Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem without monotonicity condition

Abstract

We show the existence and uniqueness of a continuous viscosity solution of a system of partial differential equations (PDEs for short) without assuming the usual monotonicity conditions on the driver function as in Hamadène and Morlais’s article [Applied Mathematics & Optimization 67 (2013), 163–196]. Our method strongly relies on the link between PDEs and reflected backward stochastic differential equations with interconnected obstacles for which we already know that the solution exists and is unique for general drivers.

Keywords

Partial differential equations interconnected obstacles viscosity solution multi-modes switching HJB system reflected backward stochastic differential equations

1. Introduction

The main objective of this paper is to study the problem of existence and uniqueness of a solution in viscosity sense ${(u^{i})}_{i = 1, m}$ of the following system of partial differential equations with obstacles which depend on the solution: $\forall i \in I : = {1, \dots, m}$ , $\begin{matrix} (1.1) & \{\begin{matrix} min {u^{i} (t, x) - {max}_{j \in I^{- i}} {u^{j} (t, x) - g_{i j} (t, x)}; \\ - \partial_{t} u^{i} (t, x) - L u^{i} (t, x) - f_{i} (t, x, {(u^{k} (t, x))}_{k = 1, \dots, m}, (σ^{⊤} D_{x} u^{i}) (t, x))} = 0; \\ u^{i} (T, x) = h_{i} (x) \end{matrix} \end{matrix}$ where $I^{- i} : = I - {i}$ and $L$ is an infinitesimal generator which has the following form $\begin{matrix} (1.2) & L φ (t, x) : = b {(t, x)}^{⊤} . D_{x} φ (t, x) + \frac{1}{2} Tr [(σ σ^{⊤}) (t, x) D_{x x}^{2} φ (t, x)] \end{matrix}$ and which is associated with the stochastic process $X^{t, x}$ solution of the SDE (1.3).

As pointed out previously, in (1.1), the obstable of $u^{i}$ is the function ${max}_{j \in I^{- i}} {u^{j} (t, x) - g_{i j} (t, x)}$ which actually depends on the solution ${(u^{i})}_{i = 1, m}$ , which means that the obstacles are interconnected.

This problem is related to the optimal stochastic switching control problem which can be described, through an example, as follows: Let us consider a power plant which has several modes of production and which the manager puts in a specific mode according to its profitability which depends on the electricity price in the energy market evolving according to the following stochastic differential equation $\begin{matrix} (1.3) & d X_{s}^{t, x} = b (s, X_{s}^{t, x}) d s + σ (s, X_{s}^{t, x}) d B_{s}, s ⩾ t and X_{t}^{t, x} = x . \end{matrix}$ The aim of the manager is to maximize her global profit over an horizon $[0, T]$ by optimally choosing controls of the form $δ : = {(θ_{k}, α_{k})}_{k ⩾ 0}$ where ${(θ_{k})}_{k ⩾ 0}$ is an increasing sequence of stopping times at which the manager switches the system across the different operating modes and ${(α_{k})}_{k ⩾ 0}$ is a sequence of random variables with values in ${1, \dots, m}$ which stand for the modes to which the production is switched. Namely for any $k ⩾ 1$ , at $θ_{k}$ , the manager switches the production from $θ_{k - 1}$ to $θ_{k}$ ( $θ_{0}$ and $α_{0}$ are the starting time and mode respectively). However, switching the plant from the mode i to the mode j is not free. It generates expenditures, which amount to $g_{i j} (s, X_{s}^{t, x})$ at time s. When the plant is run under a strategy δ, its yield is given by $\begin{matrix} J (δ; t, x) : = E [\int_{t}^{T} f^{δ} (s, X_{s}^{t, x}) d s - A_{T}^{δ} + h^{δ} (X_{T}^{t, x})] \end{matrix}$ where:

$f^{δ} (s, X_{s}^{t, x})$ is the instantaneous payoff of the plant when run under δ and $h^{δ} (X_{T}^{t, x})$ is the terminal payoff;

the quantity $A_{T}^{δ}$ stands for the total switching cost when the strategy δ is implemented.

The problem is to find an optimal management strategy

δ^{*}

, i.e., which satisfies

J (δ^{*}; t, x) = {sup}_{δ} J (δ; t, x)

. This latter quantity is nothing but the fair price of the power plant in the energy market.

In (1.1), if for any $i \in I$ , $f_{i}$ does not depend on ${(u^{k})}_{k = 1, m}$ and $D_{x} u^{i}$ , the system reduces to the Hamilton–Jacobi–Bellman one associated with the switching problem and it is shown in [4,7], etc. that it has a unique solution ${(u^{i})}_{i = 1, m}$ which satisfies $\begin{matrix} u^{i} (t, x) = sup {J (δ; t, x), δ \in A_{t}^{i}}, \end{matrix}$ where $A_{t}^{i}$ is the set of admissible strategies which from mode i at time t.

The main tool to tackle system (1.1) is to use with the following system of reflected backward stochastic differential equations (RBSDEs for short) with interconnected obstacles: $\forall i \in {1, \dots, m}$ and $s \in [t, T]$ , $\begin{matrix} (1.4) & \{\begin{matrix} Y^{i, t, x} is continuous, \\ Y_{s}^{i, t, x} = h_{i} (X_{T}^{t, x}) + \int_{s}^{T} f_{i} (r, X_{r}^{t, x}, {(Y_{r}^{k, t, x})}_{k = 1, \dots, m}, Z_{r}^{i, t, x}) d r \\ + K_{T}^{i, t, x} - K_{s}^{i, t, x} - \int_{s}^{T} Z_{r}^{i, t, x} d B_{r}, \\ Y_{s}^{i, t, x} ⩾ {max}_{j \in I^{- i}} (Y_{s}^{j, t, x} - g_{i j} (s, X_{s}^{t, x})), \\ \int_{t}^{T} [Y_{s}^{i, t, x} - {max}_{j \in I^{- i}} (Y_{s}^{j, t, x} - g_{i j} (s, X_{s}^{t, x}))] d K_{s}^{i, t, x} = 0 . \end{matrix} \end{matrix}$ Note that the generators ${(f_{i})}_{i \in I}$ of the RBSDE system (1.4) have a general form, i.e., depend on ${(y_{i})}_{i \in I}$ and $z_{i}$ . More precisely, this system can be related to switching problems with utility functions, knightian uncertainty, recursive utilities, i.e., the present utility depends also on the future utility, etc. The notion of recursive utility was first introduced by Duffie and Epstein (see [3]) to allow a separation between risk aversion and intertemporal substitution. In 1997, El Karoui et al. (see e.g. [5]) considered the case when the standard generators $f_{i}$ can depend on $z_{i}$ .

This system of RBSDEs has been investigated in several papers including ([1,7–9], etc.). In [1], the authors proved that it has a unique solution ${(Y^{i, t, x}, Z^{i, t, x}, K^{i, t, x})}_{i \in I}$ if the functions ${(f_{i})}_{i \in I}$ are merely Lipschitz w.r.t. $({(y_{l})}_{l = 1, m}, z)$ .

Concerning now the system of PDEs (1.1), Hamadene et al. proved in [7] that, additionally to the Lipschitz property mentioned above, if for any $i \in I$ and $k \in I^{- i}$ , $f_{i} (t, x, {(y_{l})}_{l = 1, m}, z)$ is non-decreasing w.r.t. $y_{k}$ (see assumption (H4)-(i) below), then system (1.4) has a unique solution ${(u^{i})}_{i = 1, m}$ in the class of continuous functions with polynomial growth and which is given by: $\begin{matrix} (1.5) & \forall i \in I, u^{i} (t, x) = Y_{t}^{i, t, x}, (t, x) \in [0, T] \times R^{k}, \end{matrix}$ where ${(Y^{i, t, x})}_{i \in I}$ is the first component of the solution of the system of reflected BSDEs (1.4). The same result is obtained if, instead of ${(f_{i})}_{i \in I}$ , their opposites ${(- f_{i})}_{i \in I}$ verify the previous monotonicity property (see (H4)-(ii)). However without assuming one of either monotonicity conditions on the drivers ${(f_{i})}_{i \in I}$ , the problem of existence and uniqueness of the solution in viscosity sense of system (1.1) remains open. In this paper, we show that system (1.1) has a unique solution without assuming the previous monotonicity properties on the drivers ${(f_{i})}_{i \in I}$ . This is the main novelty of this work. As a consequence, we fill in the gap between the probabilistic framework and the PDEs one. Our method relies on the link between reflected BSDEs and PDEs with obstacles in the Markovian framework of randomness.

The paper is organized as follows. In Section 2, we formulate accurately the problem. In Section 3, we show that Feynman–Kac formula holds for the components ${(Y^{i; t, x})}_{i \in I}$ of the solution of (1.4), i.e., the representation (1.5) holds true. In Section 4, we show that the functions ${(u^{i})}_{i \in I}$ are continuous and are the unique viscosity solution of (1.1) in the class of functions with polynomial growth. The proof is deeply related to the fact that system (1.4) of RBSDEs has a unique solution.

2. Preliminaries and notations

Let $T > 0$ be a given time horizon and $(Ω, F, P)$ be a probability space on which is defined a standard d-dimensional Brownian motion $B = {(B_{t})}_{t ⩽ T}$ whose natural filtration is ${(F_{t}^{0} : = σ (B_{s}, s ⩽ t))}_{t ⩽ T}$ and $F = {(F_{t})}_{0 ⩽ t ⩽ T}$ is its augmentation with the $P$ -null sets of $F$ . Then ${(F_{t})}_{0 ⩽ t ⩽ T}$ is right continuous and complete.

We now introduce the following spaces:

$P$ is the σ-algebra of $F$ -progressively measurable sets on $[0, T] \times Ω$ ;

$S^{2}$ is the set of $P$ -measurable, continuous, $R$ -valued processes $Y = {(Y_{s})}_{s ⩽ T}$ such that $E [{sup}_{s ⩽ T} | Y_{s} |^{2}] < \infty$ ;

$A^{2}$ is the subset of $S^{2}$ of non decreasing processes $K = {(K_{t})}_{t ⩽ T}$ such that $K_{0} = 0$ ;

$H^{2, l}$ ( $l ⩾ 1$ ) is the set of $P$ -measurable and $R^{l}$ -valued processes $Z : = {(Z_{s})}_{s ⩽ T}$ such that $E [\int_{0}^{T} | Z_{s} |_{R^{l}}^{2} d s] < \infty$ .

Next, for any given $(t, x) \in [0, T] \times R^{k}$ (k is a positive integer), we consider the following standard stochastic differential equation (SDE): $\begin{matrix} (2.1) & \{\begin{matrix} d X_{s}^{t, x} = b (s, X_{s}^{t, x}) d s + σ (s, X_{s}^{t, x}) d B_{s}, & s \in [t, T] \\ X_{s}^{t, x} = x, & 0 ⩽ s ⩽ t \end{matrix} \end{matrix}$ where $b : [0, T] \times R^{k} \to R^{k}$ and $σ : [0, T] \times R^{k} \to R^{k \times d}$ are two continuous functions and Lipschitz w.r.t. x, i.e., there exists a positive constant C such that $\begin{matrix} (2.2) & | b (t, x) - b (t, x^{'}) | + | σ (t, x) - σ (t, x^{'}) | ⩽ C | x - x^{'} |, \forall (t, x, x^{'}) \in [0, T] \times R^{k + k} . \end{matrix}$ Note that the continuity of b, σ and (2.2) imply the existence of a constant C such that $\begin{matrix} (2.3) & | b (t, x) | + | σ (t, x) | ⩽ C (1 + | x |), \forall (t, x) \in [0, T] \times R^{k} . \end{matrix}$ Conditions (2.2) and (2.3) ensure, for any $(t, x) \in [0, T] \times R^{k}$ , the existence and uniqueness of a solution ${X_{s}^{t, x}, t ⩽ s ⩽ T}$ to the SDE (2.1) (see [11] for more details). Moreover, it satisfies the following estimate: $\forall p ⩾ 1$ , $\begin{matrix} (2.4) & E [sup_{s ⩽ T} {| X_{s}^{t, x} |}^{p}] ⩽ C (1 + | x |^{p}) . \end{matrix}$

Next let us introduce the following deterministic functions ${(f_{i})}_{i = 1, \dots, m}$ , ${(h_{i})}_{i = 1, \dots, m}$ and ${(g_{i j})}_{i, j = 1, \dots, m}$ defined as follows: for any $i, j \in {1, \dots, m}$ ,

$f_{i} : (t, x, \vec{y}, z) \in [0, T] \times R^{k + m + d} ⟼ f_{i} (t, x, \vec{y}, z) \in R$ ( $\vec{y} : = (y^{1}, \dots, y^{m})$ );

$g_{i j} : (t, x) \in [0, T] \times R^{k} ⟼ g_{i j} (t, x) \in R$ ;

$h_{i} : x \in R^{k} ⟼ h_{i} (x) \in R$ .

Additionally we assume that they satisfy:

(H1)

For any $i \in {1, \dots, m}$ ,

The function $(t, x) \mapsto f_{i} (t, x, \vec{y}, z)$ is continuous, uniformly w.r.t. the variables $(\vec{y}, z)$ .

The function $f_{i}$ is Lipschitz continuous with respect to the variables $(\vec{y}, z)$ uniformly in $(t, x)$ , i.e., there exists a positive constant $C_{i}$ such that for any $(t, x) \in [0, T] \times R^{k}$ , $(\vec{y}, z)$ and $({\vec{y}}_{1}, z_{1})$ elements of $R^{m + d}$ : $\begin{matrix} (2.5) & | f_{i} (t, x, \vec{y}, z) - f_{i} (t, x, {\vec{y}}_{1}, z_{1}) | ⩽ C_{i} (| \vec{y} - {\vec{y}}_{1} | + | z - z_{1} |) . \end{matrix}$

The mapping $(t, x) \mapsto f_{i} (t, x, 0, \dots, 0)$ has polynomial growth in x, i.e., there exist two constants $C > 0$ and $p ⩾ 1$ such that for any $(t, x) \in [0, T] \times R^{k}$ , $\begin{matrix} (2.6) & | f_{i} (t, x, 0, \dots, 0) | ⩽ C (1 + | x |^{p}) . \end{matrix}$

(H2)

For all $i, j \in {1, \dots, m}$ , $g_{i i} = 0$ , and if $i \neq j$ then $g_{i j} (t, x)$ is non-negative, jointly continuous in $(t, x)$ with polynomial growth and satisfy the following non free loop property:

For any $(t, x) \in [0, T] \times R^{k}$ , for any sequence of indices $i_{1}, \dots, i_{k}$ such that $i_{1} = i_{k}$ and $card {i_{1}, \dots, i_{k}} = k - 1$ ( $k ⩾ 3$ ) we have $\begin{matrix} (2.7) & g_{i_{1} i_{2}} (t, x) + g_{i_{2} i_{3}} (t, x) + \dots + g_{i_{k - 1} i_{k}} (t, x) > 0 . \end{matrix}$

(H3)

For $i \in {1, \dots, m}$ , the function $h_{i}$ , which stands for the terminal condition, is continuous with polynomial growth and satisfies the following consistency condition: $\begin{matrix} (2.8) & \forall x \in R^{k}, h_{i} (x) ⩾ max_{j \in I^{- i}} {h_{j} (x) - g_{i j} (T, x)} . \end{matrix}$

(H4)-(i)

For any $i \in I$ and $j \in I^{- i}$ , the mapping $w \in R ⟼ f_{i} (t, x, y^{1}, \dots, y^{j - 1}, w, y^{j + 1}, \dots, y^{m}, z)$ is non-decreasing whenever the other components $(t, x, y^{1}, \dots, y^{j - 1}, y^{j + 1}, \dots, y^{m}, z)$ are fixed.

(H4)-(ii)

The functions ${(- f_{i})}_{i \in I}$ verify (H4)-(i).

The main objective of this paper is to study the system (1.1).

We now define the notion of a solution ${(u^{i})}_{i \in I}$ of system (1.1), in viscosity sense, which is the following:

Definition 2.1.
(i) Let $\vec{u} : = {(u^{i})}_{i \in I}$ be a function of $C ([0, T] \times R^{k}; R^{m})$ . We say that $\vec{u}$ is a viscosity supersolution (resp. subsolution) of (1.1) if: $\forall i \in {1, \dots, m}$ ,
$u^{i} (T, x) ⩾ (resp. ⩽) h_{i} (x)$ , $\forall x \in R^{k}$ ;

if $ϕ \in C^{1, 2} ([0, T] \times R^{k})$ is such that $(t, x) \in [0, T) \times R^{k}$ is a local minimum (resp. maximum) point of $u^{i} - ϕ$ then $\begin{array}{l} min {u^{i} (t, x) - max_{j \in I^{- i}} (u^{j} (t, x) - g_{i j} (t, x)); \\ - \partial_{t} ϕ (t, x) - L ϕ (t, x) - f_{i} (t, x, {(u^{k} (t, x))}_{k = 1, \dots, m}, (σ^{⊤} D_{x} ϕ) (t, x))} ⩾ (resp. ⩽) 0 . \end{array}$
(ii) We say that $\vec{u} : = {(u^{i})}_{i \in I}$ is a viscosity solution of (1.1) if it is both a supersolution and subsolution of (1.1).

3. Connection with systems of reflected BSDEs with oblique reflection

The viscosity solution of system (1.1) is deeply connected (one can see [7] for more details) with the following system of reflected BSDEs with interconnected obstacles (or oblique reflection) associated with $({(f_{i})}_{i \in I}, {(g_{i j})}_{i, j \in I}, {(h_{i})}_{i \in I})$ : $\forall i = 1, \dots, m$ and $s \in [t, T]$ , $\begin{matrix} (3.1) & \{\begin{matrix} Y^{i, t, x} \in S^{2}, Z^{i, t, x} \in H^{2, d} and K^{i, t, x} \in A^{2}; \\ Y_{s}^{i, t, x} = h_{i} (X_{T}^{t, x}) + \int_{s}^{T} f_{i} (r, X_{r}^{t, x}, {(Y_{r}^{k, t, x})}_{k = 1, \dots m}, Z_{r}^{i, t, x}) d r \\ + K_{T}^{i, t, x} - K_{s}^{i, t, x} - \int_{s}^{T} Z_{r}^{i, t, x} d B_{r}, \\ Y_{s}^{i, t, x} ⩾ {max}_{j \in I^{- i}} (Y_{s}^{j, t, x} - g_{i j} (s, X_{s}^{t, x})) and \\ \int_{t}^{T} (Y_{s}^{i, t, x} - {max}_{j \in I^{- i}} (Y_{s}^{j, t, x} - g_{i j} (s, X_{s}^{t, x}))) d K_{s}^{i, t, x} = 0 . \end{matrix} \end{matrix}$ This system (3.1) of reflected BSDEs is considered in several works (see e.g. [1,7–9], etc.). Under (H1)–(H3) and (H4)-(i) as well, this system has been considered first in [8] where issues of existence and uniqueness of the solution, and comparison of the solutions, are considered (see Theorem 3.2, Theorem 4.2 for point i) and Corollary 3.4. in [8] or Remark 1, pp. 190 in [7] for point ii)). Actually it is shown:

Theorem 3.1.

Assume that the deterministic functions ${(f_{i})}_{i \in I}$ , ${(g_{i j})}_{i, j \in I}$ and ${(h_{i})}_{i \in I}$ verify Assumptions (H1)–(H3) and (H4)-(i). Then system ( 3.1 ) has a unique solution ${(Y^{i}, Z^{i}, K^{i})}_{i \in I}$ .

If ${({\bar{f}}_{i})}_{i \in I}$ , ${({\bar{g}}_{i j})}_{i, j \in I}$ and ${({\bar{h}}_{i})}_{i \in I}$ are other functions satisfying (H1)–(H3) and (H4)-(i) and, moreover, for any $i, j \in I$ , $\begin{matrix} f_{i} ⩽ {\bar{f}}_{i}, h_{i} ⩽ {\bar{h}}_{i} and g_{i j} ⩾ {\bar{g}}_{i j} . \end{matrix}$
Then for any $i \in I$ , $Y^{i} ⩽ {\bar{Y}}^{i}$ where ${({\bar{Y}}^{i}, {\bar{Z}}^{i}, {\bar{K}}^{i})}_{i \in I}$ is the solution of the system associated with ${({\bar{f}}_{i})}_{i \in I}$ , ${({\bar{g}}_{i j})}_{i, j \in I}$ and ${({\bar{h}}_{i})}_{i \in I}$ .

In [1], Chassagneux et al. have also considered system (3.1) without assuming Assumption (H4)-(i). Mainly their idea is the following:

Let $\vec{Γ} : = {(Γ^{i})}_{i = 1, \dots, m} \in H^{2, m}$ and let us consider the following mapping: $\begin{array}{l} (3.2) & Θ : H^{2, m} \to H^{2, m} \\ (3.3) & \vec{Γ} \mapsto Θ (\vec{Γ}) : = {(Y^{Γ, i})}_{i = 1, \dots, m} \end{array}$ where ${(Y^{Γ, i}, Z^{Γ, i}, K^{Γ, i})}_{i \in I} \in {(S^{2} \times H^{2, d} \times A^{2})}^{m}$ (we omit the dependence on t, x of $Y^{Γ, i}$ , $Z^{Γ, i}$ , $K^{Γ, i}$ as no confusion is possible) is the unique solution of the following system of reflected BSDEs with interconnected obstacles (or oblique reflection): $\forall i \in I$ , $\begin{matrix} (3.4) & \{\begin{matrix} Y_{s}^{Γ, i} = h_{i} (X_{T}^{t, x}) + \int_{s}^{T} f_{i} (r, X_{r}^{t, x}, {\vec{Γ}}_{r}, Z_{r}^{Γ, i}) d r + K_{T}^{Γ, i} - K_{s}^{Γ, i} - \int_{s}^{T} Z_{r}^{Γ, i} d B_{r}, \forall s ⩽ T; \\ Y_{s}^{Γ, i} ⩾ {max}_{j \in I^{- i}} (Y_{s}^{Γ, j} - g_{i j} (s, X_{s}^{t, x})), \forall s ⩽ T; \\ \int_{t}^{T} (Y_{s}^{Γ, i} - {max}_{j \in I^{- i}} (Y_{s}^{Γ, j} - g_{i j} (s, X_{s}^{t, x}))) d K_{s}^{Γ, i} = 0 . \end{matrix} \end{matrix}$ Next for $α \in R$ , let us introduce the following norm on $H^{2, m}$ : ∀ $\vec{y} \in H^{2, m}$ $\begin{matrix} ‖ \vec{y} ‖_{α} : = {E [\int_{0}^{T} e^{α s} | {\vec{y}}_{s} |^{2} d s]}^{\frac{1}{2}} . \end{matrix}$ The main result in [1], is to show that Θ has a fixed point, i.e.,
Theorem 3.2 (see [1]).

Assume that the deterministic functions ${(f_{i})}_{i \in I}$ , ${(g_{i j})}_{i, j \in I}$ and ${(h_{i})}_{i \in I}$ verify Assumptions (H1)–(H3). Then there exists some appropriate positive constant $α_{0}$ (which depends on m, T and the Lipschitz constants of ${(f_{i})}_{i = 1, m}$ ) such that Θ is contraction on $(H^{2, m}, ‖ \cdot ‖_{α_{0}})$ . Therefore it has a unique fixed point ${(Y^{i})}_{i \in I}$ which, combined with the associated processes ${(Z^{i}, K^{i})}_{i \in I}$ , makes that ${(Y^{i}, Z^{i}, K^{i})}_{i \in I}$ is the unique solution of system ( 3.1 ).

Moreover the following estimate holds true: for any $^{1} \vec{Γ}$ , $^{2} \vec{Γ} \in H^{2, m}$ , $\begin{matrix} (3.5) & \forall s ⩽ T E [e^{α s} {| Y_{s}^{^{1} Γ, i} - Y_{s}^{^{2} Γ, i} |}^{2}] ⩽ \frac{2 C}{α} E [\int_{s}^{T} e^{α r} {|^{1} Γ_{r} -^{2} Γ_{r} |}^{2} d r] \end{matrix}$ where C is a common Lipschitz constant of the functions ${(f_{i})}_{i = 1, m}$ w.r.t. $(\vec{y}, z)$ and $α ⩾ C$ .

As a remark, note that in [1], the assumptions on the deterministic functions ${(f_{i})}_{i \in I}$ etc., are not exactly the same as (H1)–(H3). However the ideas of [1] can be applied under (H1)–(H3) to show the existence and uniqueness of the solution of system (3.1).

We next provide some properties of the solution of system (3.1) which will be useful later.

Proposition 3.3.
Assume (H1)–(H3). Then:
There exist deterministic functions ${(u^{i})}_{i \in I}$ of polynomial growth, defined on $[0, T] \times R^{k}$ , such that: $\begin{matrix} \forall i \in I, Y_{s}^{i, t, x} = u^{i} (s, X_{s}^{t, x}), d s \times d P on [t, T] \times R^{k} . \end{matrix}$

Assume moreover that $f_{i} (t, x, 0, 0)$ and $h_{i} (x)$ are bounded. Then the processes $Y^{i, t, x}$ and functions $u^{i}$ , $i \in I$ , are also bounded.

Proof.
First let us focus on the first point. Let $(\bar{Y}, \bar{Z})$ be the solution of the following standard BSDE: $\begin{matrix} \{\begin{matrix} \bar{Y} \in S^{2}, \bar{Z} \in H^{2, d}; \\ {\bar{Y}}_{s} = Φ (X_{T}^{t, x}) + \int_{s}^{T} Ψ (r, X_{r}^{t, x}, {\bar{Y}}_{r}, {\bar{Z}}_{r}) d r - \int_{s}^{T} {\bar{Z}}_{r} d B_{r}, \forall s ⩽ T \end{matrix} \end{matrix}$ where for any $(s, x, y, z) \in [0, T] \times R^{k + 1 + d}$ , $Ψ (s, x, y, z) : = \bar{C} m | y | + \bar{C} | z | + \sum_{i = 1, m} | f_{i} (s, x, 0, \dots, 0) |$ and $Φ (x) : = \sum_{i = 1, m} | h_{i} (x) |$ . The constant $\bar{C} : = C_{1} + \dots + C_{m}$ with, for any $i \in I$ , $C_{i}$ is the Lipschitz constant of $f_{i}$ w.r.t. $(\vec{y}, z)$ . Note that the solution of this BSDE exists and is unique by Pardoux–Peng’s result [10].

First note that since $Ψ ⩾ 0$ and $Φ ⩾ 0$ then $\bar{Y} ⩾ 0$ . Next as we are in the Markovian framework of randomness and since Φ and $Ψ (t, x, 0, 0)$ are of polynomial growth, then there exists a deterministic function $v (t, x)$ of polynomial growth (see e.g. [6]) such that: $\begin{matrix} \forall s \in [t, T], {\bar{Y}}_{s}^{t, x} = v (s, X_{s}^{t, x}) . \end{matrix}$ Next let us set, for $i \in I$ , $\begin{matrix} {\underline{Y}}_{i} = \bar{Y}, {\underline{Z}}_{i} = \bar{Z} and {\underline{K}}_{i} = 0 . \end{matrix}$ Therefore, since $g_{i j} ⩾ 0$ for any $i, j \in I$ , ${({\underline{Y}}_{i}, {\underline{Z}}_{i}, {\underline{K}}_{i})}_{i \in I}$ is a solution of the following system: for any $i \in I$ and $s ⩽ T$ , $\begin{matrix} (3.6) & \{\begin{matrix} {\underline{Y}}_{i} (s) = Φ (X_{T}^{t, x}) + \int_{s}^{T} Ψ (r, X_{r}^{t, x}, {\underline{Y}}_{i} (r), {\underline{Z}}_{i} (r)) d r + {\underline{K}}_{i} (T) - {\underline{K}}_{i} (s) - \int_{s}^{T} {\underline{Z}}_{i} (r) B_{r}; \\ {\underline{Y}}_{i} (s) ⩾ {max}_{j \neq i} {{\underline{Y}}_{j} (s) - g_{i j} (s)}; \\ \int_{0}^{T} ({\underline{Y}}_{i} (s) - {max}_{j \neq i} {{\underline{Y}}_{j} (s) - g_{i j} (s)}) d {\underline{K}}_{i} (s) = 0 . \end{matrix} \end{matrix}$ In the same way let us set for any $i \in I$ , $\begin{matrix} {\hat{Y}}_{i} = - \bar{Y}, {\hat{Z}}_{i} = - \bar{Z} and {\hat{K}}_{i} = 0, \end{matrix}$ then ${({\hat{Y}}_{i}, {\hat{Z}}_{i}, {\hat{K}}_{i})}_{i \in I}$ is a solution of the following system: for any $i \in I$ and $s ⩽ T$ , $\begin{matrix} \{\begin{matrix} {\hat{Y}}_{i} (s) = - Φ (X_{T}^{t, x}) - \int_{s}^{T} Ψ (r, X_{r}^{t, x}, - {\bar{Y}}_{r}, - {\hat{Z}}_{r}) d r + {\hat{K}}_{i} (T) - {\hat{K}}_{i} (s) - \int_{s}^{T} {\hat{Z}}_{i} (r) B_{r}; \\ {\hat{Y}}_{i} (s) ⩾ {max}_{j \neq i} {{\hat{Y}}_{j} (s) - g_{i j} (s)}; \\ \int_{0}^{T} ({\hat{Y}}_{i} (s) - {max}_{j \neq i} {{\hat{Y}}_{j} (s) - g_{i j} (s)}) d {\hat{K}}_{i} (s) = 0 . \end{matrix} \end{matrix}$ Next let us consider the following sequence of processes ${({({\tilde{Y}}_{k}^{i}, {\tilde{Z}}_{k}^{i}, {\tilde{K}}_{k}^{i})}_{i \in I})}_{k ⩾ 0}$ : $\begin{matrix} {\tilde{Y}}_{0}^{i} = 0, for all i \in I and for k ⩾ 1, {({\tilde{Y}}_{k}^{i})}_{i \in I} = Θ ({({\tilde{Y}}_{k - 1}^{i})}_{i \in I}) \end{matrix}$ where Θ is the mapping defined in (3.2) and ${\tilde{Z}}_{k}^{i}$ , ${\tilde{K}}_{k}^{i}$ are associated with ${\tilde{Y}}_{k}^{i}$ , $i \in I$ , through equation (3.4). Therefore, as Θ is a contraction (Theorem (3.2)), the sequence ${({({\tilde{Y}}_{k}^{i})}_{i \in I})}_{k ⩾ 0}$ converges to ${(Y^{i})}_{i \in I}$ in $(H^{2, m}, ‖ \cdot ‖_{α_{0}})$ . On the other hand by an induction argument on k and by using the comparison result of Theorem 3.1-ii), we have that: $\begin{matrix} (3.7) & \forall k ⩾ 0, \forall i \in I, - \bar{Y} = {\hat{Y}}_{i} ⩽ {\tilde{Y}}_{k}^{i} ⩽ {\underline{Y}}_{i} = \bar{Y} . \end{matrix}$ Indeed for $k = 0$ , this obviously holds since $\bar{Y} ⩾ 0$ . Next suppose that (3.7) holds for some $k - 1$ with $k ⩾ 1$ . Then by a linearization procedure of $f_{i}$ , which is possible since it is Lipschitz w.r.t. $(\vec{y}, z)$ , we have: for any $i \in I$ , $\begin{matrix} f_{i} (s, X_{s}^{t, x}, {({\tilde{Y}}_{k - 1}^{i} (s))}_{i \in I}, z) = f_{i} (s, X_{s}^{t, x}, 0, 0) + \sum_{l = 1, m} a_{s}^{k, i, l} {\tilde{Y}}_{k - 1}^{l} (s) + b_{s}^{k, i, l} z \end{matrix}$ where $a^{k, i, l} \in R$ and $b^{k, i, l} \in R^{d}$ are $P$ -measurable processes, bounded by the Lipschitz constant of $f_{i}$ and they are the increment rates of $f_{i}$ and actually bounded by the Lipschitz constant of $f_{i}$ . For more details, one could see [2], Appendix A.3, pp. 172. Therefore, using the induction hypothesis, we obtain: $\begin{matrix} | f_{i} (s, X_{s}^{t, x}, {({\tilde{Y}}_{k - 1}^{i} (s))}_{i \in I}, z) | ⩽ Ψ (s, X_{s}^{t, x}, {\bar{Y}}_{s}, z) . \end{matrix}$ Finally by the comparison argument of Theorem 3.1-ii) (see also [8], Corollary 3.4, pp. 411), we get: $\forall i \in I, {\tilde{Y}}_{k}^{i} ⩽ {\underline{Y}}_{i}^{'}$ where ${({\underline{Y}}_{i}^{'}, {\underline{Z}}_{i}^{'}, {\underline{K}}_{i}^{'})}_{i \in I}$ is the unique solution of the system of type (3.1) associated with $({({\underline{f}}_{i} = Ψ (s, X_{s}^{t, x}, {\bar{Y}}_{s}, z))}_{i \in I}, {({\underline{h}}_{i} = Φ (x))}_{i \in I}, {(g_{i j} (s, X_{s}^{t, x}))}_{i, j \in I})$ . But the solution of this latter system is unique (Theorem 3.2) and by (3.6), ${({\underline{Y}}_{i}, {\underline{Z}}_{i}, {\underline{K}}_{i})}_{i \in I}$ is also a solution. Therefore for any $i \in I$ , ${\underline{Y}}_{i}^{'} = {\underline{Y}}_{i}$ and then $\forall i \in I$ , ${\tilde{Y}}_{k}^{i} ⩽ {\underline{Y}}_{i} = \bar{Y}$ . In the same way one can show that $\forall i \in I$ , ${\tilde{Y}}_{k}^{i} ⩾ {\hat{Y}}_{i} = - \bar{Y}$ . Therefore (3.7) holds true for any $k ⩾ 0$ .

Next, once more, since we are in the Markovian framework of randomness, and using an induction argument on k we deduce the existence of deterministic continuous functions of polynomial growth $u^{i, k} (t, x)$ (see e.g. [7], Corollary 2, pp.182), $i \in I$ , such that for any $i \in I$ , $(t, x) \in [0, T] \times R^{k}$ , $\begin{matrix} (3.8) & {\tilde{Y}}_{k}^{i} (s) = u^{i, k} (s, X_{s}^{t, x}), \forall s \in [t, T] . \end{matrix}$ By (3.7), in taking $s = t$ , we obtain: for any $k ⩾ 0$ , $i \in I$ and $(t, x) \in [0, T] \times R^{k}$ , $\begin{matrix} (3.9) & | u^{i, k} (t, x) | ⩽ v (t, x) . \end{matrix}$ Next by using the inequality (3.5) at $s = t$ , we deduce that for any $i \in I$ , $k, p ⩾ 1$ $\begin{matrix} (3.10) & \begin{matrix} {| u^{i, k} (t, x) - u^{i, p} (t, x) |}^{2} & ⩽ \frac{2 C}{α_{0}} E [\int_{t}^{T} e^{α_{0} (r - t)} \sum_{j = 1, m} {| {\tilde{Y}}_{k - 1}^{j} (r) - {\tilde{Y}}_{p - 1}^{j} (r) |}^{2} d r] \\ ⩽ \frac{2 C}{α_{0}} E [\int_{t}^{T} e^{α_{0} (r - t)} \sum_{j = 1, m} {| u^{j, k - 1} (r, X_{r}^{t, x}) - u^{j, p - 1} (r, X_{r}^{t, x}) |}^{2} d r] . \end{matrix} \end{matrix}$ As ${({({\tilde{Y}}_{k}^{i})}_{i \in I})}_{k}$ is a Cauchy sequence in $(H^{2, m}, ‖ \cdot ‖_{α_{0}})$ , then ${({(u^{i, k})}_{i \in I})}_{k}$ is a Cauchy sequence pointwisely. This implies the existence of deterministic functions ${(u^{i})}_{i \in I}$ such that for any $i \in I$ and $(t, x) \in [0, T] \times R^{k}$ , $u^{i, k} (t, x)$ converges w.r.t. k to $u^{i} (t, x)$ . Moreover by (3.9), $u^{i}$ is of polynomial growth since v is so and finally by (3.8), $Y_{s}^{i, t, x} = u^{i} (s, X_{s}^{t, x})$ , $d s \times d P$ on $[t, T] \times R^{k}$ .

We now deal with the second point. Assume that $f_{i} (t, x, 0, 0)$ and $h_{i} (x)$ are bounded. Then the solution $\bar{Y}$ is bounded. This is obtained by a change of probability, applying Girsanov’s theorem and by multiplying both hand-sides of the equation by $e^{- m \bar{C} s}$ , conditionning and taking into account of the inequality $\bar{Y} ⩾ 0$ . Therefore the deterministic function v is a also bounded. Consequently, $u^{i, k}$ are uniformly bounded and so are $u^{i}$ , $i \in I$ . □
Remark 3.4.
At this point we do not know whether the functions $u^{i}$ , $i \in I$ , are continuous or not. However we will show later that they can be chosen continuous.

4. The main result: Existence and uniqueness of the viscosity solution for system of PDEs with interconnected obstacles

In this section, we study the existence and uniqueness in viscosity sense of the solution of the system of m partial differential equations with interconnected obstacles (1.1). The candidate to be the solution are the functions $(u^{1}, \dots, u^{m})$ defined in Proposition 3.3 by which we represent ${(Y^{i})}_{i \in I}$ . So, firstly we are going to show that those functions $u^{i}$ , $i \in I$ , can be chosen continuous.

Proposition 4.1.

Assume that (H1)–(H3) hold. Then we can choose the functions $u^{i}$ , $i \in I$ , defined in Proposition 3.3 , continuous in $(t, x)$ and of polynomial growth.

Proof.

It will be given in two steps. In the first one we are going to suppose moreover that $h_{i}$ and $f_{i} (t, x, 0, 0)$ , $i \in I$ , are bounded. Later on we deal with the general case, i.e., without assuming the boundedness of those latter functions.

Step 1: Suppose that for any $i \in I$ , $h_{i}$ and $f_{i} (t, x, 0, 0)$ are bounded.

Recall the continuous functions $u^{i, k}$ , $i \in I$ and $k ⩾ 0$ , defined in (3.8). By (3.10) they verify: $\forall k ⩾ 1$ , $i \in I$ and $(t, x) \in [0, T] \times R^{k}$ , $\begin{matrix} (4.1) & {| u^{i, k} (t, x) - u^{i, p} (t, x) |}^{2} ⩽ \frac{2 C}{α} E [\int_{t}^{T} e^{α (r - t)} \sum_{j = 1, m} {| u^{j, k - 1} (r, X_{r}^{t, x}) - u^{j, p - 1} (r, X_{r}^{t, x}) |}^{2} d r] \end{matrix}$ where, as pointed out in (3.5), C is the uniform Lipschitz constant of $f_{i}^{'} s$ w.r.t. $(\vec{y}, z)$ and $α ⩾ C$ .

On the other hand we know, by Proposition 3.3-ii), that $u^{i, k}$ are uniformly bounded for any $i \in I$ and $k ⩾ 0$ . Now let us take $α = C$ and let η be a constant such that $2 C^{- 1} m (e^{C η} - 1) = \frac{3}{4}$ and finally let us set $\begin{matrix} {‖ u^{i, k} - u^{i, p} ‖}_{\infty, η} : = sup_{(t, x) \in [T - η, T] \times R^{k}} | u^{i, k} (t, x) - u^{i, p} (t, x) | . \end{matrix}$ Going back to (3.10) and taking the summations over all i, we deduce that for any $k, p ⩾ 1$ , $\begin{array}{l} \sum_{i = 1, m} {‖ u^{i, k} - u^{i, p} ‖}_{\infty, η}^{2} & ⩽ \underset{= \frac{3}{4}}{\underset{︸}{2 m C^{- 1} (e^{C η} - 1)}} \sum_{i = 1, m} {‖ u^{i, k - 1} - u^{i, p - 1} ‖}_{\infty, η}^{2} \\ ⩽ \frac{3}{4} \sum_{i = 1, m} {‖ u^{i, k - 1} - u^{i, p - 1} ‖}_{\infty, η}^{2}, \end{array}$ which means that the sequence ${({(u^{i, k})}_{i \in I})}_{k ⩾ 0}$ is uniformly convergent in $[T - η, T] \times R^{k}$ . Thus, their limits, i.e., the functions ${(u^{i})}_{i \in I}$ are also continuous on the set $[T - η, T] \times R^{k}$ .

Next let $t \in [T - 2 η, T - η]$ , then once more by (4.1) we have: $\begin{array}{l} {| u^{i, k} (t, x) - u^{i, p} (t, x) |}^{2} \\ ⩽ 2 C α^{- 1} E [\int_{t}^{T - η} e^{α (r - t)} \sum_{j = 1, m} {| u^{j, k - 1} (r, X_{r}^{t, x}) - u^{j, p - 1} (r, X_{r}^{t, x}) |}^{2} d r] \\ (4.2) & + 2 C α^{- 1} E [\int_{T - η}^{T} e^{α (r - t)} \sum_{j = 1, m} {| u^{j, k - 1} (r, X_{r}^{t, x}) - u^{j, p - 1} (r, X_{r}^{t, x}) |}^{2} d r] . \end{array}$ Then, if we choose $t = T - 2 η$ and set $\begin{matrix} {‖ u^{i, k} - u^{i, p} ‖}_{\infty, 2 η} : = sup_{(t, x) \in [T - 2 η, T - η] \times R^{k}} | u^{i, k} (t, x) - u^{i, p} (t, x) |, \end{matrix}$ we obtain: $\begin{array}{l} \sum_{i = 1, m} {‖ u^{i, k} - u^{i, p} ‖}_{\infty, 2 η}^{2} & ⩽ 2 m C^{- 1} ((e^{C η} - 1) {‖ u^{i, k - 1} - u^{i, p - 1} ‖}_{\infty, 2 η}^{2} \\ + (e^{2 C η} - e^{C η}) \sum_{i = 1, m} {‖ u^{i, k - 1} - u^{i, p - 1} ‖}_{\infty, η}^{2}) \\ ⩽ \frac{3}{4} \sum_{i = 1, m} {‖ u^{i, k - 1} - u^{i, p - 1} ‖}_{\infty, 2 η}^{2} \\ + 2 m C^{- 1} (e^{2 C η} - e^{C η}) \sum_{i = 1, m} {‖ u^{i, k - 1} - u^{i, p - 1} ‖}_{\infty, η}^{2} . \end{array}$ It implies that $\begin{matrix} \underset{k, p \to \infty}{lim sup} \sum_{i = 1, m} {‖ u^{i, k} - u^{i, p} ‖}_{\infty, 2 η}^{2} ⩽ \frac{3}{4} \underset{k, p \to \infty}{lim sup} \sum_{i = 1, m} {‖ u^{i, k - 1} - u^{i, p - 1} ‖}_{\infty, 2 η}^{2} \end{matrix}$ since ${lim sup}_{k, p \to \infty} \sum_{i = 1, m} ‖ u^{i, k - 1} - u^{i, p - 1} ‖_{\infty, η}^{2} = 0$ . Therefore $\begin{matrix} \underset{k, p \to \infty}{lim sup} \sum_{i = 1, m} {‖ u^{i, k} - u^{i, p} ‖}_{\infty, 2 η}^{2} = 0 . \end{matrix}$ Consequently the sequence ${({(u^{i, k})}_{i \in I})}_{k ⩾ 0}$ is uniformly convergent in $[T - 2 η, T - η] \times R^{k}$ . Thus, their limits, the functions ${(u^{i})}_{i \in I}$ are also continuous in $[T - 2 η, T - η] \times R^{k}$ , which implies that ${(u^{i})}_{i \in I}$ are continuous in $[T - 2 η, T] \times R^{k}$ . Continuing now this reasoning as many times as necessary on $[T - 3 η, T - 2 η]$ , $[T - 4 η, T - 3 η]$ etc. we obtain the continuity of ${(u^{i})}_{i \in I}$ in $[0, T] \times R^{k}$ , since η is fixed.

Step 2: We now deal with the general case. Firstly by (H1)-iii), (H2) and (H3), there exist two constants C and $p \in N$ such $f_{i} (t, x, 0, \dots, 0)$ , $h_{i} (x)$ and $g_{i j} (t, x)$ are of polynomial growth, i.e., for any $(t, x) \in [0, T] \times R^{k}$ , $\begin{matrix} (4.3) & | f_{i} (t, x, 0, \dots, 0) | + | h_{i} (x) | + | g_{i j} (t, x) | ⩽ C (1 + | x |^{p}) . \end{matrix}$ To proceed for $s \in [t, T]$ let us define, $\begin{matrix} {\overline{Y}}_{s}^{i} : = Y_{s}^{i} φ (X_{s}^{t, x}), \end{matrix}$ where for $x \in R$ , $φ (x) : = \frac{1}{{(1 + | x |^{2})}^{p}}$ (p is the same constant as in (4.3)). Then by the integration-by-parts formula we have: $\begin{array}{l} d {\overline{Y}}_{s}^{i} & = φ (X_{s}^{t, x}) d Y_{s}^{i} + Y_{s}^{i} d φ (X_{s}^{t, x}) + d {⟨ Y^{i}, φ (X^{t, x}) ⟩}_{s} \\ = φ (X_{s}^{t, x}) {- f_{i} (s, X_{s}^{t, x}, {(Y_{s}^{k})}_{k = 1, \dots, m}, Z_{s}^{i}) d s - d K_{s}^{i} + Z_{s}^{i} d B_{s}} \\ + Y_{s}^{i} {L φ (X_{s}^{t, x}) d s + D_{x} φ (X_{s}^{t, x}) σ (s, X_{s}^{t, x}) d B_{s}} + Z_{s}^{i} D_{x} φ (X_{s}^{t, x}) σ (s, X_{s}^{t, x}) d s \\ = {- φ (X_{s}^{t, x}) f_{i} (s, X_{s}^{t, x}, {(Y_{s}^{k})}_{k = 1, \dots, m}, Z_{s}^{i}) + L φ (X_{s}^{t, x}) Y_{s}^{i} + D_{x} φ (X_{s}^{t, x}) σ (s, X_{s}^{t, x}) Z_{s}^{i}} d s \\ - φ (X_{s}^{t, x}) d K_{s}^{i} + {Z_{s}^{i} φ (X_{s}^{t, x}) + Y_{s}^{i} D_{x} φ (X_{s}^{t, x}) σ (s, X_{s}^{t, x})} d B_{s}, \end{array}$ where $L φ$ is given in (1.2). Next let us set, for $s \in [t, T]$ , $\begin{array}{l} d {\overline{K}}_{s}^{i} : = φ (X_{s}^{t, x}) d K_{s}^{i} and {\overline{Z}}_{s}^{i} : = Z_{s}^{i} φ (X_{s}^{t, x}) + Y_{s}^{i} D_{x} φ (X_{s}^{t, x}) σ (s, X_{s}^{t, x}) . \end{array}$ Then ${(({\overline{Y}}^{i}, {\overline{Z}}^{i}, {\overline{K}}^{i}))}_{i \in I}$ satisfies: $\forall s \in [t, T]$ , $\begin{matrix} (4.4) & \{\begin{matrix} {\overline{Y}}_{s}^{i} = {\overline{h}}_{i} (X_{T}^{t, x}) + \int_{s}^{T} {\overline{f}}_{i} (r, X_{r}^{t, x}, {({\overline{Y}}_{r}^{k})}_{k = 1, \dots, m}, {\overline{Z}}_{r}^{i}) d r + {\overline{K}}_{T}^{i} - {\overline{K}}_{s}^{i} - \int_{s}^{T} {\overline{Z}}_{r}^{i} d B_{r}, \\ {\overline{Y}}_{s}^{i} ⩾ {max}_{j \in I^{- i}} ({\overline{Y}}_{s}^{j} - {\overline{g}}_{i j} (s, X_{s}^{t, x})), \\ \int_{t}^{T} ({\overline{Y}}_{s}^{i} - {max}_{j \in I^{- i}} ({\overline{Y}}_{s}^{j} - {\overline{g}}_{i j} (s, X_{s}^{t, x}))) d {\overline{K}}_{s}^{i} = 0, \end{matrix} \end{matrix}$ where for any $i, j \in I$ , $\begin{matrix} {\overline{h}}_{i} (X_{T}^{t, x}) : = h_{i} (X_{T}^{t, x}) φ (X_{T}^{t, x}), {\overline{g}}_{i j} (s, X_{s}^{t, x}) : = g_{i j} (s, X_{s}^{t, x}) φ (X_{s}^{t, x}), \end{matrix}$ and $\begin{array}{l} {\overline{f}}_{i} (s, x, \vec{y}, z) & : = φ (x) f_{i} (s, x, φ^{- 1} (x) \vec{y}, φ^{- 1} (x) z - D_{x} φ (x) σ (s, x) φ^{- 1} (x) y^{i})) \\ - L φ (x) φ^{- 1} (x) y^{i} - D_{x} φ (x) σ (s, x) φ^{- 1} (x) {z - D_{x} φ (x) σ (s, x) φ^{- 1} (x) y^{i}} \\ (φ^{- 1} (x) = {(1 + | x |^{2})}^{p}) . \end{array}$ Here let us notice that the functions ${\overline{f}}_{i} (t, x, 0, 0)$ , ${\overline{g}}_{i j}$ and ${\overline{h}}_{i}$ are bounded. Then by the result of the first step, there exists bounded continuous functions ${({\bar{u}}^{i})}_{i \in I}$ such that for any $(t, x) \in [0, T] \times R^{k}$ , and $s \in [t, T]$ , ${\bar{Y}}_{s}^{i} = {\bar{u}}^{i} (s, X_{s}^{t, x})$ , $\forall i \in I$ . Thus for any $(t, x) \in [0, T] \times R^{k}$ , and $s \in [t, T]$ , $Y_{s}^{i} = φ^{- 1} (X_{s}^{t, x}) {\bar{u}}^{i} (s, X_{s}^{t, x})$ , $\forall i \in I$ . Then it is enough to take $u^{i} (t, x) : = φ^{- 1} (x) {\bar{u}}^{i} (t, x)$ , $(t, x) \in [0, T] \times R^{k}$ and $i \in I$ , which are continuous functions and of polynomial growth. □

We are now ready to give the main result of this paper. Let ${(Y^{i}, Z^{i}, K^{i})}_{i \in I}$ be the unique solution of (3.1) and let ${(u^{i})}_{i \in I}$ be the continuous functions with polynomial growth such that for any $(t, x) \in [0, T] \times R^{k}$ and $i \in I$ , $\begin{matrix} P -a.s., \forall s \in [t, T], Y_{s}^{i, t, x} = u^{i} (s, X_{s}^{t, x}) . \end{matrix}$ We then have:

Theorem 4.2.

The function ${(u^{i})}_{i \in I}$ is a solution in viscosity sense of system ( 1.1 ). Moreover it is unique in the class of continuous functions of polynomial growth.

Proof.

First let us show that ${(u^{i})}_{i \in I}$ is a viscosity solution of system (1.1).

Recall that ${(Y^{i}, Z^{i}, K^{i})}_{i \in I}$ is a solution of the system of reflected BSDEs with interconnected obstacles (3.1) and for any $(t, x) \in [0, T] \times R^{k}$ , $i \in I$ and $s \in [t, T]$ , $Y_{s}^{i, t, x} = u^{i} (s, X_{s}^{t, x})$ . Then ${(Y^{i}, Z^{i}, K^{i})}_{i \in I}$ verify: for any $s \in [t, T]$ and $i \in I$ , $\begin{matrix} (4.5) & \{\begin{matrix} Y_{s}^{i, t, x} = h_{i} (X_{T}^{t, x}) + \int_{s}^{T} f_{i} (r, X_{r}^{t, x}, {(u^{k} (r, X_{r}^{t, x}))}_{k \in I}, Z_{r}^{i, t, x}) d r \\ + K_{T}^{i, t, x} - K_{s}^{i, t, x} - \int_{s}^{T} Z_{r}^{i, t, x} d B_{r}, \\ Y_{s}^{i, t, x} ⩾ {max}_{j \in I^{- i}} (u^{j} (s, X_{s}^{t, x}) - g_{i j} (s, X_{s}^{t, x})) and \\ \int_{t}^{T} (Y_{r}^{i, t, x} - {max}_{j \in I^{- i}} (u^{j} (r, X_{r}^{t, x}) - g_{i j} (r, X_{r}^{t, x}))) d K_{r}^{i, t, x} = 0 . \end{matrix} \end{matrix}$ But system (4.5) is decoupled and using a result by El-Karoui et al. (Theorem 8.5 in [5]) one obtains that, for any $i_{0}$ , $u^{i_{0}}$ is a solution in viscosity sense of the following PDE with obstacle: $\begin{matrix} (4.6) & \{\begin{matrix} min {u^{i_{0}} (t, x) - {max}_{j \in I^{- i_{0}}} (u^{j} (t, x) - g_{i_{0} j} (t, x)); \\ - \partial_{t} u^{i_{0}} (t, x) - L u^{i_{0}} (t, x) - f_{i_{0}} (t, x, {(u^{k} (t, x))}_{k = 1, \dots, m}, (σ^{⊤} D_{x} u^{i_{0}}) (t, x))} = 0; \\ u^{i_{0}} (T, x) = h_{i} (x) . \end{matrix} \end{matrix}$ As $i_{0}$ is arbitrary in $I$ , then the functions ${(u^{i})}_{i \in I}$ is a solution in viscosity sense of (1.1).

Next let us show that ${(u^{i})}_{i \in I}$ is the unique solution in the class of continuous functions with polynomial growth. It is based on the uniqueness of the solution of the system of reflected BSDEs with interconnected obstacles (3.1).

So suppose that there exists another continuous with polynomial growth solution ${({\tilde{u}}^{i})}_{i = 1, \dots, m}$ of (1.1), i.e., for any $i \in I$ , $\begin{matrix} (4.7) & \{\begin{matrix} min {{\tilde{u}}^{i} (t, x) - {max}_{j \in I^{- i}} ({\tilde{u}}^{j} (t, x) - g_{i j} (t, x)); \\ - \partial_{t} {\tilde{u}}^{i} (t, x) - L {\tilde{u}}^{i} (t, x) - f_{i} (t, x, {({\tilde{u}}^{k} (t, x))}_{k = 1, \dots, m}, (σ^{⊤} D_{x} {\tilde{u}}^{i}) (t, x))} = 0; \\ {\tilde{u}}^{i} (T, x) = h_{i} (x) . \end{matrix} \end{matrix}$ Let ${({\tilde{Y}}^{i})}_{i \in I} \in H^{2, m}$ be such that for any $i \in I$ and $s \in [t, T]$ , $\begin{matrix} {\tilde{Y}}_{s}^{i, t, x} = {\tilde{u}}^{i} (s, X_{s}^{t, x}) . \end{matrix}$ Next let us define ${({\overline{Y}}^{i, t, x})}_{i \in I}$ as follows: $\begin{matrix} (4.8) & {({\overline{Y}}^{i, t, x})}_{i \in I} = Θ ({({\tilde{Y}}_{s}^{i, t, x})}_{i \in I}), \end{matrix}$ that is to say, ${({\overline{Y}}^{i, t, x}, {\overline{Z}}^{i, t, x}, {\overline{K}}^{i, t, x})}_{i \in I}$ is the solution of the following system of reflected BSDEs with oblique reflection: $\forall s \in [t, T]$ , $\forall i \in I$ $\begin{matrix} (4.9) & \{\begin{matrix} {\overline{Y}}^{i, t, x} \in S^{2}, {\overline{Z}}^{i, t, x} \in H^{2, d} and {\overline{K}}^{i, t, x} \in A^{2}; \\ {\overline{Y}}_{s}^{i, t, x} = h_{i} (X_{T}^{t, x}) + \int_{s}^{T} f_{i} (r, X_{r}^{t, x}, ({\tilde{u}}^{k} {(s, X_{s}^{t, x})}_{k = 1, \dots m}, {\overline{Z}}_{r}^{i, t, x}) d r \\ + {\overline{K}}_{T}^{i, t, x} - {\overline{K}}_{s}^{i, t, x} - \int_{s}^{T} {\overline{Z}}_{r}^{i, t, x} d B_{r}; \\ {\overline{Y}}_{s}^{i, t, x} ⩾ {max}_{j \in I^{- i}} ({\overline{Y}}_{s}^{j, t, x} - g_{i j} (s, X_{s}^{t, x})) and \\ \int_{t}^{T} ({\overline{Y}}_{s}^{i, t, x} - {max}_{j \in I^{- i}} ({\overline{Y}}_{s}^{j, t, x} - g_{i j} (s, X_{s}^{t, x}))) d {\overline{K}}_{s}^{i, t, x} = 0 . \end{matrix} \end{matrix}$ As the deterministic functions ${({\tilde{u}}^{i})}_{i = 1, \dots, m}$ are continuous and of polynomial growth, then by using a result by Hamadène–Morlais ([7], Theorem 1), one can infer the existence of deterministic continuous functions with polynomial growth ${(v^{i})}_{i = 1, \dots, m}$ such that: $\forall i \in I$ and $s \in [t, T]$ , $\begin{matrix} {\overline{Y}}_{s}^{i, t, x} = v^{i} (s, X_{s}^{t, x}) . \end{matrix}$ Moreover, ${(v^{i})}_{i = 1, \dots, m}$ is the unique viscosity solution (in the class of functions with polynomial growth) of the following system of PDEs with interconnected obstacles: $\forall i = 1, \dots, m$ $\begin{matrix} (4.10) & \{\begin{matrix} min {v^{i} (t, x) - {max}_{j \in I^{- i}} (v^{j} (t, x) - g_{i j} (t, x)); \\ - \partial_{t} v^{i} (t, x) - L v^{i} (t, x) - f_{i} (t, x, {({\tilde{u}}^{k} (t, x))}_{k = 1, \dots, m}, (σ^{⊤} D_{x} v^{i}) (t, x))} = 0; \\ v^{i} (T, x) = h_{i} (x) . \end{matrix} \end{matrix}$ Let us notice that in system (4.10), in the arguments of $f_{i}$ we have ${\tilde{u}}^{k}$ and not $v^{k}$ . On the other hand the functions $(t, x) \mapsto f_{i} (t, x, {({\tilde{u}}^{k} (t, x))}_{k = 1, \dots, m}, z)$ , $i \in I$ , are continuous uniformly w.r.t. z, i.e., they satisfy (H1-(i)). This property is needed in order to use the results of [7]).

Now as the functions ${({\tilde{u}}^{i})}_{i = 1, \dots, m}$ solve system (4.10), hence by uniqueness of the solution of this system (4.10) (see [7], Thm. 1, pp. 175), one deduces that $\begin{matrix} v^{i} = {\tilde{u}}^{i} and then {\tilde{Y}}^{i, t, x} = {\overline{Y}}^{i, t, x}, \forall i \in I . \end{matrix}$ Therefore ${({\tilde{Y}}_{s}^{i, t, x})}_{i \in I}$ verify $\begin{matrix} {({\tilde{Y}}^{i, t, x})}_{i \in I} = Θ ({({\tilde{Y}}_{s}^{i, t, x})}_{i \in I}) . \end{matrix}$ But ${(Y^{i})}_{i \in I}$ is the unique fixed point of Θ in $(H^{2, m}, ‖ \cdot ‖_{α_{0}})$ then we have that for any $s \in [t, T]$ and $i \in I$ , ${\tilde{Y}}_{s}^{i, t, x} = Y_{s}^{i, t, x}$ . Henceforth, in taking $s = t$ , we obtain that for any $i \in I$ and $(t, x) \in [0, T] \times R^{k}$ , ${\tilde{u}}^{i} (t, x) = u^{i} (t, x)$ . Thus ${(u^{i})}_{i = 1, \dots, m}$ is the unique solution of system (4.10) in the class of continuous functions with polynomial growth. □

References

J.-F.

Chassagneux,

Elie and

Kharroubi, A note on existence and uniqueness for solutions of multidimensional reflected bsdes, Electronic Communications in Probability 16 (2011), 120–128. doi:10.1214/ECP.v16-1614.

Djehiche,

Hamadene and

M.A.

Morlais, Viscosity solutions of systems of variational inequalities with interconnected bilateral obstacles, Funkcialaj Ekvacioj 58 (2015), 135–175. doi:10.1619/fesi.58.135.

Duffie and

Epstein, Stochastic differential utility, Econometrica 60 (1992), 353–394. doi:10.2307/2951600.

El Asri and

Hamadène, The finite horizon optimal multi-modes switching problem: The viscosity solution approach, Applied Mathematics & Optimization 60(2) (2009), 213–235. doi:10.1007/s00245-009-9071-3.

El Karoui,

Kapoudjian,

Pardoux,

Peng and

M.C.

Quenez, Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, The Annals of Probability (1997), 702–737.

El Karoui,

Peng and

M.C.

Quenez, Backward stochastic differential equations in finance, Mathematical finance 7(1) (1997), 1–71.

Hamadène and

M.-A.

Morlais, Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem, Applied Mathematics & Optimization 67(2) (2013), 163–196. doi:10.1007/s00245-012-9184-y.

Hamadène and

Zhang, Switching problem and related system of reflected backward sdes, Stochastic Processes and their applications 120(4) (2010), 403–426. doi:10.1016/j.spa.2010.01.003.

Hu and

Tang, Multi-dimensional bsde with oblique reflection and optimal switching, Probability Theory and Related Fields 147(1) (2010), 89–121. doi:10.1007/s00440-009-0202-1.

10.

Pardoux and

Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters 14(1) (1990), 55–61. doi:10.1016/0167-6911(90)90082-6.

11.

Revuz and

Yor, Continuous Martingales and Brownian Motion, Vol. 293, Springer Science & Business Media, 2013.