}\big{(}t,x,(v^{k\ast})_{k=1,\dots,q}(t,x),(\sigma^{\mathtt{T}}D_{x})\phi(t,x)\big{)}+\varepsilon^{\prime}.$
(4.18)

Now, supposing, without loss of generality, that
$\varepsilon\leq\varepsilon^{\prime}$ and
$\eta_{\varepsilon}\leq\eta_{\varepsilon}^{{}^{\prime\prime}}$, we have that
inequalities (4.13) and (4.14) hold true for all $(t,x)\in
B_{\eta_{\varepsilon}}$.

######  Remark 4.6.

In a similar manner, it is possible to obtain a parallel result as in Lemmas
4.4 and 4.5 for $v^{i}_{\ast}$ in lieu of $v^{i\ast}$. Namely, it can be
proved that under Assumption ((H2)) the mapping

$(t,x)\longmapsto
f_{i}\big{(}t,x,(v^{k}_{*})_{k=1,\dots,q}(t,x),\big{(}\sigma^{\mathtt{T}}D_{x}\big{)}\phi(t,x)\big{)}$

is l.s.c., and if

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t_{0},x_{0})<f_{i}(t_{0},x_{0},(v^{k}_{\ast})_{k=1,\dots,q}(t_{0},x_{0}),(\sigma^{\mathtt{T}}D_{x})\phi(t_{0},x_{0})),$
(4.19)

then there exists $\varepsilon>0$ and $\eta_{\varepsilon}$ such that for all
$(t,x)\in B_{\eta_{\varepsilon}}(t_{0},x_{0})$:

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t,x)\leq
f_{i}\big{(}t,x,(v^{k}_{*})_{k=1,\dots,q}(t,x),\big{(}\sigma^{\mathtt{T}}D_{x}\big{)}\phi(t,x))-\varepsilon.$

The proofs are very similar as the proofs given in the aforementioned lemmas,
so shall omit them.

Finally, we recall two comparison results for BSDE and RBSDE that we have
borrowed from Lemma 4.1 and Proposition 4.2, in Dumitrescu et al. [11].

######  Lemma 4.7.

Fix $t_{0}\in[0,T]$ and let $\theta$ be a stopping time with values in
$[t_{0},T]$. Consider two random variables $\xi_{1}$ and $\xi_{2}\in
L^{2}(\mathcal{F}_{\theta})$ and two drivers (a.k.a generators) $f_{1}$,
$f_{2}$ such that $f_{2}$ satisfies (H2) with Lipschitz constant $C>0$. For
$i=1,2$, let $(Y^{i}_{t},Z^{i}_{t})$ be the solution in
$\mathscr{S}^{2}\times\mathcal{H}^{2}$ of the BSDE with associated data
$(f_{i},\xi_{i})$, and terminal time $\theta$. In this case, $f_{i}$ and
$\xi_{i}$ represent the driver and terminal condition, respectively. Suppose
that for some $\epsilon>0$ we have

$1_{\\{t_{0}\leq t\leq\theta\\}}(t)f_{1}(t,Y^{1}_{t},Z^{1}_{t})\geq
1_{\\{t_{0}\leq t\leq\theta\\}}(t)f_{2}(t,Y^{1}_{t},Z^{1}_{t}),\,\,dt\otimes
d\mathbb{P}-a.e.\text{ and
}\xi_{1}\geq\xi_{2}+\varepsilon,\,\,\mathbb{P}-a.s.$

Then we have $Y^{1}_{t}\geq Y^{2}_{t}+\varepsilon e^{-CT}$, $\mathbb{P}$-a.s.
for each $t\in[t_{0},\theta]$.

######  Lemma 4.8 (A comparison result between a BSDE and a RBSDE).

Fix $t_{0}\in[0,T]$ and let $\theta$ be a stopping time on $[t_{0},T]$.
Consider the random variable $\xi_{1}\in L^{2}(\mathcal{F}_{\theta})$ and a
driver $f_{1}$. Let $(Y^{1}_{t},Z^{1}_{t})$ be the associated BSDE solution
with driver $f_{1}$, terminal time $\theta$ and terminal condition $\xi_{1}$.
Consider also $g_{2}(\cdot)\in\mathscr{S}^{2}$ and let $f_{2}$ be a driver
satisfying (H2) with Lipschitz constant $C>0$. Assume the existence of the
solution $Y^{2}_{t}$ of the associated RBSDE with driver $f_{2}$, terminal
time $\theta$ and obstacle $g_{2}$, and assume that

$\begin{array}[]{l}1_{\\{t_{0}\leq
t\leq\theta\\}}(t)f_{1}(t,Y^{1}_{t},Z^{1}_{t})\geq 1_{\\{t_{0}\leq
t\leq\theta\\}}(t)f_{2}(t,Y^{1}_{t},Z^{1}_{t}),\,\,dt\otimes
d\mathbb{P}-a.e.\\\ \text{ and }\\\ 1_{\\{t_{0}\leq
t\leq\theta\\}}(t)Y^{1}_{t}\geq 1_{\\{t_{0}\leq
t\leq\theta\\}}(t)(g_{2}({t})+\varepsilon),\forall t\geq
0,\mathbb{P}-a.s.\end{array}$

where $\varepsilon$ is a positive constant. Then, we have $Y^{1}_{t}\geq
Y^{2}_{t}+\varepsilon e^{-CT}$, $\mathbb{P}-a.s.$, for each
$t\in[t_{0},\theta]$.

We now give the main result of this section.

######  Theorem 4.9.

The function $\mathbf{u}:=(u^{1},\ldots,u^{q})$, where for each
$i=1,\ldots,q$, $u^{i}$ is defined as in (4.5), is a weak viscosity solution
of the system (4.6).

Proof. Step 1: Viscosity sub-solution property on $[0,T)\times\mathbb{R}^{r}$.

Let $\phi\in C^{1,2}([0,T]\times\mathbb{R}^{r})$ and
$(t_{0},x_{0})\in[0,T)\times\mathbb{R}^{r}$ be such that $\phi(t,x)\geq
u^{i\ast}(t,x)$, for all $(t,x)\in[0,T)\times\mathbb{R}^{r}$ and
$\phi(t_{0},x_{0})=u^{i\ast}(t_{0},x_{0}$). Without loss of generality, we can
assume that the minimum of $\phi-u^{i\ast}$ attained at $(t_{0},x_{0})$ is
strict. We need to show that if

$\phi(t_{0},x_{0})=u^{i\ast}(t_{0},x_{0})>\max\limits_{k\in\mathcal{I}^{-i}}\big{(}u^{k\ast}(t_{0},x_{0})-g_{ik}(t_{0},x_{0})\big{)}^{\ast}$
(4.20)

then

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t_{0},x_{0})-f_{i}\big{(}t_{0},x_{0},(u^{k*})_{k=1,\dots,q}(t_{0},x_{0}),(\sigma^{\mathtt{T}}D_{x})\phi(t_{0},x_{0})\big{)}\leq
0.$ (4.21)

We proceed by contradiction; i.e. we shall assume

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t_{0},x_{0})-f_{i}\big{(}t_{0},x_{0},(u^{k*})_{k=1,\dots,q}(t_{0},x_{0}),(\sigma^{\mathtt{T}}D_{x})\phi(t_{0},x_{0})\big{)}>0,$

then by Lemma 4.5 there exists $\varepsilon>0$ and $\eta_{\varepsilon}>0$ such
that, for all $(t,x)\in B_{\eta_{\varepsilon}}(t_{0},x_{0})$, we have both

$\phi(t,x)\geq\max\limits_{k\in\mathcal{I}^{-i}}\big{(}u^{k\ast}(t,x)-g_{ik}(t,x)\big{)}+\varepsilon\geq\max\limits_{k\in\mathcal{I}^{-i}}\big{(}u^{k}(t,x)-g_{ik}(t,x)\big{)}+\varepsilon,$
(4.22)

since $u^{k\ast}\geq u^{k}$, and

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t,x)-f_{i}(t,x,(u^{k*})_{k=1,\dots,q}(t,x),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t,x))\geq\varepsilon.$
(4.23)

By definition of $u^{i\ast}$, there exists a sequence $(t_{m},x_{m})_{m\geq
0}$ in $B_{\eta_{\varepsilon}}(t_{0},x_{0})$, such that
$(t_{m},x_{m})\rightarrow(t_{0},x_{0})$ and $u^{i}(t_{m},x_{m})\rightarrow
u^{i\ast}(t_{0},x_{0})$. Now let us fix $m$ and take the associated state
process $X^{t_{m},x_{m}}$ defined in (4.3) and define the stopping time
$\theta^{m}$ as

$\theta^{m}:=(t_{0}+\eta_{\varepsilon})\wedge\inf\left\\{s\geq
t_{m}:\left|X^{t_{m},x_{m}}_{s}-x_{0}\right|\geq\eta_{\varepsilon}\right\\}.$
(4.24)

Applying Itô’s lemma to $\phi(s,X^{t_{m},x_{m}}_{s})$, it can be seen that

$\left(\phi(s,X^{t_{m},x_{m}}_{s}),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(s,X^{t_{m},x_{m}}_{s});\hskip
5.69046ptt_{m}\leq s\leq\theta^{m}\right)$

is the solution of the BSDE with coefficient
$-\left(\partial_{s}+\mathcal{L}\right)\phi(s,x)$, terminal time $\theta^{m}$
and terminal value $\phi(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})$. The idea
is to compare this BSDE with the solution
$(\underline{Y}^{i,t_{m},x_{m}}_{s})_{t_{m}\leq s\leq\theta^{m}}$ of the RBSDE
with coefficient $f_{i}$, barrier
$\max_{k\in\mathcal{I}^{-i}}\\{u^{k}-g_{ik}\\}$ and terminal condition
$u^{i\ast}(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})$. Note that by definition
of $\theta^{m}$ and inequality (4.23), we have

$\begin{array}[]{ll}-\left(\partial_{s}+\mathcal{L}\right)\phi\big{(}s,X^{t_{m},x_{m}}_{s}\big{)}\geq
f_{i}\left(s,X^{t_{m},x_{m}}_{s},(u^{k*})_{k=1,\dots,q}(s,X^{t_{m},x_{m}}_{s}),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(s,X^{t_{m},x_{m}}_{s})\right)+\varepsilon&\\\
\hskip 119.50148pt\geq
f_{i}\left(s,X^{t_{m},x_{m}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{m},x_{m}}_{s}),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(s,X^{t_{m},x_{m}}_{s})\right)+\varepsilon\end{array}$

for each $t_{m}\leq s\leq\theta^{m}$, where to reach the last inequality we
use that $u^{\ast}\geq u$, the monotonicity property (H2)-(H2)(iv) and the
Remark 4.1. It remains to compare the solution $\phi(s,X^{t_{m},x_{m}}_{s})$
of the BSDE with the barrier
$\max_{k\in\mathcal{I}^{-i}}\\{u^{k}(s,X^{t_{m},x_{m}}_{s})-g_{ik}(s,X^{t_{m},x_{m}}_{s})\\}\mathbf{1}_{[s<\theta^{m}]}+u^{i\ast}(s,X^{t_{m},x_{m}}_{s})\mathbf{1}_{[s=\theta^{m}]}$
of the RBSDE for $t_{m}\leq s\leq\theta^{m}$. From inequality (4.22) and
definition of $\theta^{m}$ we derive that

$\begin{array}[]{ll}\phi(s,X^{t_{m},x_{m}}_{s})\geq\max\limits_{k\in\mathcal{I}^{-i}}\left(u^{k\ast}(s,X^{t_{m},x_{m}}_{s})-g_{ik}(s,X^{t_{m},x_{m}}_{s})\right)+\varepsilon&\\\
\hskip
62.59596pt\geq\max\limits_{k\in\mathcal{I}^{-i}}\left(u^{k}(s,X^{t_{m},x_{m}}_{s})-g_{ik}(s,X^{t_{m},x_{m}}_{s})\right)+\varepsilon\hskip
14.22636pt\text{for}\hskip 8.5359ptt_{m}\leq s<\theta^{m}.\end{array}$ (4.25)

On the other hand, to show that the inequality holds at $\theta^{m}$, we
recall that the minimum $(t_{0},x_{0})$ is strict and hence there exists
$\gamma_{\varepsilon}$ such that

$\phi(t,x)-u^{i\ast}(t,x)\geq\gamma_{\varepsilon}\hskip
8.5359pt\text{on}\hskip 2.84544pt[0,T]\times\mathbb{R}^{r}\setminus
B_{\eta_{\varepsilon}}({t_{0},x_{0}}).$

In particular, we have

$\phi(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})\geq
u^{i\ast}(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})+\gamma_{\varepsilon}.$
(4.26)

Therefore, from (4.25), (4.26) and letting
$\delta_{\varepsilon}:=\min(\varepsilon,\gamma_{\varepsilon})$, we get

$\phi(s,X^{t_{m},x_{m}}_{s})\geq\max\limits_{k\in\mathcal{I}^{-i}}\left(u^{k}(s,X^{t_{m},x_{m}}_{s})-g_{ik}(s,X^{t_{m},x_{m}}_{s})+\delta_{\varepsilon}\right)\mathbf{1}_{[s<\theta^{m}]}+\left(u^{i\ast}(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})+\delta_{\varepsilon}\right)\mathbf{1}_{[s=\theta^{m}]}$

for $t_{m}\leq s\leq\theta^{m}\hskip 2.84544pt\text{a.s.}$. Thus, by the
comparison result in Lemma 4.8, we have

$\phi(s,X^{t_{m},x_{m}}_{s})\geq\underline{Y}^{i,t_{m},x_{m}}_{s}+\delta_{\varepsilon}K\hskip
28.45274pt\text{for}\hskip 8.5359ptt_{m}\leq s\leq\theta^{m}$

where $K$ is a positive constant which only depends on $T$ and the Lipschitz
constant of $f_{i}$. In particular, for $t=t_{m}$, we have

$\phi(t_{m},x_{m})\geq\underline{Y}^{i,t_{m},x_{m}}_{t_{m}}+\delta_{\varepsilon}K.$

Now, since $u^{i}(t_{m},x_{m})\rightarrow u^{i\ast}(t_{0},x_{0})$ and $\phi$
is continuous with $\phi(t_{0},x_{0})=u^{i\ast}(t_{0},x_{0})$, for $m$
sufficiently large we have both

$|u^{i}(t_{m},x_{m})-u^{i\ast}(t_{0},x_{0})|\leq\frac{1}{4}\delta_{\varepsilon}K$
(4.27)

and

$|u^{i\ast}(t_{0},x_{0})-\phi(t_{m},x_{m})|\leq\frac{\delta_{\varepsilon}K}{4},$
(4.28)

whence
$|\phi(t_{m},x_{m})-u^{i}(t_{m},x_{m})|\leq\frac{1}{2}\delta_{\varepsilon}K$,
and hence

$u^{i}(t_{m},x_{m})\geq\underline{Y}^{i,t_{m},x_{m}}_{t_{m}}+\frac{1}{4}\delta_{\varepsilon}K.$
(4.29)

But $u^{i\ast}(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})\geq
u^{i}(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})$, then by comparison theorem
$u^{i}(s,X^{t_{m},x_{m}}_{s})=Y^{i,t_{m},x_{m}}_{s}\leq\underline{Y}^{i,t_{m},x_{m}}_{s}$
for $t_{m}\leq s\leq\theta^{m}$. Thus, for $s=t_{m}$, we get
$u^{i}(t_{m},x_{m})\leq\underline{Y}^{i,t_{m},x_{m}}_{t_{m}}$ that produces a
contradiction with (4.29). Therefore (4.21) holds true and then also the
viscosity subsolution property in $[0,T)\times\mathbb{R}^{r}$.

Step 2: Viscosity super-solution property on $[0,T)\times\mathbb{R}^{r}$.

Let $(t_{0},x_{0})\in[0,T)\times\mathbb{R}^{r}$ and $\phi\in
C^{1,2}([0,T]\times\mathbb{R}^{r})$ be such that
$\phi(t_{0},x_{0})=u_{\ast}^{i}(t_{0},x_{0})$ and $\phi(t,x)\leq
u^{i}_{\ast}(t,x)$, for all $(t,x)\in[0,T]\times\mathbb{R}^{r}$. As stated
above, we can suppose that the maximum is strict in $(t_{0},x_{0})$. Since by
construction $u^{i}\geq\max_{k\in\mathcal{I}^{-i}}\left(u^{k}-g_{ik}\right)$,
then it is easy to see that
$u^{i}_{\ast}(t_{0},x_{0})\geq(\max_{k\in\mathcal{I}^{-i}}(u^{k}_{\ast}-g_{ik}))_{\ast}(t_{0},x_{0})$.
Now, we show that

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t_{0},x_{0})-f_{i}\big{(}t_{0},x_{0},(u^{k}_{*})_{k=1,\dots,q}(t_{0},x_{0}),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t_{0},x_{0})\big{)}\geq
0.$

Similar to the subsolution case, we shall proceed by contradiction, namely,
suppose that

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t_{0},x_{0})-f_{i}\big{(}t_{0},x_{0},(u^{k}_{*})_{k=1,\dots,q}(t_{0},x_{0}),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t_{0},x_{0})\big{)}<0,$

then by Remark 4.6 there exists $\varepsilon>0$ and $\eta_{\varepsilon}>0$
such that, for all $(t,x)\in B_{\eta_{\varepsilon}}(t_{0},x_{0})$, we have

$-\left(\partial_{t}+\mathcal{L}\right)\phi(t,x)-f_{i}\big{(}t,x,(u^{k}_{*})_{k=1,\dots,q}(t,x),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(t,x)\big{)}\leq-\varepsilon.$
(4.30)

Let $(t_{m},x_{m})_{m\geq 1}$ be a sequence in
$B_{\eta_{\varepsilon}}(t_{0},x_{0})$ such that
$(t_{m},x_{m})\rightarrow(t_{0},x_{0})$ and $u^{i}(t_{m},x_{m})\rightarrow
u_{\ast}^{i}(t_{0},x_{0})$. We introduce the state process $X^{t_{m},x_{m}}$
and define the stopping time $\theta^{m}$ as in (4.24). Next, we apply Itô’s
formula to $\phi(s,X^{t_{m},x_{m}}_{s})$ in order to obtain

$\left(\phi(s,X^{t_{m},x_{m}}_{s}),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(s,X^{t_{m},x_{m}}_{s});t_{m}\leq
s\leq\theta^{m}\right)$

is the solution of the BSDE associated with terminal time $\theta^{m}$,
terminal value $\phi(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})$ and driver
$(-\left(\partial_{t}+\mathcal{L}\right)\phi(s,X^{t_{m},x_{m}}_{s}))_{s\in[t_{m},\theta_{m}]}$.
Then by definition of $\theta^{m}$ and inequality (4.30), we get

$\begin{array}[]{ll}-\left(\partial_{t}+\mathcal{L}\right)\phi\left(s,X^{t_{m},x_{m}}_{s}\right)\leq
f_{i}\big{(}s,X^{t_{m},x_{m}}_{s},(u^{k}_{*})_{k=1,\dots,q}(s,X^{t_{m},x_{m}}_{s}),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(s,X^{t_{m},x_{m}}_{s})\big{)}-\varepsilon&\\\
\hskip 108.12054pt\leq
f_{i}\big{(}s,X^{t_{m},x_{m}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{m},x_{m}}_{s}),\left(\sigma^{\mathtt{T}}D_{x}\right)\phi(s,X^{t_{m},x_{m}}_{s})\big{)}-\varepsilon\end{array}$
(4.31)

for $t_{m}\leq s\leq\theta^{m}$ a.s., where to reach the last inequality we
use the monotonicity property (H2)-(iv) and Remark 4.1 and that $u^{j}\geq
u^{j}_{\ast}$ for $j=1,\ldots,q$. It remains to compare the terminal
conditions of the BSDEs with coefficients
$-\left(\partial_{t}+\mathcal{L}\right)\phi$ and $f_{i}$ respectively. Since
the maximum $(t_{0},x_{0})$ is strict, there exists $\gamma_{\varepsilon}$
(which depends on $\eta_{\varepsilon}$) such that
$u_{\ast}^{i}(t,x)\geq\phi(t,x)+\gamma_{\varepsilon}$ on
$[0,T]\times\mathbb{R}^{r}\setminus B_{\eta_{\varepsilon}}(t_{0},x_{0})$,
which implies

$\phi(\theta_{m},X^{t_{m},x_{m}}_{\theta^{m}})\leq
u^{i}_{\ast}(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})-\gamma_{\varepsilon}.$

Thus using inequality (4.31) and the comparison result for BSDEs, Lemma 4.7,
we derive that

$\phi(s,X^{t_{m},x_{m}}_{s})\leq\bar{Y}^{i,t_{m},x_{m}}_{s},\hskip
8.5359pt\text{for}\hskip 5.69046ptt_{m}\leq s\leq\theta^{m}$

and therefore, in $s=t_{m}$, we have
$\phi(t_{m},x_{m})\leq\bar{Y}^{i,t_{m},x_{m}}_{t_{m}}$. As above mentioned, we
can assume that $m$ is sufficient large so that
$|\phi(t_{m},x_{m})-u^{i}(t_{m},x_{m})|\leq\frac{\delta_{\varepsilon}K}{2}$.
We thus get

$u^{i}(t_{m},x_{m})-\frac{\gamma_{\varepsilon}K}{2}\leq\phi(t_{m},x_{m})\leq\bar{Y}^{i,t_{m},x_{m}}_{t_{m}}$

and hence

$u^{i}(t_{m},x_{m})<\bar{Y}^{i,t_{m},x_{m}}_{t_{m}}.$ (4.32)

But $u^{i}_{\ast}(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})\leq
u^{i}(\theta^{m},X^{t_{m},x_{m}}_{\theta^{m}})$, then by Lemma 4.7 we get
$\bar{Y}^{i,t_{m},x_{m}}_{s}\leq
Y^{i,t_{m},x_{m}}_{s}=u^{i}(s,X^{t_{m},x_{m}}_{s})$ for $t_{m}\leq
s\leq\theta^{m}$, and thus $\bar{Y}^{i,t_{m},x_{m}}_{t_{m}}\leq
u^{i}(t_{m},x_{m})$, which is a contradiction with (4.32). Therefore the
viscosity supersolution property in $[0,T)\times\mathbb{R}^{r}$ holds true.

Step 3: Subsolution property at $(T,x)$.

We now show that for any $i=1,...,m$,

$\min\big{\\{}u^{i\ast}(T,x_{0})-h_{i}(x_{0});\hskip
5.69046ptu^{i\ast}(T,x_{0})-\max\limits_{j\in\mathcal{I}^{-i}}\left(u^{j\ast}-g_{ij})\right)^{\ast}(T,x_{0})\big{\\}}\leq
0.$

We follow here the same idea as in Bouchard [2] (see also Theorem 1 in
Hamadène and Morlais [17]). We reason by contradiction, namely, we assume that

$\min\big{\\{}u^{i\ast}(T,x_{0})-h_{i}(x_{0});\hskip
5.69046ptu^{i\ast}(T,x_{0})-\max\limits_{j\in\mathcal{I}^{-i}}\left(u^{j\ast}-g_{ij}\right)^{\ast}(T,x_{0})\big{\\}}=2\varepsilon>0.$
(4.33)

Let $(t_{k},x_{k})$ be a sequence in $[0,T)\times\mathbb{R}^{k}$ such that

$(t_{k},x_{k})\rightarrow(T,x_{0})\hskip 8.5359pt\text{and}\hskip
8.5359ptu^{i}(t_{k},x_{k})\rightarrow u^{i\ast}(T,x_{0})\hskip
8.5359pt\text{as}\hskip 5.69046ptk\rightarrow\infty.$ (4.34)

Since $u^{i\ast}$ is u.s.c and of polynomial growth, we can find a sequence
$\left(\varphi_{n}\right)_{n\geq 0}$ of functions of
$C^{1,2}([0,T]\times\mathbb{R}^{k})$ and neighborhood $B_{n}$ of $(T,x_{0})$
such that $\varphi^{n}\rightarrow u^{i\ast}$, and hence from the inequality
(4.33) we have

$\min\big{\\{}\varphi^{n}(t,x)-h_{i}(x);\hskip
5.69046pt\varphi^{n}(t,x)-\max\limits_{j\in\mathcal{I}^{-i}}\left(u^{j\ast}-g_{ij}\right)^{\ast}(t,x)\big{\\}}\geq\varepsilon\hskip
8.5359pt\text{for all}\hskip 5.69046pt(t,x)\in B_{n},$ (4.35)

for $n$ large enough. On the other hand, after possibly passing to a sub-
sequence of $(t_{k},x_{k})_{k\geq 1}$ we can assume that the previous
inequality holds on $B^{k}_{n}:=[t_{k},T]\times B(x_{k},\delta^{k}_{n})$ for
some $\delta^{k}_{n}\in(0,1)$ small enough in such a way that
$B^{k}_{n}\subset B_{n}$. Since $u^{i\ast}$ is locally bounded (recall it has
polynomial growth), there exists $\zeta>0$ such that
$\left|u^{i\ast}\right|\leq\zeta$ on $B_{n}$. We can then assume that
$\varphi^{n}\geq-2\zeta$ on $B_{n}$. Next we define

$\tilde{\varphi}^{n}_{k}(t,x):=\varphi^{n}(t,x)+\frac{4\zeta\left|x-x_{k}\right|^{2}}{(\delta^{k}_{n})^{2}}+\sqrt{T-t}.$

Note that $\tilde{\varphi}^{n}_{k}\geq\varphi^{n}$ and

$\left(u^{i\ast}-\tilde{\varphi}^{n}_{k}\right)(t,x)\leq-\zeta\hskip
8.5359pt\text{for}\hskip 5.69046pt(t,x)\in[t_{k},T]\times\partial
B(x_{k},\delta^{k}_{n}).$ (4.36)

Since $\partial_{t}(\sqrt{T-t})\rightarrow-\infty$ as $t\rightarrow T$, we can
choose $t_{k}$ close enough to $T$ to ensure that

$-\left(\partial_{t}+\mathcal{L}\right)\tilde{\varphi}^{n}_{k}(t,x)\geq
0\hskip 8.5359pt\text{on}\hskip 8.5359ptB^{k}_{n}.$ (4.37)

Next let us consider the following stopping times

$\theta^{k}_{n}:=\inf\\{s\geq t_{k}:(s,X^{t_{k},x_{k}}_{s})\in
B^{k^{c}}_{n}\\}\wedge T\hskip 8.5359pt$ (4.38)

and

$\vartheta^{\varepsilon}_{k}:=\inf\\{s\geq
t_{k},u^{i}(s,X^{t_{k},x_{k}}_{s})\leq\max_{j\in\mathcal{I}^{-i}}(u^{j}(s,X^{t_{k},x_{k}}_{s})-g_{ij}(s,X^{t_{k},x_{k}}_{s}))+\frac{\varepsilon}{4}\\}\wedge
T$ (4.39)

where $B^{k^{c}}_{n}$ is the complement of $B^{k}_{n}$.

First note that for a subsequence $\\{k\\}$,
$\mathbb{P}[\vartheta^{\varepsilon}_{k}>t_{k}]=1$. Actually from (4.33), we
have

$\displaystyle u^{i\ast}(T,x_{0})$
$\displaystyle\geq\max\limits_{j\in\mathcal{I}^{-i}}\left(u^{j\ast}-g_{ij}\right)^{\ast}(T,x_{0})+2\varepsilon$
$\displaystyle\geq\max\limits_{j\in\mathcal{I}^{-i}}\left(u^{j}-g_{ij}\right)^{\ast}(T,x_{0})+2\varepsilon.$

Therefore taking into account of (4.34), at least for a subsequence, for any
$k\geq 1$,

$\displaystyle
u^{i\ast}(t_{k},x_{k})\geq\max\limits_{j\in\mathcal{I}^{-i}}\left(u^{j}-g_{ij}\right)(t_{k},x_{k})+\varepsilon.$

Now let us stick to this subsequence. If
$\mathbb{P}[\vartheta^{\varepsilon}_{k}=t_{k}]>0$, then by the càdlàg property
of the processes which define $\vartheta^{\varepsilon}_{k}$ we have
$u^{i}({t_{k},x_{k}})\leq\max_{j\in\mathcal{I}^{-i}}(u^{j}({t_{k},x_{k}})-g_{ij}({t_{k},x_{k}}))+\frac{\varepsilon}{4}$,
which contradicts the previous inequality and then the claim is valid.

On the other hand the property which characterizes the jumps of $Y^{i}$ in the
definition (3.1), implies that on $[t_{k},\vartheta^{\varepsilon}_{k}]$ the
process $Y^{i}$ is continuous and $dK^{i}_{s}=0$ for
$s\in[t_{k},\vartheta^{\varepsilon}_{k}]$. Applying now Itô’s formula to the
process
$(\tilde{\varphi}^{n}_{k}(s,X_{s}))_{s\in[t_{k},\theta^{k}_{n}\wedge\vartheta^{\varepsilon}_{k}]}$
and taking expectation, we obtain

$\begin{array}[]{ll}\tilde{\varphi}^{n}_{k}(t_{k},x_{k})=\mathbb{E}\bigg{[}\tilde{\varphi}^{n}_{k}(\theta^{k}_{n}\wedge\vartheta^{\varepsilon}_{k},X^{t_{k},x_{k}}_{\theta^{k}_{n}\wedge\vartheta^{\varepsilon}_{k}})-\displaystyle\int^{\theta^{k}_{n}\wedge\vartheta^{\varepsilon}_{k}}_{t_{k}}\left(\partial_{t}+\mathcal{L}\right)\tilde{\varphi}^{n}_{k}(s,X^{t_{k},x_{k}}_{s})ds\bigg{]}&\\\
&\\\ \hskip
31.2982pt\geq\mathbb{E}\big{[}\tilde{\varphi}^{n}_{k}(\theta^{k}_{n},X^{t_{k},x_{k}}_{\theta^{k}_{n}})\mathbf{1}_{[\theta^{k}_{n}\leq\vartheta^{\varepsilon}_{k}]}+\tilde{\varphi}^{n}_{k}(\vartheta^{\varepsilon}_{k},X^{t_{k},x_{k}}_{\vartheta^{\varepsilon}_{k}})\mathbf{1}_{[\theta^{k}_{n}>\vartheta^{\varepsilon}_{k}]}\big{]}\hskip
28.45274pt\text{by (\ref{T3})}&\\\ &\\\ \hskip
31.2982pt=\mathbb{E}\big{[}\big{\\{}\tilde{\varphi}^{n}_{k}(\theta^{k}_{n},X^{t_{k},x_{k}}_{\theta^{k}_{n}})\mathbf{1}_{[\theta^{k}_{n}<T]}+\tilde{\varphi}^{n}_{k}(\theta^{k}_{n},X^{t_{k},x_{k}}_{\theta^{k}_{n}})\mathbf{1}_{[\theta^{k}_{n}=T]}\big{\\}}\mathbf{1}_{[\theta^{k}_{n}\leq\vartheta^{\varepsilon}_{k}]}+\tilde{\varphi}^{n}_{k}(\vartheta^{\varepsilon}_{k},X^{t_{k},x_{k}}_{\vartheta^{\varepsilon}_{k}})\mathbf{1}_{[\theta^{k}_{n}>\vartheta^{\varepsilon}_{k}]}\big{]}&\\\
&\\\ \hskip
31.2982pt\geq\mathbb{E}\Big{[}\big{\\{}\big{(}u^{i\ast}(\theta^{k}_{n},X^{t_{k},x_{k}}_{\theta^{k}_{n}})+\zeta\big{)}\mathbf{1}_{[\theta^{k}_{n}<T]}+\big{(}h_{i}(T,X^{t_{k},x_{k}}_{T})+\varepsilon\big{)}\mathbf{1}_{[\theta^{k}_{n}=T]}\big{\\}}\mathbf{1}_{[\theta^{k}_{n}\leq\vartheta^{\varepsilon}_{k}]}&\\\
&\\\ \hskip
76.82234pt+\Big{\\{}\max\limits_{j\in\mathcal{I}^{-i}}\big{(}u^{j\ast}-g_{ij}\big{)}^{\ast}(\vartheta^{\varepsilon}_{k},X^{t_{k},x_{k}}_{\vartheta^{\varepsilon}_{k}})+\varepsilon\Big{\\}}\mathbf{1}_{[\theta^{k}_{n}>\vartheta^{\varepsilon}_{k}]}\Big{]}\hskip
28.45274pt\text{by (\ref{T2}) and (\ref{T1})}&\\\ &\\\ \hskip
31.2982pt\geq\mathbb{E}\Big{[}\big{\\{}\big{(}u^{i}(\theta^{k}_{n},X^{t_{k},x_{k}}_{\theta^{k}_{n}})+\zeta\big{)}\mathbf{1}_{[\theta^{k}_{n}<T]}+\big{(}h_{i}(T,X^{t_{k},x_{k}}_{T})+\varepsilon\big{)}\mathbf{1}_{[\theta^{k}_{n}=T]}\big{\\}}\mathbf{1}_{[\theta^{k}_{n}\leq\vartheta^{\varepsilon}_{k}]}&\\\
&\\\ \hskip
76.82234pt+\Big{\\{}\max\limits_{j\in\mathcal{I}^{-i}}\big{(}u^{j}(\vartheta^{\varepsilon}_{k},X^{t_{k},x_{k}}_{\vartheta^{\varepsilon}_{k}})-g_{ij}(\vartheta^{\varepsilon}_{k},X^{t_{k},x_{k}}_{\vartheta^{\varepsilon}_{k}})\big{)}+\varepsilon\Big{\\}}\mathbf{1}_{[\theta^{k}_{n}>\vartheta^{\varepsilon}_{k}]}\Big{]}\end{array}$

$\begin{array}[]{ll}\hskip
31.2982pt\geq\mathbb{E}\Big{[}\big{\\{}\big{(}u^{i}(\theta^{k}_{n},X^{t_{k},x_{k}}_{\theta^{k}_{n}})+\zeta\big{)}\mathbf{1}_{[\theta^{k}_{n}<T]}+\big{(}h_{i}(T,X^{t_{k},x_{k}}_{T})+\varepsilon\big{)}\mathbf{1}_{[\theta^{k}_{n}=T]}\big{\\}}\mathbf{1}_{[\theta^{k}_{n}\leq\vartheta^{\varepsilon}_{k}]}+\big{\\{}u^{i}(\vartheta^{\varepsilon}_{k},X^{t_{k},x_{k}}_{\vartheta^{\varepsilon}_{k}})+\frac{3\varepsilon}{4}\big{\\}}\mathbf{1}_{[\theta^{k}_{n}>\vartheta^{\varepsilon}_{k}]}\Big{]}\hskip
28.45274pt\text{by (\ref{stoppingtime2})}&\\\ &\\\ \hskip
31.2982pt\geq\mathbb{E}\big{[}{u^{i}(\theta^{k}_{n}\wedge\vartheta^{\varepsilon}_{k},X^{t_{k},x_{k}}_{\theta^{k}_{n}\wedge\vartheta^{\varepsilon}_{k}})}\big{]}+\left(\zeta\wedge\frac{3\varepsilon}{4}\right)&\\\
&\\\ \hskip
31.2982pt=\mathbb{E}\big{[}u^{i}(t_{k},x_{k})\big{]}-\mathbb{E}\bigg{[}\displaystyle\int^{\theta^{k}_{n}\wedge\vartheta^{\varepsilon}_{k}}_{t_{k}}f_{i}(s,X^{t_{k},x_{k}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{k},x_{k}}_{s}),Z^{i,t_{k},x_{k}}_{s})ds\bigg{]}+\left(\zeta\wedge\frac{3\varepsilon}{4}\right)\end{array}$
(4.40)

where the last equality is due to the fact that the process
$Y^{i}_{.}=u^{i}(\cdot,X_{.})$, stopped at time
$\theta^{k}_{n}\wedge\vartheta_{k}$, solves a RBSDE system of the type (3.1)
with data given by
$((f_{i})_{i\in\mathcal{I}},(h_{i})_{i\in\mathcal{I}},(g_{ij})_{i\in\mathcal{I}})$,
and the last inequality is obtained by monotonicity property of $f_{i}$ and
since $u^{j\ast}\geq u^{j}$ for $j\in\mathcal{I}^{-i}$. Besides, note that by
definition of $\theta^{k}_{n}\wedge\vartheta_{k}$ we have $dK^{i,t,x}=0$ on
$[t_{k},\vartheta^{\varepsilon}_{k}]$. Next, we have that both
$(u^{j})_{j=1,\ldots,m}$ and
$(t,x)\rightarrow\big{\|}Z^{i,t,x}_{\cdot}\big{\|}_{\mathcal{H}^{2,d}}(t,x)$
are of polynomial growth. Thus by Assumption (H2)-(i),(iii) and inequality
(4.4) we deduce that

$\lim\limits_{k\rightarrow\infty}\mathbb{E}\bigg{[}\displaystyle\int^{\theta^{k}_{n}\wedge\vartheta^{\varepsilon}_{k}}_{t_{k}}f_{i}(s,X^{t_{k},x_{k}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{k},x_{k}}_{s}),Z^{i,t_{k},x_{k}}_{s})ds\bigg{]}=0,$
(4.41)

and hence taking the limit in both hand sides of the inequality (4) as
$k\rightarrow\infty$ yields

$\begin{array}[]{ll}\varphi^{n}(T,x_{0})=\lim\limits_{k\rightarrow\infty}\left[\varphi^{n}(t_{k},x_{k})+\sqrt{T-t_{k}}\right]=\lim\limits_{k\rightarrow\infty}\tilde{\varphi}^{n}_{k}(t_{k},x_{k})&\\\
&\\\ \hskip
88.2037pt\geq\lim\limits_{k\rightarrow\infty}u^{i}(t_{k},x_{k})+\left(\zeta\wedge\frac{3\varepsilon}{4}\right)=u^{i\ast}(T,x_{0})+\left(\zeta\wedge\frac{3\varepsilon}{4}\right).\end{array}$
(4.42)

Therefore, taking $n$ large enough and recalling that $\varphi^{n}\rightarrow
u^{i\ast}$ pointwisely, we get a contradiction. Thus for any
$x\in\mathbb{R}^{k}$ and $i\in\mathcal{I}$ we have

$\min\Big{\\{}u^{i\ast}(T,x)-h_{i}(x);\hskip
5.69046ptu^{i\ast}(T,x)-\max\limits_{j\in\mathcal{I}^{-i}}\left(u^{j\ast}-g_{ij}\right)^{\ast}(T,x)\Big{\\}}\leq
0.$ (4.43)

which is the claim.

Step 4: Supersolution property at $(T,x_{0})$.

We are going to show that

$\min\Big{\\{}u^{i}_{\ast}(T,x_{0})-h_{i}(x_{0});\hskip
5.69046ptu^{i}_{\ast}(T,x_{0})-\big{(}\max\limits_{j\in\mathcal{I}^{-i}}\left(u^{j}_{\ast}(T,x_{0})-g_{ij}(T,x_{0})\right)\big{)}_{\ast}\Big{\\}}\geq
0.$ (4.44)

Let $(t_{k},x_{k})_{k\geq 1}$ be a sequence in $[0,T)\times\mathbb{R}^{d}$
such that

$(t_{k},x_{k})\rightarrow(T,x_{0})\hskip 5.69046pt\text{and}\hskip
5.69046ptu^{i}(t_{k},x_{k})\rightarrow u^{i}_{\ast}(T,x_{0})\hskip
11.38092pt\text{as}\hskip 11.38092ptk\rightarrow\infty.$ (4.45)

Since $u^{i}(t,x)$ is deterministic, we have from the definition of $u^{i}$
that

$\begin{array}[]{ll}u^{i}(t_{k},x_{k})=\mathbb{E}\bigg{[}h_{i}(X^{t_{k},x_{k}}_{T})+\displaystyle\int^{T}_{t_{k}}f_{i}(s,X^{t_{k},x_{k}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{k},x_{k}}_{s}),Z^{i,t_{k},x_{k}}_{s})ds+K^{i}_{T}-K^{i}_{t_{k}}\bigg{]}&\\\
&\\\ \hskip
48.36958pt\geq\mathbb{E}\bigg{[}h_{i}(X^{t_{k},x_{k}}_{T})+\displaystyle\int^{T}_{t_{k}}f_{i}(s,X^{t_{k},x_{k}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{k},x_{k}}_{s}),Z^{i,t_{k},x_{k}}_{s})ds\bigg{]}\end{array}$
(4.46)

where we have used that $dK^{i,t,x}\geq 0$ on $[t_{k},T]$. Next taking the
limit in both hand sides as $k\rightarrow\infty$, using that $h_{i}$ is
continuous and arguing similarly to (4.41) we have

$\displaystyle u^{i}_{\ast}(T,x_{0})$
$\displaystyle\geq\lim_{k\rightarrow\infty}\mathbb{E}\bigg{[}h_{i}(X^{t_{k},x_{k}}_{T})+\int^{T}_{t_{k}}f_{i}(s,X^{t_{k},x_{k}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{k},x_{k}}_{s}),Z^{i,t_{k},x_{k}}_{s})ds\bigg{]}$
$\displaystyle=\mathbb{E}\big{[}h_{i}(X^{T,x_{0}}_{T})\big{]}=h_{i}(x_{0}),$

that is, $u^{i}_{\ast}(T,x_{0})\geq h_{i}(x_{0})$. On the other hand, setting
$\tau_{k}=(T+t_{k})/2$, considering the RBSDE (3.1) on $[t_{k},\tau_{k}]$,
taking expectation to obtain

$\begin{array}[]{ll}u^{i}(t_{k},x_{k})\geq\mathbb{E}\bigg{[}u^{i}(\tau_{k},X^{t_{k},x_{k}}_{\tau_{k}})+\displaystyle\int^{\tau_{k}}_{t_{k}}f_{i}(s,X^{t_{k},x_{k}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{k},x_{k}}_{s}),Z^{i,t_{k},x_{k}}_{s})ds\bigg{]}&\\\
&\\\ \hskip
45.52458pt\geq\mathbb{E}\bigg{[}\max\limits_{j\in\mathcal{I}^{-i}}\big{(}u^{j}(\tau_{k},X^{t_{k},x_{k}}_{\tau_{k}})-g_{ij}(\tau_{k},X^{t_{k},x_{k}}_{\tau_{k}})\big{)}+\int^{\tau_{k}}_{t_{k}}f_{i}(s,X^{t_{k},x_{k}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{k},x_{k}}_{s}),Z^{i,t_{k},x_{k}}_{s})ds\bigg{]}&\\\
&\\\ \hskip
45.52458pt\geq\mathbb{E}\bigg{[}(\max\limits_{j\in\mathcal{I}^{-i}}\big{(}u^{j}_{*}-g_{ij})\big{)}_{*}(\tau_{k},X^{t_{k},x_{k}}_{\tau_{k}})+\int^{\tau_{k}}_{t_{k}}f_{i}(s,X^{t_{k},x_{k}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{k},x_{k}}_{s}),Z^{i,t_{k},x_{k}}_{s})ds\bigg{]}\end{array}$
(4.47)

since $dK^{i,t,x}\geq 0$ and
$u^{i}(\tau_{k},X^{t_{k},x_{k}}_{\tau_{k}})\geq\max\limits_{j\in\mathcal{I}^{-i}}\big{(}u^{j}(\tau_{k},X^{t_{k},x_{k}}_{\tau_{k}})-g_{ij}(\tau_{k},X^{t_{k},x_{k}}_{\tau_{k}})\big{)}$.
It implies that

$\displaystyle\lim_{k\rightarrow\infty}u^{i}(t_{k},x_{k})$
$\displaystyle\geq\liminf_{k\rightarrow\infty}\mathbb{E}\bigg{[}(\max\limits_{j\in\mathcal{I}^{-i}}\big{(}u^{j}_{*}-g_{ij})\big{)}_{*}(\tau_{k},X^{t_{k},x_{k}}_{\tau_{k}})$
$\displaystyle\hskip
56.9055pt+\int^{\tau_{k}}_{t_{k}}f_{i}(s,X^{t_{k},x_{k}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{k},x_{k}}_{s}),Z^{i,t_{k},x_{k}}_{s})ds\bigg{]}$
$\displaystyle\geq\mathbb{E}\bigg{[}\liminf_{k}(\max\limits_{j\in\mathcal{I}^{-i}}\big{(}u^{j}_{*}-g_{ij})\big{)}_{*}(\tau_{k},X^{t_{k},x_{k}}_{\tau_{k}})$
$\displaystyle\hskip
56.9055pt+\int^{\tau_{k}}_{t_{k}}f_{i}(s,X^{t_{k},x_{k}}_{s},(u^{k})_{k=1,\dots,q}(s,X^{t_{k},x_{k}}_{s}),Z^{i,t_{k},x_{k}}_{s})ds\bigg{]}$
$\displaystyle\geq(\max\limits_{j\in\mathcal{I}^{-i}}\big{(}u^{j}_{*}-g_{ij})\big{)}_{*}(t_{k},x_{k}).$

The second inequality stems from Fatou’s Lemma while the third one is due to
the fact that
$(\max\limits_{j\in\mathcal{I}^{-i}}\big{(}u^{j}_{*}-g_{ij})\big{)}_{*}$ is
lower semicontinuous and by (4.45), at least for a subsequence,
$((\tau_{k},X^{t_{k},x_{k}}_{\tau_{k}}))_{k}\rightarrow(T,x)$
$\mathbb{P}-a.s.$. Thus

$\min\Big{\\{}u^{i}_{\ast}(T,x_{0})-h_{i}(x_{0});\hskip
5.69046ptu^{i}_{\ast}(T,x_{0})-\big{(}\max\limits_{j\in\mathcal{I}^{-i}}\left(u^{j}_{\ast}(T,x_{0})-g_{ij}(T,x_{0})\right)\big{)}_{\ast}\Big{\\}}\geq
0$ (4.48)

which is the claim. The proof is now complete. $\blacksquare$

######  Remark 4.10.

If the switching costs $g_{ij}$ are continuous, conditions (4.10) and (4.9),
read respectively as:

$\min\big{\\{}v^{i\ast}(T,x_{0})-h_{i}(x_{0});\hskip
5.69046ptu^{i\ast}(T,x_{0})-\max\limits_{j\in\mathcal{I}^{-i}}\left(v^{j\ast}-g_{ij}\right)(T,x_{0})\big{\\}}\leq
0$

and

$\min\Big{\\{}v^{i}_{\ast}(T,x_{0})-h_{i}(x_{0});\hskip
5.69046ptv^{i}_{\ast}(T,x_{0})-\big{(}\max\limits_{j\in\mathcal{I}^{-i}}\left(v^{j}_{\ast}-g_{ij}\right)\big{)}(T,x_{0})\Big{\\}}\geq
0.$

Therefore $v^{i}_{\ast}(T,x_{0})\geq h_{i}(x_{0})$ and by the non free-loop
property one deduces that $v^{i\ast}(T,x_{0})\leq h_{i}(x_{0})$ which implies
that $v^{i}(T,x_{0})=h_{i}(x_{0})$. For more details one can see e.g. Hamadène
and Morlais [17]. $\blacksquare$

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