There are three main results in this thesis. The first is existence and uniqueness of the solution in viscosity sense for a system of nonlinear m variational integral-partial differential equations with interconnected obstacles. From the probabilistic point of view, this system is related to optimal stochastic switching problem when the noise is driven by a Lévy process. As a by-product we obtain that the value function of the switching problem is continuous and unique solution of its associated Hamilton-Jacobi-Bellman system of equations. Next, we study a general class of min-max and max-min nonlinear second-order integral-partial variational inequalities with interconnected bilateral obstacles, related to a multiple modes zero-sum switching game with jumps. Using Perron's method and by the help of systems of penalized unilateral reflected backward SDEs with jumps, we construct a continuous with polynomial growth viscosity solution, and a comparison result yields the uniqueness of the solution. At last, we deal with the solutions of systems of PDEs with bilateral inter-connected obstacles of min-max and max-min types in the Brownian framework. These systems arise naturally in stochastic switching zero-sum game problems. We show that when the switching costs of one side are smooth, the solutions of the min-max and max-min systems coincide. Furthermore, this solution is identified as the value function of the zero-sum switching game.

III

IV

Résumé

Cette thèse est composée de trois parties. Dans la première nous montrons l'existence et l'unicité de la solution continue et à croissance polynomiale, au sens viscosité, du système non linéaire de m équations variationnelles de type intégro-différentiel à obstacles unilatéraux interconnectés. Ce système est lié au problème du switching optimal stochastique lorsque le bruit est dirigé par un processus de Lévy. Un cas particulier du système correspond en effet à l'équation d'Hamilton-Jacobi-Bellman associé au problème du switching et la solution de ce système n'est rien d'autre que la fonction valeur du problème. Ensuite, nous étudions un système d'équations intégro-différentielles à obstacles bilatéraux interconnectés. Nous montrons l'existence et l'unicité des solutions continus à croissance polynomiale, au sens viscosité, des systèmes min-max et max-min. La démarche conjugue les systèmes d'EDSR réfléchies ainsi que la méthode de Perron. Dans la dernière partie nous montrons l'égalité des solutions des systèmes max-min et min-max d'EDP lorsque le bruit est uniquement de type diffusion. Nous montrons que si les coûts de switching sont assez réguliers alors ces solutions coincident. De plus elles sont caractérisées comme fonction valeur du jeu de switching de somme nulle.

V VI 4 On the identity of min-max and max-min solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles.

4.1 Notations and first results . . . . . . . . . . . .

The differential from of this equation is

-dY t = g(t, Y t , Z t )dt -Z t dB t , Y T = ξ. (1.1.2)
Hereafter g is called the coefficient and ξ the terminal value of the BSDE. The BSDE (1.1.1) has a unique solution under the standard assumptions as follows:

(H1)

           (i) (g(t, 0, 0)) t≤T ∈ H 2 m
(ii) g is uniformly Lipschitz with respect to (y, z), i.e., there exists a constant C ≥ 0 such that for any (y, y ′ , z, z ′ ) :

|g(ω, t, y, z)g(ω, t, y ′ , z ′ )| ≤ C(|yy ′ | + |zz ′ |), dt ⊗ dPa.e.

Theorem 1.1.1. (Pardoux and Peng [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF]) Under the assumption (H1), there exists a unique solution (Y, Z) of the BSDE with parameters (g, ξ).

Using Itô's formula we obtain the following a priori estimate.

Proposition 1.1.1. Let (Y, Z) be a solution of BSDE (1.1.1). Then there exists a constant c > 0 such that

E[ sup 0≤t≤T |Y t | 2 + T 0 |Z t | 2 dt] ≤ cE[|ξ| 2 + T 0 |g(t, 0, 0)| 2 dt]. (1.1.3)
When the coefficient is linear, we can get explicitly the component Y of the solution.

1 Proposition 1.1.2. (El Karoui, Peng, and Quenez [START_REF] El Karoui | Les aspects probabilistes du contrôle stochastique[END_REF]) Let (β, µ) be a bounded (R, R d )-valued progressively measurable process, φ be an element of H 2 1 (0, T ) and ξ ∈ L where (Γ t,s ) s≥t is the adjoint process defined by the forward linear SDE ∀s ∈ [t, T ], dΓ t,s = Γ t,s (β s ds + µ s dB s ) and Γ t,t = 1.

(ii) If ξ and φ are both non-negative, then the process (Y t ) t≤T is non-negative.

In one-dimensional case, i.e., when m = 1, we have a comparison result between the Y 's as soon as we can compare the associated coefficient and terminal values. More precisely, Theorem 1.1.2. (El Karoui, Peng, and Quenez [START_REF] El Karoui | Les aspects probabilistes du contrôle stochastique[END_REF]) Let us consider the solutions (Y, Z) and (Y ′ , Z ′ ) of two BSDEs associated with parameters (g, ξ) and (g ′ , ξ ′ ). We assume that g satisfies (H1), and When the coefficients of the BSDE are deterministic functions of a diffusion process, the solution (Y, Z) is also a deterministic function of the same process. If, in addition, a certain regularity on the coefficients is introduced, it is possible to relate these functions with the pair (solution, gradient) of some semi-linear PDE. The basic framework is the following: the randomness of the coefficient and the terminal value of a Markvian BSDE comes from a diffusion process (X t,x s ) s∈[t,T ] , which is the strong solution of a standard SDE: dX t,x s = b(s, X t,x s )ds + σ(s, X t,x s )dB s , t ≤ s ≤ T X t,x s = x, s ∈ [0, t].

(g ′ (s, Y ′ s , Z ′ s )) s≤T is element of H 2 1 . If ξ ≤ ξ ′ P -a.
(1.1.5)

For any given (t, x) ∈ [0, T ] × R d , we will denote by (Y t,x s , Z t,x s ) s∈[t,T ] the solution of the following BSDE:

-dY s = g(s, X t,x s , Y s , Z s )ds -Z s dB s , s ≤ T ; Y T = Ψ(X t,x T ).

(1. 1.6) In order to have good estimates of the solution, we assume that the following condition is satisfied. (v) The mapping x → (g(t, x, 0, 0), Ψ(x)) is continuous.

Theorem 1.1.3. (Dellacherie and Meyer [START_REF] Dellacherie | Probabilités et Potentiel[END_REF]) Under (H2), there exist two measurable deterministic functions u(t, x) and d(t, x) such that the solution (Y t,x , Z t,x ) of BSDE (1.1.6) is given by ∀t ≤ s ≤ T, Y t,x s = u(s, X t,x s ) and Z t,x s = d(s, X t,x s )σ(s, X t,x s ), ds ⊗ dPa.e..

Let us now consider the following system of semilinear parabolic PDEs, where u is a R m -valued function, defined on [0, T ] × R d satisfying ∂u ∂t + Lu(t, x) + g(t, x, u(t, x), D σ u(t, x)) = 0 ∀(t, x) ∈ [0, T ] × R d , u(T, x) = Ψ(x), ∀x ∈ R d .

(1.1.7)

L is a second-order differential operator given by

L := 1 2 d i,j=1 (σσ * ) i,j ∂ 2 ∂x i ∂x j + d i=1 b i ∂ ∂x i , D σ u := Duσ. (1.1.8)
Under the assumptions (H2) on the coefficients, we can only consider the solution of PDE (1.1.7) in viscosity sense. Moreover, we need to make the following restriction: for 1 ≤ i ≤ m, the i-th coordinate of g, denoted by g i , depends only ont the i-th row of the matrix z. Therefore, the equation (1.1.7) can be written as ∂ui ∂t + Lu i (t, x) + g i (t, x, u(t, x), Du i σ(t, x)) = 0, i = 1,

• • • m, u(T, x) = Ψ(x), ∀x ∈ R d .
Now let us introduce the definition of a viscosity solution:

Definition 1.1.2. Assume u ∈ C[0, T ] × R d ; R m
) and u(T, x) = Ψ(x), for all x ∈ R d . u is called a viscosity subsolution (resp. supersolution) of PDE (1.1.7) if , for any

1 ≤ i ≤ m, φ ∈ C 1,2 ([0, T ] × R d )
and (t, x) ∈ [0, T ] × R d such that φ(t, x) = u(t, x) and u(t, x) is a local maximum (resp. minimum) of u iφ, ∂φ ∂t + Lφ(t, x) + g i (t, x, u(t, x), (Dφσ)(t, x)) ≤ 0 (resp. ≥ 0).

Moreover, u ∈ C([0, T ] × R d ; R m
) is called a viscosity solution of PDE (1.1.7) if it is both a viscosity subsolution and a viscosity supersolution.

We now give the probabilistic interpretation of the viscosity solution of PDE (1.1.7) using (Y t,x s , Z t,x s ) solution of the BSDE (1.1.5): Theorem 1.1.4. (Pardoux and Peng [START_REF] Pardoux | Backward stochastic differential equations and quasilinear parabolic partial differential equations[END_REF]) Under Assumptions (H2), u := Y t,x t is a viscosity solution of PDE (1.1.7) and there exist two constants C and p, such that

|u(t, x)| ≤ C(1 + |x| p ), ∀(t, x) ∈ [0, T ] × R d .

Systems of Integro-PDEs with Interconnected Obstacles and Multi-Modes Switching Problem Driven by Lévy Process

Let us introduce the following spaces:

S 2 := {{ϕ t , 0 ≤ t ≤ T } is an IR-valued, F t -
ℓ 2 := {x = (x n ) n≥1 is an IR-valued sequence s.t. x 2 := ∞ i=1
x 2 i < ∞}; H 2 (ℓ 2 ) := {ϕ = (ϕ t ) t≤T = ((ϕ n t ) n≥1 ) t≤T s.t. ∀n ≥ 1, ϕ n is P-predictable process and

E( T 0 ϕ t 2 dt) = ∞ i=1 E( T 0 |ϕ i t | 2 dt) < ∞};
L 2 := {ϕ is an IR-valued, F T -random variable such that E|ϕ| 2 < ∞}.

Recalling some results for RBSDE

We first recall the Lévy-Khintchine formula of a Lévy process (L t ) t≤T whose characteristic exponent is Ψ, i.e., ∀θ ∈ IR, E(e iθLt ) = e tΨ (θ) with

Ψ(θ) = iaθ - 1 2 ̟ 2 θ 2 + I R (e iθx -1 -iθx (|x|<1) )Π(dx) = iaθ - 1 2 ̟ 2 θ 2 + |x|>1 (e iθx -1)Π(dx) + 0<|x|<1 (e iθx -1 -iθx)Π(dx)
where a ∈ IR, ̟ ≥ 0 and Π is a measure concentrated on IR, setting Π(0) = 0, so that the domain of integration is the whole space IR and not only E := IR\{0}, called the Lévy measure of X, satisfying:

(i) I R (1 ∧ x 2 )Π(dx) < ∞;
(ii) ∃ǫ > 0, λ > 0 s.t. (-ǫ,ǫ) c e λ|x| Π(dx) < +∞.

Those conditions (i)-(ii) imply that the Lévy process (L t ) t≤T have moments of all orders. On the other hand we have,

+∞ -∞ |x| i Π(dx) < ∞, ∀i ≥ 2.
(1.2.1)

Following [START_REF] Nualart | Chaotic and predictable representations for Lévy processes[END_REF] we define, for every i ≥ 1, the so-called power-jump processes L (i) and their compensated version Y (i) , also called Teugels martingales, as follows:

L (1) t = L t L (i) t = s≤t (∆L s ) i , t ≤ T and i ≥ 2 Y (i) t = L (i) t -tE(L (i) 1 ).
Note that for any t ≤ T , E(L (i) t ) = t I R x i Π(dx) is finite for any i ≥ 2 ( [START_REF] Meyer | Probabilités et potentiel[END_REF], pp. [START_REF] Hamadène | On the starting and stopping problem: application in reversible investments[END_REF]). An orthonormalization procedure can be applied to the martingales Y (i) in order to obtain a set of pairwise strongly orthonormal martingales (H (i) ) i=∞ i=1 such that each H (i) is a linear combination of the (Y (j) ) j=1,i , i.e., H (i) = c i,i Y (i) + ... + c i,1 Y (1) .

It has been shown in [START_REF] Nualart | Chaotic and predictable representations for Lévy processes[END_REF] that the coefficients c i,k correspond to the orthonormalization of the polynomials 1, x, x 2 , ... with respect to the measure ν(dx) = x 2 Π(dx)+̟ 2 δ 0 (dx), where δ 0 is the Diracmeasure in 0. Specifically the polynomials (q i ) i≥0 defined by

q i-1 (x) = c i,i x i-1 + c i,i-1 x i-2 + ... + c i,1 , i ≥ 1 satisfy I R
q n (x)q m (x)ν(dx) = δ nm , , ∀n, m ≥ 0.

Next let us set p i (x) = xq i-1 (x) = c i,i x i + c i,i-1 x i-1 + ... + c i,1 x pi (x) = x(q i-1 (x)q i-1 (0)) = c i,i x i + c i,i-1 x i-1 + ... + c i,2 x 2 .

Then for any i ≥ 1 and t ≤ T we have: (2) ]tc i,1 E(L 1 )

H (i) t = 0<s≤t {c i,i (∆L s ) i + ... + c i,2 (∆L s ) 2 } + c i,1 L t -tE[c i,i (L 1 ) (i) + ... + c i,2 (L 1 )
= q i-1 (0)L t + 0<s≤t pi (∆L s ) -tE[ 0<s≤1 pi (∆X s )]tq i-1 (0)E(L 1 ).

As a consequence, ∆H

t = p i (∆L t ) for each i ≥ 1. In the particular case of i = 1, we obtain

H (1) t = c 1,1 (L t -tE(L 1 )) where c 1,1 = [ I R x 2 Π(dx) + ̟ 2 ] -1 2 and E[L 1 ] = a + |x|≥1 xΠ(dx).
Finally note that for any i, j ≥ 1 the predictable quadratic variation process is < H (i) , H (j) > t = δ ij t, ∀t ≤ T .

The main result in [START_REF] Nualart | Chaotic and predictable representations for Lévy processes[END_REF] is the following representation property.

Theorem 1.2.1. ( [START_REF] Nualart | Chaotic and predictable representations for Lévy processes[END_REF], Remark 2). Let ζ be a random variable of L 2 , then there exists a process Z = (Z i ) i≥1 that belongs to H 2 (ℓ 2 ) such that:

ζ = E(ζ) + i≥1 T 0 Z i s dH (i) s .
As a consequence of Theorem 2.1.1, as in the framework of Brownian noise only, one can study standard BSDEs or reflected ones. The result below related to existence and uniqueness of a solution for a reflected BSDE driven by a Lévy process, is proved in [START_REF] Ren | Generalized reflected BSDE driven by a Levy Processes and an obstacle problem for PDIE with a nonlinear Neumann boundary condition[END_REF]. Actually let us introduce a triplet (f, ξ, S) that satisfies: Assumptions (A1) (i) ξ a random variable of L 2 which stands for the terminal value ; (ii) f : [0, T ] × Ω × IR × ℓ 2 -→ IR is a function such that the process (f (t, 0, 0)) t≤T belongs to H 2 and there exists a constant κ > 0 verifying

|f (t, y, z) -f (t, y ′ , z ′ )| ≤ κ(|y -y ′ | + z -z ′ ℓ 2
), for every t, y, y ′ , z and z ′ . (iii) S := (S t ) 0≤t≤T is a process of S 2 such that S T ≤ ξ, Pa.s., and whose jumps are inaccessible stopping times. This in particular implies that for any t ≤ T , S p t = S t-, where S p is the predictable projection of S (see e.g. [START_REF] Dellacherie | Capacités et processus stochastiques[END_REF], pp.58 for more details).

In [START_REF] Ren | Generalized reflected BSDE driven by a Levy Processes and an obstacle problem for PDIE with a nonlinear Neumann boundary condition[END_REF], the authors have proved the following result related to existence and uniqueness of the solution of a reflected BSDE whose noise is driven by a Lévy process. Theorem 1.2.2. Assume that the triplet (f, ξ, S) satisfies Assumptions (A1), then there exists a unique triplet of processes (Y, U, K) := ((Y t , U t , K t )) t≤T with values in IR × ℓ 2 × IR + such that:

       (Y, U, K) ∈ S 2 × H(ℓ 2 ) × A 2 ; Y t = ξ + T t f (s, Y s , U s )ds + K T -K t - ∞ i=1 T t U i s dH (i) s , ∀t ≤ T ; Y t ≥ S t , ∀ 0 ≤ t ≤ T,
T 0 (Y t -S t )dK t = 0, Pa.s.

(1.2.

2)

The triplet (Y, U, K) is called the solution of the reflected BSDE associated with (f, ξ, S).

Let us now introduce the following assumption on the process V .

Assumptions (A2):

The process V = (V i t ) i≥1 verifies:

∞ i=1 V i t p i (△L t ) > -1 dP ⊗ dt -a.e
and there exists a constant C such that:

∞ i=1 |V i t | 2 ≤ C, dP ⊗ dt -a.e.
We will give now a comparison theorem for RBSDE driven by a Lévy process.

Theorem 1.2.3. For i = 1, 2, let (ξ i , S i , f i ) be a triple which satisfies the same Assumptions as in Theorem 1.1 and let (Y i t , K i t , U i t ) t≤T be the solution of the RBSDE associated with (ξ i , S i , f i ). Assume that:

(i) Pa.s, ξ 1 ≥ ξ 2 and ∀t ∈ [0, T ], f 1 (t, y, u) ≥ f 2 (t, y, u), S 1 t ≥ S 2 t ; (ii) For any U 1 , U 2 ∈ H 2 (l 2 ), there exists (V U 1 ,U 2 j ) j≥1 which depends on U 1 and U 2 , satisfies (A2) and such that f 1 verifies: Next we are going to make a connection between reflected BSDEs and their associated PDEs with obstacle. Consider the following SDE: X t,x s = x + s t b(r, X t,x r )dr + s t σ(r, X t,x r-)dL r , ∀t ≤ s ≤ T, (1.2.3) and X t,x s = x if s ≤ t, where we assume that the functions b and σ are jointly continuous, Lipschitz continuous w.r.t. x uniformly in t, i.e., there exists a constant C ≥ 0 such that for any t ∈ [0, T ],

f 1 (t, Y 2 t , U 1 t ) -f 1 (t, Y 2 t , U 2 t ) ≥ V U 1 ,U
x,x ′ ∈ IR, it holds,

|σ(t, x) -σ(t, x ′ )| + |b(t, x) -b(t, x ′ )| ≤ C|x -x ′ |. (1.2.4)
σ is uniformly bounded, b is of linear growth, i.e., there exists a constant C > 0, such that for all (t, x)

∈ [0, T ] × R, |b(t, x)| ≤ C(1 + |x|), |σ(t, x)| ≤ C. (1.2.5)
Under the above conditions, the process X t,x exists and is unique (see e.g. [START_REF] Laukajtys | Penalization methods for reflecting stochastic differential equations with jumps[END_REF]), and satisfies:

∀p ≥ 1, E[sup s≤T |X t,x s | p ] ≤ C(1 + |x| p ). (1.2.6)
Next let us consider the following functions:

h : x ∈ IR → h(x) ∈ IR; f : (t, x, y, u) ∈ [0, T ] × IR × IR × l 2 → f (t, x, y, u) ∈ IR; Ψ : (t, x) ∈ [0, T ] × IR → Ψ ∈ IR,
which satisfy the following assumptions:

Assumptions (A3):

(i) h, Ψ and f (t, x, 0, 0) are jointly continuous and of polynomial growth, which we denote as h, Ψ and f (t, x, 0, 0) ∈ Π p , i.e., there exist positive constants C and p such that: ∀(t, x) ∈ [0, T ] × IR, (ii) the mapping (y, z) → f (t, x, y, z) is Lipschitz continuous uniformly in (t, x) ; (iii) For any x ∈ IR, h(x) ≥ Ψ(T, x).

(iv) The generator satisfies, f (t, x, y, u) = h(t, x, y,

i≥1 θ i t u i ), ∀(t, x, y, u) ∈ [0, T ] × IR × IR × ℓ 2
where the mapping η ∈ IR -→ h(t, x, y, η) is non decreasing, and there exists a constant C > 0, such that ∀t ∈ [0, T ], z, z ′ ∈ R, x, y ∈ R, |h(t, x, y, z)h(t, x, y, z ′ )| ≤ C|zz ′ |.

Further more (θ i t ) i≥1 is uniformly bounded i.e. and moreover i≥1 θ i t p i (∆L t ) > 0, dt ⊗ dPa.e.. Noting that the assumption (A3)(iv) satisfies the assumption (ii) in Theorem 1.7, which allows us to use comparison theorem in the proof of Theorem 1.8.

In the case of Markovian setting, i.e. when randomness stems from an exogenous process (X t,x s ) s≤T , Yong Ren and Mohamed El Otmani have shown in [START_REF] Ren | Generalized reflected BSDE driven by a Levy Processes and an obstacle problem for PDIE with a nonlinear Neumann boundary condition[END_REF] the relationship between RBSDE and IPDE. Let (t, x) ∈ [0, T ] × IR k be fixed and let us consider the following reflected BSDE:

       (Y t,x , U t,x , K t,x ) ∈ S 2 × H(ℓ 2 ) × A 2 ; Y t,x s = h(X t,x T ) + T s f (r, X t,x r , Y t,x r , Z t,x r )dr + K t,x T -K t,x s - ∞ i=1 T s Z t,x,i r dH (i)
r , s ≤ T ;

∀s ≤ T, Y t,x s ≥ Ψ(s, X t,x s ) and T 0 (Y t,x s -Ψ(s, X t,x s ))dK t,x s = 0, Pa.s..

(1.2.7)

There exists a continuous deterministic function u(t, x) which satisfies ∀(t, x) ∈ [0, T ] × IR k , ∀s ∈ [t, T ], Y t,x s = u(s, X t,x s ).

(1.2.8)

Consider now the following IPDE: min u(t, x) -Ψ(t, x); -∂ t u(t, x) -Lu(t, x)f (t, x, u(t, x), Φ(u)(t, x)) = 0 u(T, x) = h(x) (1.2.9)

where L is the generator which has the following expression:

Lu(t, x) = (E[L 1 ]σ(t, x) + b(t, x))∂ x u(t, x) + 1 2 σ(t, x) 2 ̟ 2 ∂ 2
xx u(t, x) + I R [u(t, x + σ(t, x)y)u(t, x) -∂ x u(t, x)σ(t, x)y]Π(dy)

(1.2.10) and Φ(u)(t, x) = 1 c1,1 ∂ x u(t, x)σ(t, x) k=1 + I R (u(t, x + σ(t, x)y)u(t, x) -∂ x u(t, x)y)p k (y)Π(dy) k≥1 .

(1.2.11)

Theorem 1.2.4. Under Assumption (A3), the function u defined in (1.2.8) is continuous and is a viscosity solution of (1.2.9).

Motivation

In this paper, we study the existence and uniqueness of a solution to the system of integro-partial differential equations (IPDEs in short) of the form:

∀i = 1, • • • , m,      min{u i (t, x) -max j =i (u i (t, x) -g ij (t, x)); -∂ui ∂t (t, x) -Lu i (t, x) -f i (t, x, u 1 , u 2 , • • • , u m )} = 0 u i (T, x) = h i (x) (1.2.12)
where L is a generator defined in (1.2.10) and associated with a stochastic differential equation whose noise is driven by a Lévy process defined on a filtered probability space (Ω, F, (F) t≤T , P ) and then L is a non local operator.

This system is related to a stochastic optimal switching problem since a particular case is actually its associated Hamilton-Jacobi-Bellman system.

The multi-modes switching problem of interest is related to investment of a capital in the most profitable economy in the globalization. More precisely, consider an agent that aims at investing a capital in one of several economies denoted by ǫ 1 , • • • , ǫ m . His objective is to obtain the best return for the investment. The capital is invested in the economy ǫ i up to the time when the agent makes the decision to switch it from ǫ i to ǫ j (i = j) because there is no longer enough profitability in ǫ i . Moving the capital from ǫ i to ǫ j incures expenditures which amount to g ij . Therefore, the agent should deal with two main problem: what are the optimal successive times to move the capital, and when the decision to switch from current economy is made, in which new economy will the capital be invested. More precisely, let (a s ) s∈[0,T ] be the following pure jump process:

a s := α 0 {θ0} (s) + ∞ j=1 α j-1 ]θj-1,θj ] (s), ∀s ≤ T,
where {θ j } j≥0 is an increasing sequence of stopping times with values in [0, T ] and (α j ) j≥0 are random variable with values in A := {1, . . . , m} (the set of modes to which the controller can switch) such for any j ≥ 0, α j is F θj -measurable. The pair Υ = ((θ j ) j≥0 , (α j ) j≥0 ) is called a strategy of switching and when it satisfies P [θ n < T, ∀n ≥ 0] = 0 it is said admissible. Finally we denote by A i t the set of admissible strategies such that α 0 = i and θ 0 = t.

Assume next that for any i = 1, . . . , m, f i (t, x, (y i ) i=1,...,m ) = f i (t, x), i.e., f i does not depend on (y i ) i=1,m . Let Υ be an admissible strategy of A i t with which one associates a payoff given by:

J a (t, x) = J(Υ)(t, x) := E[ T t f a(s) (s, X t,x s )ds - j≥1 g αj-1,αj (θ j , X θj ) {θj <T } + h a T (X t,x T )]
where f a(s) (s, X t,x s ) = i∈A f i (s, X t,x s )1 [a(s)=i] , s ∈ [t, T ], (resp. h a T (X t,x T ) = i∈A h i (X t,x T )1 [a T =i] ) is the instantaneous (resp. terminal) payoff when the strategy a (or Υ) is implemented while g iℓ is the switching cost function when moving from mode i to mode ℓ (i, ℓ ∈ A, i = ℓ). Next let us define the optimal payoff when starting from mode i ∈ A at time t by u i (t, x) := inf

Υ∈A i t J(Υ)(t, x)
(1.2.13)

A similar problem has been already considered by Biswas et al. [START_REF] Biswas | Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion processes[END_REF], however one should emphazise that in that work, the switching costs are constant and do not depend on (t, x). This latter feature makes the problem easier to handle since one can directly work with the functions u i defined in (1.2.13).

Optimal switching problems are well documented in the literature (see e.g. [START_REF] Biswas | Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion processes[END_REF][START_REF] Carmona | Valuation of energy storage: An optimal switching approach[END_REF][START_REF] Arnarson | A PDE approach to regularity of solutions to finite horizon optimal switching problems[END_REF][START_REF] Chassagneux | A note on existence and uniqueness for solutions of multidimensional reflected BSDEs[END_REF][START_REF] Hamadène | On the starting and stopping problem: application in reversible investments[END_REF][START_REF] Hamadène | Switching problem and related system of reflected backward SDEs[END_REF][START_REF] El Asri | The finite horizon optimal multi-modes switching problem: the viscosity solution approach[END_REF][START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF][START_REF] Pham | Optimal switching over multiple regimes[END_REF][START_REF] Vathana | A mixed singular/switching control problem for a dividend policy with reversible technology investment[END_REF][START_REF] Zervos | A problem of sequential entry and exit decisions combined with discretionary stopping[END_REF][START_REF] Duckworth | A model for investment decisions with switching costs[END_REF] etc. and the references therein), especially in connection with mathematical finance, energy market, etc.

Main results

The main objective and novelty of this paper is to study system (1.2.12) without the restrictions made by Biswas et al., [START_REF] Biswas | Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion processes[END_REF] i.e., to allow the switching costs g ij to depend on (t, x) and to show that (1.2.12) has a unique solution. First let us introduce the following functions f i , h i and g ij , i, j ∈ A:

f i : [0, T ] × IR k × IR m × ℓ 2 -→ IR (t, x, (y i ) i=1,m , u) -→ f i (t, x, (y i ) i=1,m , u) h i (resp. g ij ) : [0, T ] × IR k -→ IR (t, x) -→ h i (t, x) (resp. g ij (t, x))
which satisfy:

Assumption (A4) (I) For any i ∈ A: (i) the mapping (t, x) → f i (t, x, -→ y , u) is continuous uniformly with respect to ( -→ y , u) where -→ y = (y i ) i=1,m ;

(ii) the mapping ( -→ y , u)

→ f i (t, x, -→ y , u) is Lipschiz continuous uniformly w.r.t. (t, x) ; (iii) f i (t, x, 0, 0) is of polynomial growth w.r.t. (t, x). (iv) For any U 1 , U 2 ∈ H 2 (l 2 ), X t , Y t ∈ S 2 , there exists (V U 1 ,U 2 ,i j
) j≥1 which depends on U 1 and U 2 , satisfies (A2) such that :

f i (t, X t , Y t , U 1 t ) -f i (t, X t , Y t , U 2 t ) ≥ V U 1 ,U 2 ,i , (U 1 -U 2 ) p t , dP ⊗ dt -a.e.;
(v) For any i ∈ A, for any k = i, the mapping

y k → f i (t, x, y 1 , • • • , y k-1 , y k , y k+1 , • • • , y m , u) is non- decreasing whenever the other components (t, x, y 1 , • • • , y k-1 , y k+1 , • • • , y m , u) are fixed.
(II) ∀i, j ∈ A, g ii ≡ 0 and for i = j, g jk (t, x) is non-negative, continuous with polynomial growth and satisfy the following non-free loop property:

∀(t, x) ∈ [0, T ] × R and for any sequence of indices i 1 , • • • , i k such that i 1 = i k and card{i 1 , • • • , i k } = k -1 we have: g i1i2 (t, x) + g i2i3 (t, x) + • • • + g i k i1 (t, x) > 0, ∀(t, x) ∈ [0, T ] × IR k .
(III) ∀i ∈ A, h i is continuous with polynomial growth and satisfies the following coherance conditions:

h i (x) ≥ max j∈A -i (h j (x) -g ij (T, x)), ∀x ∈ IR.
Our method is based on the link of (1.2.12) with systems of reflected BSDEs with inter-connected obstacles driven by a Lévy process, i.e., systems of the following form: ∀j = 1, . . . , m, ∀s ≤ T ,

               Y j,x,t s = h j (X t,x T ) + T s f j (r, X t,x r , (Y k,t,x r ) k∈A , (U j,x,t,i r ) i≥1 )dr - ∞ i=1 T s U j,x,t,i r dH (i) r + K j,x,t T -K j,x,t s , s ≤ T Y j,x,t s ≥ max k =j {Y k,x,t s -g jk (s, X t,x s )}, ∀s ≤ T ; [Y j,x,t s -max k =j {Y k,x,t s -g jk (s, X t,x s )}]dK j,x,t s = 0. (1.2.14)
Under assumption (A4) on the data (f i ) i=1,...,m , (h i ) i=1,...,m and (g ij ) i,j=1,...,m we show existence and uniqueness of F t -adapted processes (Y j,x,t s , (U j,x,t,i s ) i≥1 , K j,x,t s ) s≤T which satisfy (1.2.14). The proof is given in two steps.

Step 1: Let us consider the following BSDEs :

Ȳs = max j=1,m h j (X t,x T ) + T s max j=1,m f j (r, X t,x r , Ȳr , • • • , Ȳr , Ūr )dr - ∞ i=1 T s Ū i r dH (i) r and Y s = min j=1,m h j (X t,x T ) + T s min j=1,m f j (r, X t,x r , Y r , • • • , Y r , U r )dr - ∞ i=1 T s U i r dH (i) r .
Foe n ≥ 0 define (Y j,n , U j,n , K j,n ) by:

                       Y j,n ∈ S 2 , U j,n ∈ H 2 (ℓ 2 ), K j,n ∈ A 2 Y j,0 = Y Y j,n s = h j (X t,x T ) + T s f j (r, X t,x r , Y 1,n-1 r , • • • , Y j-1,n-1 r , Y j,n r , Y j+1,n-1 r , • • • , Y m,n-1 , U j,n r )dr - ∞ i=1 T s U i,j,n r dH (i) r + K j,n T -K j,n s , s ≤ T ; Y j,n s ≥ max k∈Aj (Y k,n-1 s -g jk (s, X t,x s )), ∀s ≤ T ; T 0 [Y j,n r -max k∈Aj (Y k,n-1 r -g jk (r, X t,x r ))]dK j r = 0 (1.2.15) For i = 1, • • • , m
, by induction we have: ∀n, j, ∀s ≤ T, Y j,n-1 s ≤ Y j,n s ≤ Ȳs , Pa.s., and

E[ sup s∈[0,T ] | Ȳs | 2 ] < ∞.
The sequence (Y j,n ) n≥0 has a limit denoted by Y j for j = 1, • • • , m. By the monotonic limit theorem in [START_REF] Fan | Reflected backward stochastic differential equations driven by a Levy process[END_REF], Y j ∈ S 2 and there exist

U j ∈ H 2 (ℓ 2 ), K j ∈ A 2 , such that      Y j s = h j (X t,x T ) + T s f j (r, X t,x r , -→ Y r , U j r )dr - ∞ i=1 T s U i,j r dH (i) r + K j T -K j s , s ≤ T ; Y j s ≥ max k∈Aj (Y k s -g jk (s, X t,x s )), s ≤ T, (1.2.16)
where for any j ∈ A, U j is the weak limit of (U j,n ) n≥1 in H 2 (ℓ 2 ) and for any stopping time τ , K j τ is the weak limit of K j,n τ in L 2 (Ω, F τ , P ). Finally note that K j is predictable since the processes K n,j are so, ∀n ≥ 1.

Let us now consider the following RBSDE:

             Ŷ j ∈ S 2 , Û j ∈ H 2 (ℓ 2 ), Kj ∈ A 2 ; Ŷ j s = h j (X t,x T ) + T s f j (r, X t,x r , Y 1 r , • • • , Ŷ j-1 r , Ŷ j r , Ŷ j+1 r , • • • , Y m r , Û j r )dr - ∞ i=1 T s Û i,j r dH (i) r + Kj T -Kj s , s ≤ T ; Ŷ j s ≥ max k∈Aj (Y k s -g jk (s, X t,x s )), ∀s ≤ T ; T 0 [ Ŷ j r--max k∈Aj (Y k r--g jk (r, X t,x r-))]d Kj r = 0.
(1.2.17)

Using Tanaka-Meyer's formula (see e.g. [START_REF] Protter | Stochastic integration and differential equations[END_REF], pp.216) on ( Ŷ j -Y j ) + between s and T , we can show: P -a.s., Ŷ j ≤ Y j for any j ∈ A. On the other hand, since ∀j ∈ A, Y j,n-1 ≤ Y j , we have

max k∈Aj (Y k,n-1 s -g jk (s, X t,x s )) ≤ max k∈Aj (Y k s -g jk (s, X t,x s )), ∀s ≤ T.
Then by Comparison Theorem, we obtain Y j,n ≤ Ŷ j , thus by taking limit, Y j ≤ Ŷ j which implies Y j = Ŷ j , ∀j ∈ A.

Next using Itô's formula with Y j -Ŷ j we obtain, for any s ∈ [0, T ],

(Y j s -Ŷ j s ) 2 = (Y j 0 -Ŷ j 0 ) 2 + 2 s 0 (Y j r--Ŷ j r-)d(Y j r -Ŷ j r ) + ∞ i=1 ∞ k=1 s 0 (U i,j r -Û i,j r )(U k,j r -Û k,j r )d[H i , H k ] r .
As Y j = Ŷ j and taking expectation in both-hand sides of the previous equality to obtain

E[ T 0 i≥1 (U i,j r -Û i,j r ) 2 dr] = 0.
It implies that U j = Û j , dt ⊗ dP and finally K j = Kj for any j ∈ A.

Next by the assumptions on g ij , we can show that the predictable process K j is continuous since it is predictable. As j is arbitrary in A, then the processes K j is continuous and taking into account (1.2.17), we deduce that the triples (Y j , U j , K j ), j ∈ A, is a solution for system (1.2.14).

Step 2: Now we deal with the general case, and we introduce the operator Θ :

[H 2 ] m → [H 2 ] m , Γ → Y , such that:      Y j s = h j (X t,x T ) + T s f j (r, X t,x r , Γ r , U j r )dr - ∞ i=1 T s U i,j r dH (i) r + K j T -K j s , ∀s ≤ T. Y j s ≥ max k∈Aj {Y k s -g jk (s, Y j s )}, ∀s ≤ T ; T 0 [Y j s -max k∈Aj {Y k s -g jk (s, Y j s )}]dK j s = 0 (1.2.18)
By

Step 1, we have the existence of Y j , j ∈ A. To get the uniqueness to the solution of (1.2.18), let

Γ := ((Γ i s ) s∈[0,T ] ) i∈A such that ∀i ∈ A, Γ i ∈ H 2 . For s ≤ T , let V a(.) s = h a(T ) (X t,x T ) + T s f a(r) (r, X t,x r , -→ Γ r , N a r )dr - ∞ i=1 T s N a,i r dH (i) r -A a (T, X t,x T ) + A a (s, X t,x s ).
We can prove that Y j s = V a * s = ess sup It follows that Θ is well defined. Next let us define the following norm:

Y 2,β := (E[ T 0 e βs |Y s | 2 ds]) 1 2
.

Then we prove that ,

Θ(Γ 1 ) -Θ(Γ 2 ) 2,β ≤ 2LT m β Γ 1 -Γ 2 2,β (1.2.19)
For β large enough, Θ is contraction on the Banach space (([H 2 ]) m , . 2,β ), then the fixed point theorem ensures the existence of a unique Y such that Θ(Y ) = Y , which is the unique solution of system of RBSDE (1.2.14). On the other hand there exist deterministic functions (u j (t, x)) j∈A of polynomial growth such that:

∀s ∈ [t, T ], Y j,x,t s = u j (s, X t,x s ). (1.2.20)
The next main result is the existence and uniqueness of a solution for the system of PDEs (1.2.12) with interconected obstacles. For this objective we use its link with the system of RBSDEs (1.2.14). However we are led to make, hereafter, the following additional assumption.

Assumption (A5):

For any i ∈ A, f i does not depend on the variable u ∈ ℓ 2 .

In the Brownian framework of noise, the link between systems of PDEs with interconnected obstacles and systems of reflected BSDEs with oblique reflection has been already stated in several papers (see e.g. [START_REF] El Karoui | Les aspects probabilistes du contrôle stochastique[END_REF]). Therefore in this paper we extend this link to the setting where the noise is driven by a Lévy process. Recall the system of IPDEs: ∀i ∈ A,

min{u i (t, x) -max j∈Ai (u j (t, x) -g ij (t, x)); -∂ui ∂t (t, x) -Lu i (t, x) -f i (t, x, u 1 , u 2 , • • • , u m )} = 0; u i (T, x) = h i (x) (1.2.21) where Lu(t, x) = (E[L 1 ]σ(t, x) + b(t, x))∂ x u(t, x) + 1 2 T r[(σσ T )̟ 2 D 2 xx u(t, x)]+ I R [u(t, x + σ(t, x)y) -u i (t, x) -∂u ∂x (t, x)σ(t, x
)y]Π(dy). We are going to give the definition of viscosity solution of (1.2.21). So let us define by I 1,δ , I 2,δ the following non local terms:

I(t, x, φ) := I R [φ(t, x + σ(t, x)y) -φ(t, x) - ∂φ ∂x (t, x)σ(t, x)y]Π(dy); I 1 δ (t, x, φ) = |y|≤δ [φ(t, x + σ(t, x)y) -φ(t, x) - ∂φ ∂x (t, x)σ(t, x)y]Π(dy); I 2 δ (t, x, q, φ) = |y|≥δ [φ(t, x + σ(t, x)y) -φ(t, x) -qσ(t, x)y]Π(dy); L φ u(t, x) = (E[L 1 ]σ(t, x) + b(t, x))∂ x φ(t, x) + 1 2 T r[(σσ T )̟ 2 D 2 xx φ(t, x)]+ I 1 δ (t, x, φ) + I 2 δ (t,
x, ∇φ, u). By Lemma 5.1 in Appendix, I 1 δ (t, x, φ) and I 2 δ (t, x, q, φ) verify the Assumption (NLT) which is introduced by Barles et al. ([6]).

Next, we give two definitions of the viscosity solution of (1.2.21), and according to [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF] (pp.571), they are equivalent. For locally bounded function u: (t, x) ∈ [0, T ] × R → u(t, x) ∈ R, we define its lower semi-continuous (lsc for short) envelope u * , and upper semi-continuous (usc for short) envelope u * as following:

u * (t, x) = lim

(t ′ ,x ′ )→(t,x), t ′ <T u(t ′ , x ′ ), u * (t, x) = lim (t ′ ,x ′ )→(t,x), t ′ <T u(t ′ , x ′ ). Definition 1.2.1. A function (u 1 , • • • , u m ) : [0, T ] × R → R m ∈ Π g such that for any i ∈ A, u i is lsc (resp. usc), is said to be a viscosity subsolution of (1.2.21) (resp. supsolution) if for any i ∈ A, u i (T, x) ≤ h i (x) (resp. u i (T, x) ≥ h i (x)); and for any test function ϕ ∈ Π g C 1,2 ([0, T ] × R), if (t 0 , x 0 ) ∈ [0, T ] × R is global maximum (resp. minimum) point of u i -ϕ, min{u i (t 0 , x 0 ) -max j∈Ai (u j (t 0 , x 0 ) -g ij (t 0 , x 0 )); ∂ϕ i ∂t (t 0 , x 0 ) -Lϕ i (t 0 , x 0 ) -f i ((t 0 , x 0 , u 1 (t 0 , x 0 ), • • • , u i-1 (t 0 , x 0 ), u i (t 0 , x 0 ), • • • , u m (t 0 , x 0 ))} ≤ 0 (resp. ≥ 0), (u i ) m i=1 is called a viscosity solution of (1.2.21) if (u i * ) m i=1 (resp. (u i * ) m i=1 ) is a viscosity supersolution (resp. subsolution) of (1.2.21). Definition 1.2.2. A function (u 1 , • • • , u m ) : [0, T ] × R → R m ∈ Π g such that for any i ∈ A, u i is lsc (resp. usc), is said to be a viscosity subsolution of (1.2.21) (resp. supsolution) if u i (T, x) ≤ h i (x) (resp. u i (T, x) ≥ h i (x)); and for any test function ϕ ∈ C 1,2 ([0, T ] × R), if (t 0 , x 0 ) ∈ [0, T ] × R is a maximum (resp. minimum) point of u i -ϕ on [0, T ] × B(x 0 , Cδ)
, where C is the bound of σ, and δ > 0,

min{u i (t 0 , x 0 ) -max j∈Ai (u j (t 0 , x 0 ) -g ij (t 0 , x 0 )); ∂ϕ i ∂t (t 0 , x 0 ) -L ϕ u i (t 0 , x 0 ) -f i ((t 0 , x 0 , u 1 (t 0 , x 0 ), • • • , u i-1 (t 0 , x 0 ), u i (t 0 , x 0 ), • • • , u m (t 0 , x 0 ))} ≤ 0 (resp. ≥ 0). (u i ) m i=1 is called a viscosity solution of (1.2.21) if (u i * ) m i=1 (resp. (u i * ) m i=1
) is a viscosity supersolution (resp. subsolution) of (1.2.21).

Using the first definition, we can prove the following lemma:

Lemma 1.2.1. Let (u i ) m
i=1 be a supersolution of (1.2.21) then ∀γ ≥ 0, ∃λ 0 > 0 which does not depend on θ such that ∀λ ≥ λ 0 and θ > 0,

-→ v = (u i (t, x) + θe -λt |x| 2γ+2 ) m i=1 is supersolution of (1.2.21). Remark 1.2.1. If (u i ) m
i=1 is a viscosity subsolution of (1.2.21) which belongs to Π g , i.e. for some γ > 0 and C > 0,

|u i (t, x)| ≤ C(1 + |x| γ ), ∀(t, x) ∈ [0, T ] × IR k and i ∈ A.
Then there exists λ 0 > 0 such that for any λ ≥ λ 0 and θ > 0,

-→ v (t, x) = (u i (t, x) -θe -λt (1 + |x| 2γ+2 )) m i=1
is subsolution of (1.2.21).

The next theorem shows the relationship between (1.2.21) and (1.2.14), and so the existence of the viscosity solution for (1.2.21).

Theorem 1.2.5. The function (u j (t, x)) j∈A defined in (1.2.20), is a viscosity solution of (1.2.21), with polynomial growth.

For the sake of clarity, we divide the proof into two steps.

Step1. First we will show that (u j ) j∈A is a supersolution of (1.2.21). For all j ∈ A, as u j is lsc, so u j * = u j . Consider the sequence of function:

u n j (t, x) = Y j,n,t,x t
, where Y j,n,t,x t is the unique solution of

                           Y j,x,t,0 s = min j∈A H (j) (X t,x T ) + T s min j∈A f j (r, X t,x r , Y j,x,t,0 r , • • • , Y j,x,t,0 r )dr - ∞ i=1 T s U j,x,t,i,0 r dH (i) r Y j,x,t,n s = H (j) (X t,x T ) + T s f j (r, X t,x r , Y 1,x,t,n-1 r , • • • , Y i-1,x,t,n-1 r , Y i,x,t,n r , • • • , Y m,x,t,n-1 r )dr - ∞ i=1 T s U j,x,t,i,n r dH (i) r + K j,x,t,n T -K j,x,t,n s n = 1, 2, • • • , m Y j,x,t,n s ≥ max k∈Aj {Y k,x,t,n-1 s -g jk (s, X t,x s )}, ∀s ≤ T ; (Y j,x,t,n s -max k∈Aj {Y k,x,t,n-1 s -g jk (s, X t,x s )})dK j,x,t,n s = 0. (1.2.22)
By theorem 1.3 and induction, u n j (t, x) is the unique viscosity solution of

            
-∂u j,0 ∂t (t, x) -Lu j,0 (t, x)min j∈A f j (t, x, u j,0 ) = 0; u j,0 (T, x) = min j∈A h j (x); min{u j,n (t, x)max k∈Aj (u j,n-1 (t, x)g jk (t, x));

-∂u j,n ∂t (t, x) -Lu j,n (t, x) -f j (t, x, u 1,n-1 , • • • , u j-1,n-1 , u j,n , • • • , u m,n-1 )} = 0, u j,n (T, x) = h j (x).
(1.2.23) Also we know that, ∀j ∈ A, u n j ր u j , and for any n = 1, 2, • • • , u n j is continuous with polynomial growth. This together with the monotonic condition on f j , i.e. for any i ∈ A, for any k = i, the mapping

y k → f i (t, x, y 1 , • • • , y k-1 , y k , y k+1 , • • • , y m , u) is nondecreasing whenever the other components (t, x, y 1 , • • • , y k-1 , y k+1 , • • • , y m ,
u) are fixed, using the similar way with Theorem 1 in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF], we can show that

- ∂φ ∂t (t, x) -L φ u j (t, x) -f j (t, x, u 1 (t, x), • • • , u j-1 (t, x), u j (t, x), • • • , u m (t, x)) ≥ 0.
We have know that ∀j ∈ A, u j ≥ max k∈Aj (u k (t, x)g jk (t, x)) and u j (T, x) = h j (x), so (u j ) m j=1 is a supersolution of (1.2.21).

Step2. Next we show that (u * j ) j∈A is a subsolution of (1.2.21), using the same method as [START_REF] Hamadène | Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems[END_REF] we can prove that:

min{u * j (T, x) -h j (x); u * j (T, x) -max k∈Aj (u * k (T, x) -g jk (T, x))} = 0.
this together with the non-free loop assumption on the cost function g ij , we can show that: u * j (T, x) = h j (x), ∀j ∈ A. Noting that since u n j ր u j and u n j is continuous, we have

u * j (t, x) = lim n→∞ sup * u n j (t, x) = lim n→∞,t ′ →t,x ′ →x u n j (t ′ , x ′ ).
Besides ∀j ∈ A and n ≥ 0 we deduce from the construction of u n j that :

u n j (t, x) ≥ max l∈Aj (u n l (t, x) -g jl (t, x)),
take the limit to obtain:

∀j ∈ A, ∀x ∈ R, u * j (t, x) ≥ max l∈Aj (u * l (t, x) -g jl (t, x)). Next, fix j ∈ A, for (t, x) ∈ [0, T [×R such that u * j (t, x) -max l∈Aj (u * l (t, x) -g jl (t, x)) > 0. (1.2.24)
By the same way as Step 1, we have:

- ∂φ ∂t (t, x) -L φ u * ,j (t, x) -f j (t, x, u * ,1 (t, x), • • • , u * ,j-1 (t, x), u * ,j (t, x), • • • , u * ,m (t, x)) ≤ 0.
This together with (1.2.24) shows that (u * j ) m j=1 is a subsolution of (1.2.21).

The second main result is a comparison theorem of subsolution and supersolution, from which we can get the continuity and uniqueness of the viscosity soluion of (1.2.21).

Theorem 1.2.6. Let (u j ) j∈A be a subsolution of (1.2.21), (v j ) j∈A be a supsolution of (1.2.21)

such that ∀j ∈ A, u j , v j ∈ Π g , then ∀(t, x) ∈ [0, T ] × R, u j (t, x) ≤ v j (t, x).
The proof is based on Jensen-Ishii's Lemma [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF]. For ( t, x) which is the maximum point of u j (t, x)w j (t, x), for ε > 0 define test function as follows:

Φ j ε (t, x, y) := u j (t, x) -w j (t, y) - |x -y| 2 ε -ψ(t, x), where ψ(t, x) := ρ|x -x| 4 + |t -t| 2 .
Let (t ε , x ε , y ε ) be such that

Φ j ε (t ε , x ε , y ε ) = max (t,x,y)∈[0,T ]×R 2 Φ j ε (t, x, y).
Then we proved two facts:

(i) lim ε (u j (t ε , x ε ), w j (t ε , y ε )) = (u j ( t, x), w j ( t, x)). (ii) l := I 1,δ (t ε , x ε , φ x ) + I 2,δ (t ε , x ε , q ε u , u j ) ≤ I 1,δ (t ε , y ε , -φ y ) + I 2,δ (t ε , y ε , q ε w , w j ) + O( |xε-yε| 2 ε ) + o ε (1) + 1 ε o δ (1) + o ρ (1)
. These with Jensen-Ishii's Lemma, by contradiction and doubling variable technique we can prove that

∀(t, x) ∈ [0, T ] × R, u j (t, x) ≤ v j (t, x).
The last main result is the existence and uniqueness of systems of IPDE (1.2.21), when (-f j ) j∈A verify [A4] (1)-(iv), i.e. ∀j ∈ A, k ∈ A j f j is non-increasing in y k , which we rewrite as Assumption (A4'):

(1) For any i ∈ A:

(i) the mapping (t, x) → f i (t, x, -→ y ) is continuous uniformly with respect to -→ y where -→ y = (y i ) i=1,m ;

(ii)the mapping -→ y → f i (t, x, -→ y ) is Lipschiz continuous uniformly w.r.t. (t, x) ;

(iii) f i (t, x, 0) is of polynomial growth w.r.t. (t, x).

(iv) For any i ∈ A, for any k = i, the mapping

y k → f i (t, x, y 1 , • • • , y k-1 , y k , y k+1 , • • • , y m ) is nonin- creasing whenever the other components (t, x, y 1 , • • • , y k-1 , y k+1 , • • • , y m ) are fixed.
(2) ∀i, j ∈ A, g ii ≡ 0 and for i = j, g jk (t, x) is non-negative, continuous with polynomial growth and satisfy the following non-free loop property:

∀(t, x) ∈ [0, T ] × R and for any sequence of indices i 1 , • • • , i k such that i 1 = i k and card{i 1 , • • • , i k } = k -1 we have: g i1i2 (t, x) + g i2i3 (t, x) + • • • + g i k i1 (t, x) > 0, ∀(t, x) ∈ [0, T ] × IR k .
(3) ∀i ∈ A, h i is continuous with polynomial growth and satisfies the following coherance conditions:

h i (x) ≥ max j∈A -i (h j (x) -g ij (T, x)), ∀x ∈ IR. Theorem 1.2.7. If (f j ) j∈A verify [A4 ′ ], then systems of IPDE (1.2.21
) has a unique continuous viscosity solution (u j ) j∈A with polynomial growth.

1.3 Viscosity solution of system of variational inequalities with interconnected bilateral obstacles and connections to multiple modes switching game of jump-diffusion processes

Preliminaries

Let (Ω, F, (F t ) t≥0 , P ) be a stochastic basis such that F 0 contains all P -null elements of F, and F t + ε>0 F t+ε = F t , t ≥ 0, and suppose that the filtration is generated by the following two mutually independent process: -a d-dimensional standard Brownian motion (W t ) t≥0 -a Poisson random measure N on R + × E, where E R l -{0} is equipped with its Borel field B E , with compensator ν(dtde) = dtn(de), such that { N ((0, t]×A) = (N -ν)((0, t]×A)} 0≤t≤T is and F t -martingale for all A ∈ B E satisfying n(A) < ∞. n is assumed to be a σ-finite measure on (E, B E ) satisfying:

E (1 ∧ x 2 )n(dx) < ∞. (1.3.1)
Let T be a fixed positive constant and A 1 (resp. A 2 ) denote the set of switching modes for player 1 (resp. player 2). Let m 1 (resp. m 2 ) be the cardinal of the set A 1 (resp. A 2 ) and for (i, j)

∈ A 1 × A 2 , A 1 i := A 1 -{i} and A 2 j := A 2 -{j}. Next, for -→ y = (y kl ) (k,l)∈A 1 ×A 2 ∈ R m1×m2 .
For any y 1 ∈ R, denote by [ -→ y i,j , y 1 ] the matrix which is obtained from -→ y by replacing the element y ij with y

1 . A function Φ : (t, x) ∈ [0, T ] × R → Φ(t, x) ∈ R is called of polynomial growth if there exist two non-negative real constant C and γ such that |Φ(t, x)| ≤ C(1 + |x| γ ).
Hereafter, this class of functions is denoted by Π g .

We define the following spaces of processes, let:

P be the σ-algebra of F t -predictable subsets of Ω × [0, T ]; L 2 := {ξ is an IR-valued, F T -random variable such that ||ξ|| 2 L 2 := E|ξ| 2 < ∞}; H 2 := {{ϕ t , 0 ≤ t ≤ T } is an IR-valued, F t -progressively measurable process s.t. ||ϕ|| 2 H 2 := E( T 0 |ϕ t | 2 ) < ∞}; S 2 := {{ϕ t , 0 ≤ t ≤ T } is an IR-valued, F t -adapted RCLL process s.t. ||ϕ|| 2 S 2 := E( sup 0≤t≤T |ϕ t | 2 ) < ∞} ;
A 2 is the subspace of S

:= E( T 0 E |U t (e)| 2 n(de)dt) < ∞}.
In this paper, we investigate existence and uniqueness of viscosity solutions -→ v (t, x) := (v ij (t, x)) (i,j)∈A 1 ×A 2 of the following system of variational inequalities with upper and lower interconnected obstacles:

∀(i, j) ∈ A 1 × A 2 ,            min{(v ij -L ij [ -→ v ])(t, x), max{(v ij -U ij [ -→ v ])(t, x), -∂ t v ij (t, x) -Lv ij (t, x) -g ij (t, x, (v kl (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x v ij (t, x), B ij v ij (t, x))}} = 0 v ij (T, x) = h ij (x) (1.3.2)
where, for any (t, x) ∈ [0, T ] × R,

Lφ(t, x) := b(t, x)D x φ(t, x) + 1 2 σ 2 (t, x)D 2 xx φ(t, x) + E (φ(t, x + β(x, e)) -φ(t, x) -D x φ(t, x)β(x, e))n(de), B ij φ(t, x) = E (φ(t, x + β(x, e)) -φ(t, x))γ ij (x, e)n(de),
and

∀(i, j) ∈ A 1 × A 2 , L ij [ -→ v ])(t, x) := max k∈A 1 i {(v kj -g ik )(t, x)} and U ij [ -→ v ])(t, x) := min l∈A 2 j {(v il -g jl )(t, x)}.
Denote by

I 1 δ (t, x, φ) = |e|≤δ (φ(t, x + β(x, e)) -φ(t, x) -D x φ(t, x)β(x, e))n(de); I 2 δ (t, x, q, φ) = |e|≥δ (φ(t, x + β(x, e)) -φ(t, x) -qβ(x, e))n(de); I 1,B ij δ (t, x, φ) = |e|≤δ (φ(t, x + β(x, e)) -φ(t, x))γ ij (x, e)n(de); I 2,B ij δ (t, x, φ) = |e|≥δ (φ(t, x + β(x, e)) -φ(t, x))γ ij (x, e)n(de); I(t, x, φ) = E (φ(t, x + β(x, e)) -φ(t, x) -D x φ(t, x)β(x, e))n(de); I B ij (t, x, φ) = E (φ(t, x + β(x, e)) -φ(t, x))γ ij (x, e)n(de); L φ u(t, x) := b(t, x)D x φ(t, x) + 1 2 σ 2 (t, x)D 2 xx φ(t, x) + I 1 (t, x, φ) + I 2 (t, x, D x φ, u),
The following assumptions will be in force throughout the rest of the paper.

(A0) The functions b(t, x) and σ(t, x): [0, T ] × R → R are jointly continuous in (t, x), of linear growth in (t, x) and Lipschitz continuous w.r.t. x, meaning that there exists a non-negative constant C such that for any (t, x, x ′ ) ∈ [0, T ] × R we have:

|b(t, x)| + |σ(t, x)| ≤ C(1 + |x|), |σ(t, x) -σ(t, x ′ )| + |b(t, x) -b(t, x ′ )| ≤ C|x -x ′ |.
The function β : R × E → R is measurable, continuous in x and such that for some real K and all e ∈ E, for any x, x ′ ∈ R,

|β(x, e)| ≤ K(1 ∧ |e|), |β(x, e) -β(x ′ , e)| ≤ K|x -x ′ |(1 ∧ |e|). (A1) For any (i, j) ∈ A 1 × A 2 , g ij (t, x, -→ y , z, q) : R × R × R m1×m2 × R d × R → R, (i)
is continuous in (t, x) uniformly w.r.t. the other variables ( -→ y , z, q) and for any (t, x) the mapping (t, x) → g i,j (t, x, 0, 0, 0) is of polynomial growth.

(ii) satisfies the standard hypothesis of Lipschitz continuity w.r.t. the variables ( -→ y , z, q), i.e. ∀(t, x) ∈

[0, T ] × R, ∀( -→ y 1 , -→ y 2 ) ∈ R m1×m2 × R m1×m2 , (z 1 , z 2 ) ∈ R d+d , (q 1 , q 2 ) ∈ R × R, |g ij (t, x, -→ y 1 , z 1 , q 1 ) -g ij (t, x, -→ y 2 , z 2 , q 2 )| ≤ C(| -→ y 1 --→ y 2 | + |z 1 -z 2 | + |q 1 -q 2 |),
where, | -→ y | stands for the standard Euclidean norm of -→ y in R m1 × R m2 .

(iii) q → g ij (t, x, y, z, q) is non-decreasing, for all (t, x, y, z)

∈ [0, T ] × R × R m1×m1 × R. Futhermore, let γ ij : R × B E → R such that there exists C > 0, 0 ≤ γ ij (x, e) ≤ C(1 ∧ |e|), x ∈ R, e ∈ B E |γ ij (x, e) -γ ij (x ′ , e)| < C|x -x ′ |(1 ∧ |e|), x, x ′ ∈ R, e ∈ E.
We set

f ij (t, x, y, z, u) = g ij (t, x, y, z, E u(e)γ ij (x, e)n(de)), for (t, x, y, z, u) ∈ [0, t] × R × R m1×m2 × R × L 2 (R, B E , n).
Noting that under Assumption (A0) and (A1), by ([5]), I,

I B ij , I 1 δ , I 2 δ , I 1,B ij δ , I 2,B ij δ
satisfy the Assumption (NLT), which is given in appendix.

(A2) Monotonicity: For any (i, j) ∈ A 1 × A 2 and any (k, l) = (i, j) the mapping y k,l → g i,j (t, x, -→ y , z, u) is non-decreasing. (A3) The functions h ij (x) : R → R are continuous w.r.t. x, belong to class Π g and satisfy

∀(i, j) ∈ A 1 × A 2 and x ∈ R, max k∈A 1 i (h kj (x) -g ik (T, x)) ≤ h ij (x) ≤ min l∈A 2 j (h il (x) -g jl (T, x)),
where g ik and g jl are given in the next assumption. (A4) The no free loop property: The switching costs g ik and ḡjl are non-negative, jointly continuous in (t, x), belong to Π g and satisfy the following condition:

For any loop in A 1 ×A 2 , i.e., any sequence of pairs (i 1 , j 1 ), . . . , (i N , j N ) of Γ 1 ×Γ 2 such that (i N , j N ) = (i 1 , j 1 ), card{(i 1 , j 1 ), . . . , (i N , j N )} = N -1 and ∀ q = 1, . . . , N -1, either i q+1 = i q or j q+1 = j q , we have

∀(t, x) ∈ [0, T ] × IR k , q=1,N -1 ϕ iqiq+1 (t, x) = 0, (1.3.3)
where, ∀ q = 1, . . . , N -1, ϕ iqiq+1 (t, x) = -g iqiq+1 (t, x)1 1 iq =iq+1 + ḡjqiq+1 (t, x)1 1 jq =jq+1 .

Consider now the following SDE:

X t,x s = x + s t b(r, X t,x r )dr + s t σ(r, X t,x r )dW r + s t E β(X t,x r-, e) N (drde), s ∈ [t, T ], x ∈ R.
The existence and uniqueness of the solution X t,x s follows from [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF].

Next, we give three definitions of the viscosity solution of (1.3.2), and according to [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF] (pp.571), they are equivalent. For locally bounded function u: (t, x) ∈ [0, T ] × R → u(t, x) ∈ R, we define its lower semi-continuous (lsc for short) envelope u * , and upper semi-continuous(usc for short) envelope u * as following:

u * (t, x) = lim

(t ′ ,x ′ )→(t,x), t ′ <T u(t ′ , x ′ ), u * (t, x) = lim (t ′ ,x ′ )→(t,x), t ′ <T u(t ′ , x ′ ) Definition 1.3.1. A function -→ u = (u ij (t, x)) (i,j)∈A 1 ×A 2 : [0, T ] × R → R A 1 ×A 2 such that for any (i, j) ∈ A 1 × A 2 , u ij ∈ Π g is lsc (resp. usc),
ϕ ∈ C 1,2 ([0, T ] × R), if (t 0 , x 0 ) ∈ [0, T ] × R is a global maximum (resp. minimum) point of u i,j -ϕ,                    min{(u ij -L ij [ -→ u ])(t 0 , x 0 ), max{(u ij -U ij [ -→ u ])(t 0 , x 0 ), -∂ t ϕ(t 0 , x 0 ) -b(t 0 , x 0 )∂ x ϕ(t 0 , x 0 ) -1 2 σ 2 (t 0 , x 0 )∂ 2 xx ϕ(t 0 , x 0 ) -I(t 0 , x 0 , ϕ) -g ij (t 0 , x 0 , (u kl (t 0 , x 0 )) (k,l)∈A 1 ×A 2 , σ(t 0 , x 0 ))∂ x ϕ(t 0 , x 0 ), I B ij (t 0 , x 0 , ϕ)}} ≤ 0 (resp. ≥ 0); v ij (T, x) ≤ h ij (x) (resp. ≥). Definition 1.3.2. A function -→ u = (u ij (t, x)) (i,j)∈A 1 ×A 2 : [0, T ] × R → R A 1 ×A 2 such that for any (i, j) ∈ A 1 × A 2 , u ij ∈ Π g is lsc (resp. usc)
, is said to be a viscosity subsolution (resp. supsolution) of (1.3.2) if for any δ > 0, (t 0 , x 0 ) ∈ (0, T ) and a function ϕ ∈ C 1,2 ([0, T ]×R), such that (t 0 , x 0 ) ∈ [0, T ]×R is a maximum (resp. minimum) point of u i,jϕ on [0, T ] × B(x 0 , Kδ), where K is the bound of β,

                   min{(u ij -L ij [ -→ u ])(t 0 , x 0 ), max{(u ij -U ij [ -→ u ])(t 0 , x 0 ), -∂ t ϕ(t 0 , x 0 ) -b(t 0 , x 0 )∂ x ϕ(t 0 , x 0 ) -1 2 σ 2 (t 0 , x 0 )∂ 2 xx ϕ(t 0 , x 0 ) -I 1 δ (t 0 , x 0 , φ) -I 2 δ (t 0 , x 0 , ∂ x ϕ, u ij ) -g ij (t 0 , x 0 , (u kl (t 0 , x 0 )) (k,l)∈A 1 ×A 2 , σ(t 0 , x 0 ))∂ x ϕ(t 0 , x 0 ), I 1,B ij δ (t 0 , x 0 , ϕ) + I 2,B ij δ (t 0 , x 0 , u ij ))}} ≤ 0 (resp. ≥ 0); v ij (T, x) ≤ h ij (x) (resp. ≥).
Definition 1.3.3. (i) For a function u: [0, T ] × R → R, lsc (resp. usc), we denote J -u(t, x) the parabolic subjet (resp. J + u(t, x) the parabolic superjet) of u at (t,x)∈ [0, T ] × R, as the set of triples (p,q,M)∈ R × R × S k ; where S k is the set of symmetric real matrices of dimension k

u(t ′ , x ′ ) ≥ u(t, x) + p(t ′ -t) + q, x ′ -x + 1 2 x ′ -x, M (x ′ -x) + o(|t ′ -t| + |x ′ -x|) 2 (resp. ≤) (ii)
We denote Ju(t, x) (resp. J+ u(t, x)) the parabolic limiting superjet (resp. superjet) of u at (t,x), as the set of triples (p,q,M)∈ R × R × S k s.t.

(p, q, M ) = lim n→∞ (p n , q n , M n ), (t, x) = lim n→∞ (t n , x n )
where (p n , q n , M n ) ∈ J -u(t n , x n ) (resp.J + u(t n , x n )) and u(t, x) = lim n→∞ u(t n , x n ).

(iii) A function -→ u = (u ij (t, x)) (i,j)∈A 1 ×A 2 : [0, T ] × R → R A 1 ×A 2 such that for any (i, j) ∈ A 1 × A 2 , u ij ∈ Π g is lsc (resp. usc)
, is said to be a viscosity subsolution (resp. supsolution) of (1.3.2) if for any δ > 0, (t 0 , x 0 ) ∈ (0, T )×R and a function φ ∈ C 1,2 ([0, T ]×R), if (t 0 , x 0 ) ∈ [0, T ]×R is a maximum (resp. minimum) point of u i,jφ on (0, T ) × B(x 0 , Kδ), and if (p, q, M ) ∈ Ju i,j (t 0 , x 0 )(resp. J+ u i,j (t 0 , x 0 )) with q = D t φ(t 0 , x 0 ), p = D x φ(t 0 , x 0 ), and M ≥ D 2 xx φ(t 0 , x 0 ) (resp. M ≤ D 2 xx φ(t 0 , x 0 )), then:

                   min{(u ij -L ij [ -→ u ])(t 0 , x 0 ), max{(u ij -U ij [ -→ u ])(t 0 , x 0 ), -p -b(t 0 , x 0 )q -1 2 σ 2 (t 0 , x 0 )M -I 1 δ (t 0 , x 0 , φ) I 2 δ (t 0 , x 0 , q, u ij ) -g ij (t 0 , x 0 , (u kl (t 0 , x 0 )) (k,l)∈A 1 ×A 2 , σ(t 0 , x 0 )q, I 1,B ij δ (t 0 , x 0 , φ) + I 2,B ij δ (t 0 , x 0 , u ij ))}} ≤ 0 (resp. ≥ 0); v ij (T, x) ≤ h ij (x) (resp. ≥). Definition 1.3.4. A function -→ u = (u ij (t, x)) (i,j)∈A 1 ×A 2 such that for any (i, j) ∈ A 1 × A 2 , u ij ∈ Π g , is called a viscosity solution of (1.3.2) if (u ij * (t, x)) (i,j)∈A 1 ×A 2 (resp. (u * ij (t, x)) (i,j)∈A 1 ×A 2
) is a viscosity supersolution (resp. subsolution) of (1.3.2).

Two approximating schemes

For n, m ≥ 0, let (Y i,j,n,m , Z i,j,n,m , U i,j,n,m ) (i,j)∈A 1 ×A 2 be the solution of the following system of BSDEs.

       (Y i,j,n,m , Z i,j,n,m , U i,j,n,m ) ∈ S 2 × H 2 × H 2 ( N ); dY i,j,n,m s = -f i,j,n,m (s, X t,x s , (Y k,l,n,m s ) (k,l)∈A 1 ×A 2 , Z i,j,n,m s , U i,j,n,m s )ds +Z i,j,n,m s dB s + E U i,j,n,m s (e) N (dsde), s ≤ T. Y i,j,n,m T = h i,j (X t,x T ), (1.3.4)
where,

f i,j,n,m (s, X t,x s , (y ij ) (ij)∈A 1 ×A 2 , z s , u s ) := g i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z s , E u s (e)γ ij (X t,x
s , e)n(de))

= g i,j (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z s , E u s (e)γ ij (X t,x s , e)n(de))

+ n(y ijmax

k∈A 1 i {y kj -g ik (s, X t,x s )}) --m(y ij -min l∈A 2 j {y il -g jl (s, X t,x s )}) + .
Let us recall that under Assumption (A1), the solution (Y i,j,n,m , Z i,j,n,m , U i,j,n,m ) (i,j)∈A 1 ×A 2 of (1.3.4) exists and is unique (see [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF]). By the assumption(A1)(iii), we have the comparison theorem for BSDE with jumps (see [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF] Theorem 2.4). The we have:

Proposition 1.3.1.
For any (i, j) ∈ A 1 × A 2 and n, m ≥ 0 we have

P -a.s., Y i,j,n,m ≤ Y i,j,n+1,m and Y i,j,n,m+1 ≤ Y i,j,n,m , (i, j) ∈ A 1 × A 2 . (1.3.5)
Moreover, for any (i, j) ∈ A 1 × A 2 and n, m ≥ 0, there exists a deterministic continuous function v i,j,n,m ∈ Π g such that, for any t ≤ T ,

Y i,j,n,m s = v i,j,n,m (s, X t,x s ), s ∈ [t, T ]. (1.3.6)
Finally, for any (i, j) ∈ A 1 × A 2 and n, m ≥ 0,

v i,j,n,m (t, x) ≤ v i,j,n+1,m (t, x) and v i,j,n,m+1 (t, x) ≤ v i,j,n,m (t, x), (t, x) ∈ [0, T ] × R (1.3.7)
The proof of first claim is based on the result by Xuehong Zhu (2010) ( [START_REF] Zhu | Backward stochastic viability property with jumps and applications to the comparison theorem for multidimensional BSDEs with jumps[END_REF], Theorem 3.1) related to the comparison of solutions of multi-dimensional BSDEs. The second claim is just the representation of solutions of standard BSDEs with jumps by deterministic functions in the Markovian framework (see [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF]). The inequalities of (1.3.7) are obtained by taking s = t in (1.3.5) in view of the representation (1.3.6) of Y i,j,n,m by v i,j,n,m and X t,x . Now we will show two approximation schemes obtained from the sequence Y i,j,m,n , (i, j) ∈ A 1 ×A 2 ) n,m of the solution of system (1.3.4). The first scheme is a sequence of decreasing reflected BSDEs with interconnected lower obstacles:

∀(i, j) ∈ A 1 × A 2 ,                                  ( Ȳ i,j,m , Zi,j,m , Ū i,j,m , Ki,j,m ) ∈ S 2 × H 2 × H 2 ( N ) × A 2 ; Ȳ i,j,m s = h i,j (X t,x T ) + T s f i,j,m (r, X t,x r , ( Ȳ k,l,m r ) (k,l)∈A 1 ×A 2 , Zi,j,m r , Ū i,j,m r )dr - T s Zi,j,m r dB r - T s E Ū i,j,m r (e) N (drde) + Ki,j,m T -Ki,j,m s , s ≤ T ; Ȳ i,j,m s ≥ max k∈A 1 i { Ȳ k,j,m s -g ik (s, X t,x s )}, s ≤ T ; T 0 ( Ȳ i,j,m s -max k∈A 1 i { Ȳ k,j,m s -g ik (s, X t,x s )})d Ki,j,m s = 0, (1.3.8) where, ∀(i, j) ∈ A 1 × A 2 , m ≥ 0 and s ≤ T , f i,j,m (s, X t,x s , -→ y , z, u) :=g ij,+,m (s, X t,x s , (y kl ) (k,l)∈A 1 ×A 2 , z, E u(e)γ ij (X t,x s , e)n(de)) =g ij (s, X t,x s , (y kl ) (k,l)∈A 1 ×A 2 , z, E u ij (e)γ ij (X t,x s , e)n(de)) -m(y ij -min l∈A 2 j (y il + g jl (s, X t,x s ))) + .
Thanks to the assumption (A1)-(A3) and non free loop assumption, by Theorem (5.4.1) in appendix, the solution of (1.3.8) exists and is unique. Moreover, we have the following properties.

Proposition 1.3.2. For any (i, j) ∈ A 1 × A 2 and m ≥ 0, we have:

(i) lim n→∞ E[ sup t≤s≤T |Y i,j,n,m s -Ȳ i,j,m s | 2 ] → 0 (1.3.9)
(ii) Pa.s., Ȳ i,j,m ≥ Ȳ i,j,m+1 .

(iii)There exists a deterministic continuous functions (ū k,l,m ) (k,l)∈A 1 ×A 2 in Π g such that, for every t ≤ T ,

Ȳ i,j,m s = ūi,j,m (s, X t,x s ), s ∈ [t, T ]. (1.3.10) Moreover, ∀(i, j) ∈ A 1 × A 2 and (t, x) ∈ [0, T ] × R k , ūi,j,m (t, x) ≥ ūi,j,m+1 (t, x).
Finally, (ū i,j,m ) (i,j)∈A 1 ×A 2 is the unique viscosity solution in the class Π g of the following system of variational inequalities with inter-connected obstacles.

∀(i, j) ∈ A 1 × A 2 ,              min{ū i,j,m (t, x) -max k∈A 1 i (ū k,j,m (t, x) -g ik (t, x)); -∂ t ūi,j,m (t, x) -Lū i,j,m (t, x) g ij,+,m (t, x, (ū k,l,m (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x ūi,j,m (t, x), B ij ūi,j,m (t, x))} = 0;
ūi,j,m (T, x) = h i,j (x).

(1.3.11)

The second scheme is the increasing approximating scheme:

∀(i, j) ∈ A 1 × A 2 ,                                    (Y i,j,n , Z i,j,n , U i,j,n , K i,j,n ) ∈ S 2 × H 2 × H 2 ( N ) × A 2 ; Y i,j,n s = h i,j (X t,x T ) + T s f i,j,n (r, X t,x r , (Y k,l,n r ) (k,l)∈A 1 ×A 2 , Z i,j,n r , U i,j,n r )dr - T s Z i,j,n r dB r - T s E U i,j,n r (e) N (drde) + K i,j,n T -K i,j,n s , s ≤ T ; Y i,j,n s ≤ min l∈A 2 j {Y i,l,n s + g jl (s, X t,x s )}, s ≤ T, T 0 (Y i,j,n s -min l∈A 2 j {Y k,j,n s + g jl (s, X t,x s )})dK i,j,n s = 0, (1.3.12)
where, ∀(i, j) ∈ A 1 × A 2 , n ≥ 0 and s ≤ T ,

f i,j,n (s, X t,x s , -→ y , z, u) :=g ij,-,n (s, X t,x s , (y kl ) (k,l)∈A 1 ×A 2 , z, E u(e)γ ij (X t,x s , e)n(de)) =g ij (s, X t,x s , (y kl ) (k,l)∈A 1 ×A 2 , z, E u(e)γ ij (X t,x
s , e)n(de))

+ n(y ijmax

k∈A 1 i (Y kj -g ik (s, X t,x s ))) -.
Thanks to the assumption (A1)-(A3) and the non free loop assumption, by Theorem 5.4.1 in appendix, the solution of (1.3.12) exists and is unique.

Proposition 1.3.3. For any (i, j) ∈ A 1 × A 2 and n ≥ 0, we have:

(i) lim m→∞ E[ sup t≤s≤T |Y i,j,n,m s -Y i,j,n s | 2 ] → 0 (1.3.13)
(ii) For any n ≥ 0, Pa.s., Y i,j,n ≤ Y i,j,n+1 .

(iii)There exits a unique m 1 × m 2 -uplet of deterministic continuous functions

(u k,l,n ) (k,l)∈A 1 ×A 2 in Π g such that, for every t ≤ T , Y i,j,n s = u i,j,n (s, X t,x s ), s ∈ [t, T ]. (1.3.14) Moreover, ∀(i, j) ∈ A 1 × A 2 and (t, x) ∈ [0, T ] × R k , u i,j,n (t, x) ≤ u i,j,n+1 (t, x).
Finally, (u i,j,n ) (i,j)∈A 1 ×A 2 is the unique viscosity solution in the class Π g of the following system of variational inequalities with inter-connected obstacles.

∀(i, j) ∈ A 1 × A 2 ,              max{u i,j,n (t, x) -min l∈A 2 j
(u i,l,n (t, x) + g jl (t, x)); -∂ t u i,j,n (t, x) -Lu i,j,n (t, x) g ij,-,n (t, x, (u k,l,n (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x u i,j,n (t, x), B ij u i,j,n (t, x))} = 0; u i,j,n (T, x) = h i,j (x).

(1.3.15)

We define ūij (t, x) := lim m→∞ ūi,j,m (t, x), u ij (t, x) := lim n→∞ u i,j,n (t, x).

Then, as a by-product of Proposition 1.3.2 and 1.3.3, we have:

Corollary 1.3.1. ∀(i, j) ∈ A 1 × A 2 , the function ūij (resp. u ij ) is usc (resp. lsc). Moreover, ūij and u ij belong to Π g , for any (t, x) ∈ [0, T ] × R, u ij (t, x) ≤ ūij (t, x).

Main results

In this paper we will show the uniqueness and existence of solution for (1.3.2). To begin with, we need the following lemma for the proof of uniqueness.

Lemma 1.3.1. Let ( -→ u ) (i,j)∈A 1 ×A 2 (resp. ( -→ w ) (i,j)∈A 1 ×A 2
) be an usc subsolution (resp. lsc supersolution) of (1.3.2) which belongs to Π g . For (t, x) ∈ [0, T ] × R and let Γ(t, x) be the following set:

Γ(t, x) := {(i, j) ∈ A 1 × A 2 , u ij (t, x) -w ij (t, x) = max (k,l)∈A 1 ×A 2 (u kl (t, x) -w kl (t, x))}.
Then there exists (i 0 , j 0 ) ∈ Γ(t, x) such that

u i0j0 (t, x) > L i0j0 [ -→ u ](t, x), w i0j0 (t, x) < U i0j0 [ -→ w ](t, x).
(1.3.16)

We then prove:

Theorem 1.3.1. Let ( -→ u ) (i,j)∈A 1 ×A 2 (resp. ( -→ w ) (i,j)∈A 1 ×A 2
) is an usc subsolution (resp. lsc supersolution) of (1.3.2) which belongs to Π g . Then it holds that for any (i, j)

∈ A 1 × A 2 , u ij (t, x) ≤ w ij (t, x), (t, x) ∈ [0, T ] × R. Corollary 1.3.2. System (1.3.
2) has at most one viscosity solution belongs to Π g , and it is necessary continuous.

It will takes three steps to prove the existence.

Proposition 1.3.4. The family (ū ij ) (i,j)∈A 1 ×A 2 is a viscosity subsolution of the system (1.3.2).

Proposition 1.3.5. Let m 0 be fixed in N . Then the family (ū ij,m0 ) (i,j)∈A 1 ×A 2 is a viscosity supersolution of the system (1.3.2).

Consider now the set U m0 defined as follows.

U m0 = { -→ u := (u ij ) (i,j)∈A 1 ×A 2 s.t. -→ u is a subsolution of (1.3.2), ∀(i, j) ∈ A 1 × A 2 , ūi,j ≤ u i,j ≤ ūij,m0 }. U m0 is not empty since it contains (ū ij ) (i,j)∈A 1 ×A 2 . Next for (t, x) ∈ [0, T ] × R and (i, j) ∈ A 1 × A 2 , set: m0 u ij (t, x) = sup{u ij (t, x), (u kl ) (k,l)∈A 1 ×A 2 ∈ U m0 }.
Now we give the main result of this section, which mainly consists in adapting the Perron's method to construct a viscosity solution to (1.3.2).

Theorem 1.3.2. The family ( m0 u ij ) (i,j)∈A 1 ×A 2 does not depend on m 0 and is the unique continuous viscosity solution in the class Π g of the system (3.1.2).

1.4 On the identity of min-max and max-min solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles.

Assumptions and notations

Let T (resp. k, d) be a fixed positive constant (resp. two integers) and Γ 1 (resp. Γ 2 ) denote the set of switching modes for player 1 (resp. 2). For later use, we shall denote by Λ the cardinal of the product set Γ 1 ×Γ 2 and for (i, j) ∈ Γ 1 ×Γ 2 , (Γ 1 ) -i := Γ 1 -{i} and (Γ 2 ) -j := Γ 2 -{j}. For y = (y kl ) (k,l)∈Γ 1 ×Γ 2 ∈ IR Λ , (i, j) ∈ Γ 1 × Γ 2 , and y ∈ IR, we denote by [(y kl ) (k,l)∈Γ 1 ×Γ 2 -{i,j} , y] the matrix obtained from the matrix y = (y kl ) (k,l)∈Γ 1 ×Γ 2 by replacing the element y ij with y.

For any (i,

j) ∈ Γ 1 × Γ 2 , let b : (t, x) ∈ [0, T ] × IR k → b(t, x) ∈ IR k ; σ : (t, x) ∈ [0, T ] × IR k → σ(t, x) ∈ IR k×d ; f ij : (t, x, y, z) ∈ [0, T ] × IR k+Λ+d → f ij (t, x, y, z) ∈ IR ; g ik : (t, x) ∈ [0, T ] × IR k → g ik (t, x) ∈ IR (k ∈ (Γ 1 ) -i ); ḡjl : (t, x) ∈ [0, T ] × IR k → ḡjl (t, x) ∈ IR (l ∈ (Γ 2 ) -j ); h ij : x ∈ IR k → h ij (x) ∈ IR. A function Φ : (t, x) ∈ [0, T ] × IR k → Φ(t, x)
∈ IR is called of polynomial growth if there exist two non-negative real constants C and γ such that

|Φ(t, x)| ≤ C(1 + |x| γ ), (t, x) ∈ [0, T ] × IR k .
Hereafter, this class of functions is denoted by Π

g . Let C 1,2 ([0, T ] × IR k ) (or simply C 1,2
) denote the set of real-valued functions defined on [0, T ] × IR k , which are once (resp. twice) differentiable w.r.t. t (resp.

x) and with continuous derivatives.

The following assumptions on the data of the systems (1.4.7) and (1.4.8) are in force throughout the paper.

(H0) The functions b and σ are jointly continuous in (t, x) and Lipschitz continuous w.r.t. x uniformly in t, meaning that there exists a non-negative constant C such that for any (t, x, x

′ ) ∈ [0, T ] × R k+k we have |σ(t, x) -σ(t, x ′ )| + |b(t, x) -b(t, x ′ )| ≤ C|x -x ′ |.
Therefore, they are also of linear growth w.r.t. x, i.e., there exists a constant C such that for any

(t, x) ∈ [0, T ] × IR k , |b(t, x)| + |σ(t, x)| ≤ C(1 + |x|). (H1) Each function f ij (i) is continuous in (t,
x) uniformly w.r.t. the other variables ( y, z) and, for any (t, x), the mapping (t, x) → f ij (t, x, 0, 0) is of polynomial growth.

(ii) is Lipschitz continuous with respect to the variables (

y := (y ij ) (i,j)∈Γ1×Γ2 , z) uniformly in (t, x), i.e. ∀ (t, x) ∈ [0, T ] × R k , ∀ ( y 1 , y 2 ) ∈ R Λ × R Λ , (z 1 , z 2 ) ∈ R d × R d , |f ij (t, x, y 1 , z 1 ) -f ij (t, x, y 2 , z 2 )| ≤ C (| y 1 -y 2 | + |z 1 -z 2 |) ,
where, | y| stands for the standard Euclidean norm of y in IR Λ .

(H2) Monotonicity: Let y = (y kl ) (k,l)∈Γ 1 ×Γ 2 . For any (i, j) ∈ Γ 1 × Γ 2 and any (k, l) = (i, j) the mapping y kl → f ij (s, y, z) is non-decreasing.

(H3) The functions h ij , which are the terminal conditions in the systems (1.4.7) and (1.4.8), are continuous with respect to x, belong to class Π g and satisfy

∀(i, j) ∈ Γ 1 × Γ 2 and x ∈ IR k , max k∈(Γ 1 ) -i h kj (x) -g ik (T, x) ≤ h ij (x) ≤ min l∈(Γ 2 ) -j h il (x) + ḡjl (T, x) .
(H4) The no free loop property: The switching costs g ik and ḡjl are non-negative, jointly continuous in (t, x), belong to Π g and satisfy the following condition:

For any loop in Γ 1 × Γ 2 , i.e., any sequence of pairs (i 1 , j 1 ), . . . , (i N , j N ) of Γ 1 × Γ 2 such that (i N , j N ) = (i 1 , j 1 ), card{(i 1 , j 1 ), . . . , (i N , j N )} = N -1 and any q = 1, . . . , N -1, either i q+1 = i q or j q+1 = j q , we have

∀(t, x) ∈ [0, T ] × IR k , q=1,N -1 ϕ iqiq+1 (t, x) = 0, (1.4.1)
where, ∀ q = 1, . . . , N -1, ϕ iqiq+1 (t, x) = -g iqiq+1 (t, x)1

1 iq =iq+1 + ḡjqiq+1 (t, x)1 1 jq =jq+1 .
This assumption implies in particular that

∀ (i 1 , . . . , i N ) ∈ (Γ 1 ) N such that i N = i 1 and card{i 1 , . . . , i N } = N -1, N -1 p=1 g i k ,i k+1 > 0 (1.4.2)
and

∀ (j 1 , . . . , j N ) ∈ (Γ 2 ) N such that j N = j 1 and card{j 1 , . . . , j N } = N -1, N -1 p=1 ḡj k ,j k+1 > 0. (1.4.3)
By convention we set ḡj,j = g i,i = 0.

Conditions (1.4.2) and (1.4.3) are classical in the literature of switching problems and usually referred to as the no free loop property.

We now introduce the probabilistic tools we need later. Let (Ω, F, P ) be a fixed probability space on which is defined a standard d-dimensional Brownian motion B = (B t ) 0≤t≤T whose natural filtration is (F 0 t := σ{B s , s ≤ t}) 0≤t≤T . Let F = (F t ) 0≤t≤T be the completed filtration of (F 0 t ) 0≤t≤T with the P-null sets of F, hence (F t ) 0≤t≤T satisfies the usual conditions, i.e., it is right continuous and complete. On the other hand let P be the σ-algebra on [0, T ] × Ω of F-progressively measurable sets.

Next, let (i) H (iii) A 2 i be the subset of S 2 of non-decreasing processes K = (K t ) t≤T such that K 0 = 0. For (t, x) ∈ [0, T ] × IR k , let X t,x be the diffusion process solution of the following standard SDE:

∀s ∈ [t, T ], X t,x s = x + s t b(r, X t,x r )dr + s t σ(r, X t,x r )dB r ; X t,x s = x, s ∈ [0, t]. (1.4.4)
Under Assumption (H0) on b and σ, the process X t,x exists and is unique. Moreover, it satisfies the following estimates: For all p ≥ 1,

E[sup s≤T |X t,x s | p ] ≤ C(1 + |x| p ). (1.4.5) Its infinitesimal generator L X is given, for every (t, x) ∈ [0, T ] × IR k and φ ∈ C 1,2 , by L X φ(t, x) := 1 2 k i,j=1 (σσ * (t, x)) i,j ∂ 2 xixj φ(t, x) + i=1,k b i (t, x)∂ xi φ(t, x) = 1 2 T r[σσ ⊤ (t, x)D 2 xx φ(t, x)] + b(t, x) ⊤ D x φ(t, x).
(1.4.6)

Motivation

Let us consider the following two systems of partial differential equations (PDEs) with bilateral interconnected obstacles (i.e., the obstacles depend on the solution) of min-max and max-min types: for any

(i, j) ∈ Γ 1 × Γ 2 , (t, x) ∈ [0, T ] × IR k ,        min v ij (t, x) -L ij ( v)(t, x) ; max v ij (t, x) -U ij ( v)(t, x); -∂ t v ij -L X (v ij )(t, x) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ(t, x) ⊤ D x v ij (t, x)) = 0 ; v ij (T, x) = h ij (x) (1.4.7)
and

       max vij (t, x) -U ij ( v)(t, x) ; min vij (t, x) -L ij ( v)(t, x) -∂ t vij -L X (v ij )(t, x) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ(t, x) ⊤ D x vij (t, x)) = 0 ; vij (T, x) = h ij (x)
(1.4.8) where (i) Γ 1 and Γ 2 are finite sets (possibly different); (ii) For any (t, x)

∈ [0, T ] × IR k , v(t, x) = (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 and for any (i, j) ∈ Γ 1 × Γ 2 , L ij ( v)(t, x) = max k∈Γ 1 ,k =i {v kj (t, x) -g ik (t, x)}, U ij ( v)(t, x) = min p∈Γ 2 ,p =j {v ip (t, x) + ḡjp (t, x)}.
(iii) L X is a second order generator associated with a diffusion process described below.

The systems (1.4.7) and (1.4.8) are of min-max and max-min types respectively. The barriers L ij ( v), U ij ( v) and L ij ( v), U ij ( v) depend on the solution v and v of (1.4.7) and (1.4.8) respectively. They are related to zero-sum switching game problems since actually, specific cases of these systems, stand for the Hamilton-Jacobi-Bellman-Isaacs equations associated with those games.

Switching problems have recently attracted a lot of research activities, especially in connection with mathematical finance, commodities, and in particular energy, markets, etc (see e.g. [START_REF] Carmona | Valuation of energy storage: An optimal switching approach[END_REF][START_REF] Ludkovski | Stochastic switching games and duopolistic competition in emissions markets[END_REF][START_REF] Chassagneux | Discrete-time Approximation of Multidimensional BSDEs with oblique reflections[END_REF][START_REF] Chassagneux | A note on existence and uniqueness for solutions of multidimensional reflected BSDEs[END_REF][START_REF] Elie | Probabilistic representation and approximation for coupled systems of variational inequalities[END_REF][START_REF] Arnarson | A PDE approach to regularity of solutions to finite horizon optimal switching problems[END_REF][START_REF] Djehiche | A finite horizon optimal multiple switching problem[END_REF][START_REF] Djehiche | Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles[END_REF][START_REF] El Asri | The finite horizon optimal multi-modes switching problem: the viscosity solution approach[END_REF][START_REF] Hamadène | On the starting and stopping problem: application in reversible investments[END_REF][START_REF] Hamadène | Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems[END_REF][START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF][START_REF] Ishii | Viscosity Solutions of a System of Nonlinear Second-Order Elliptic PDEs Arising in Switching Games[END_REF][START_REF] Li | Optimal Switching Problems under Partial Information[END_REF][START_REF] Lundström | Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type[END_REF][START_REF] Lundström | Systems of Variational Inequalities for Non-Local Operators Related to Optimal Switching Problems: Existence and Uniqueness[END_REF][START_REF] Perninge | Irreversible investments with delayed reaction: an application to generation re-dispatch in power system operation[END_REF][START_REF] Zervos | A problem of sequential entry and exit decisions combined with discretionary stopping[END_REF][START_REF] Duckworth | A model for investment decisions with switching costs[END_REF][START_REF] Pham | Optimal switching over multiple regimes[END_REF][START_REF] Vathana | A mixed singular/switching control problem for a dividend policy with reversible technology investment[END_REF] and the references therein). Several points of view, mainly dealing with control problems have been considered (theoritical and applied [START_REF] Carmona | Valuation of energy storage: An optimal switching approach[END_REF][START_REF] Ludkovski | Stochastic switching games and duopolistic competition in emissions markets[END_REF][START_REF] Chassagneux | A note on existence and uniqueness for solutions of multidimensional reflected BSDEs[END_REF][START_REF] Djehiche | A finite horizon optimal multiple switching problem[END_REF][START_REF] El Asri | The finite horizon optimal multi-modes switching problem: the viscosity solution approach[END_REF][START_REF] Hamadène | On the starting and stopping problem: application in reversible investments[END_REF][START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF][START_REF] Perninge | Irreversible investments with delayed reaction: an application to generation re-dispatch in power system operation[END_REF], numerics [START_REF] Chassagneux | Discrete-time Approximation of Multidimensional BSDEs with oblique reflections[END_REF][START_REF] Hamadène | On the starting and stopping problem: application in reversible investments[END_REF], filtering and partial information [START_REF] Li | Optimal Switching Problems under Partial Information[END_REF]). However, except [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF][START_REF] Ishii | Viscosity Solutions of a System of Nonlinear Second-Order Elliptic PDEs Arising in Switching Games[END_REF], problems related to games did not attract that much interest in the literature.

In [START_REF] Djehiche | Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles[END_REF], by means of systems of reflected backward stochastic differential equations (BSDEs) with inter-connected obstacles in combination with Perron's method, Djehiche et al. ( [START_REF] Djehiche | Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles[END_REF]) have shown that each of the systems (1.4.7) and (1.4.8) has a unique continuous solution with polynomial growth, under classical assumptions on the data f ij , ḡij , g ij , h ij . The question of whether or not these solutions coincide was conjectured as an open problem, leaving a possible connection of the solution of system (1.4.7) and (1.4.8) with zero-sum switching games unanswered.

Main results

The main objective of this paper is two-fold: (i) to investigate under which additional assumptions on the data of these problems, the unique solutions of systems (1.4.7) and (1.4.8) coincide; (ii) to make a connection between this solution and the value function of the associated zero-sum switching game. Indeed, we show that if the switching costs of one side, i.e. either (ḡ ij ) (i,j)∈Γ 1 ×Γ 2 or (g ij ) (i,j)∈Γ 1 ×Γ 2 , are regular enough, then the solutions of the systems (1.4.7) and (1.4.8) coincide. Furthermore, we show that this solution has a representation as a value function of a zero-sum switching game. To the best of our knowledge these issues have not been addressed in the literature yet. The main strategy to obtain these results is to show that the barriers, which depend on the solution, are comparable and then to make use of Theorem 5.5.1 (whose proof in an appendix at the end of the paper) to conclude that the solutions of the min-max and max-min systems coincide. This comparison is obtained under a regularity assumption on (ḡ ij ) (i,j)∈Γ 1 ×Γ 2 or (g ij ) (i,j)∈Γ 1 ×Γ 2 . Theorem 5.5.1 extends a result derived in [START_REF] Hamadene | BSDEs with two reflecting barriers driven by a Brownian and a Poisson noise and related Dynkin game[END_REF] on minmax and max-min PDEs with fixed obstacles, where we relax the condition of strict separation between the obstacles. To get the result that (1.4.7) and (1.4.8) coincide, we should be able to compare the inter-connected obstacles of them, i.e.

(i) ∀(i, j) ∈ Γ 1 × Γ 2 , L ij ( v) ≤ U ij ( v) or (ii) ∀(i, j) ∈ Γ 1 × Γ 2 , L ij ( v) ≤ U ij ( v).
(1.4.9)

For that let us introduce the following assumption.

(H5):

(i) For any (i, j) ∈ Γ 1 × Γ 2 , the functions ḡij are C 1,2 . Moreover, D x ḡij and D 2 xx ḡij belong to Π g . Furthermore, for any j 1 , j 2 , j 3 ∈ Γ 2 such that |{j 1 , j 2 , j 3 }| = 3, ḡj1j3 (t, x) < ḡj1j2 (t, x) + ḡj2j3 (t, x), ∀(t, x) ∈ [0, T ] × IR k .
(ii) For any (i, j) ∈ Γ 1 × Γ 2 , the function f ij verifies the following estimate:

|f ij (t, x, y, z ij )| ≤ C(1 + |x| p )
for some real constants C and p. Proposition 1.4.1. Under Assumptions (H0)-(H5) we have, for every

(i, j) ∈ Γ 1 × Γ 2 , L ij ( v) ≤ v ij ≤ U ij ( v).
The following estimate is the key to the proof of this proposition: For every (i,

j) ∈ Γ 1 × Γ 2 and m ≥ 0, E m T 0 l∈(Γ 2 ) -j { Y ij,m s -Y il,m s -ḡjl (s, X t,x s )} + ds ≤ C(1 + |x| p ), (1.4.10)
where, Y is the unique solution of system of RBSDE (4.2.2) , and the constant C is independent of m and x. As a by product of Proposition 1.4.1 and Theorem 5.5.2 (displayed in the appendix), we have:

Theorem 1.4.1. Under Assumptions (H0)-(H5), for any (i, j) ∈ Γ 1 × Γ 2 , it holds that v ij = vij .
The next main result is the connection between this solution and the value function of the associated zero-sum switching game. Let us made the following assumption: (H6):

(i) For any (i, j) ∈ Γ 1 × Γ 2 , the function f ij does not depend on z ij . (ii) For any (i, j) ∈ Γ 1 × Γ 2 , the function f ij does not depend on ( y, z ij ).
We first describe briefly the zero-sum switching game. Assume we have two players π 1 and π 2 who intervene on a system with the help of switching strategies. An admissible switching strategy for π 1 (resp. π 2 ) is a sequence δ := (σ n , ξ n ) n≥0 (resp. ν := (τ n , ζ n ) n≥0 ) where for any n ≥ 0, (i) σ n (resp. τ n ) is an F-stopping times such that P -a.s.,

σ n ≤ σ n+1 ≤ T (resp. τ n ≤ τ n+1 ≤ T ) ; (ii) ξ n (resp. ζ n ) is a random variable with values in Γ 1 (resp. Γ 2 ) which is F σn (resp. F τn )-measurable ; (iii) P [σ n < T, ∀n ≥ 0] = P [τ n < T, ∀n ≥ 0] = 0 ; (iv) If (A δ s )
s≤T and (B ν s ) s≤T are the F-adapted RCLL processes defined by:

∀ s ∈ [t, T ), A δ s = n≥1 g ξn-1ξn (σ n , X t,x σn )1 [σn≤s] and A δ T = lim s→T A δ s , and ∀ s ∈ [t, T ), B ν s = n≥1 ḡζn-1ζn (τ n , X t,x τn )1 [τn≤s] and B ν T = lim s→T B ν s . Then, E[(A δ T ) 2 + (B ν T ) 2 ] < ∞. For any s ≤ T , A δ s (resp. B ν s )
is the cumulative switching cost at time s for π 1 (resp. π 2 ) when she implements the strategy δ (resp. ν).

Next, for t ∈ IR, i ∈ Γ 1 (resp. j ∈ Γ 2 ), we say that the admissible strategy δ

:= (σ n , ξ n ) n≥0 (resp. ν := (τ n , ζ n ) n≥0 ) belongs A i π1 (t) (resp. A i π2 (t)) if σ 0 = t, ξ 0 = i, E[(A δ T ) 2 ] < ∞ (resp. τ 0 = t, ζ 0 = j, E[(B ν T ) 2 ] < ∞).
Given an admissible strategy δ (resp. ν) of π 1 (resp. π 2 ) one associates a stochastic process (u s ) s≤T (resp. (v s ) s≤T ) which indicates along with time the current mode of π 1 (resp. π 2 ) and which is defined by:

∀s ≤ T, u s = ξ 0 1 {σ0} (s) + n≥1 ξ n-1 1 ]σn-1,σn] (s) (resp. v s = ζ 0 1 {τ0} (s) + n≥1 ζ n-1 1 ]τn-1,τn] (s)). (1.4.11) Let now δ = (σ n , ξ n ) n≥0 (resp. ν = (τ n , ζ n ) n≥0
) be a strategy for π 1 (resp. π 2 ) which belongs to A i π1 (t) (resp. A j π2 (t)). The interventions of the players are not free and generate a payoff which is a reward (resp. cost) for π 1 (resp. π 2 ) and whose expression is given by

J t (δ, ν) := E[h u T v T (X T ) + T t f (r, X t,x r , u r , v r )dr -A δ T + B ν T |F t ], (1.4.12)
where, for any (k, l) ∈ Γ 1 × Γ 2 , we set f (s, x, k, l) = f kl (s, x), since f kl is assumed to not depend on ( y, z ij ).

Theorem 1.4.2. Suppose Assumptions (H0)-( H5) and (H6)-(ii) are satisfied. Then, for any

(i 0 , j 0 ) ∈ Γ 1 × Γ 2 , v i0j0 (t, x) = ess sup δ∈A i 0 π 1 (t) ess inf ν∈A j 0 π 2 (t) J t (δ, ν) = ess inf ν∈A j 0 π 2 (t) ess sup δ∈A i 0 π 1 (t) J t (δ, ν), (1.4.13)
where J t (δ, ν) is the payoff of the switching game defined in (1.4.12).

As a by product of Theorem (1.4.1) and the uniqueness of the solution of system (1.4.7) we have the following result in the case when the functions f ij depend also on y.

Corollary 1.4.1. Suppose Assumptions (H0)-(H5) and (H6)-(i) are satisfied and let (v ij ) (i,j)∈Γ 1 ×Γ 2 be the unique solution of system (1.4.7) and (1.4.8). Then for any

(t, x) ∈ [0, T ]×IR k and (i 0 , j 0 ) ∈ Γ 1 ×Γ 2 , v i0j0 (t, x) = ess sup δ∈A i 0 π 1 ess inf ν∈A j 0 π 2 Jt (δ, ν) = ess inf ν∈A j 0 π 2 ess sup δ∈A i 0 π 1 Jt (δ, ν). (1.4.14)
where,

Jt (δ, ν) := E[h u T v T (X T ) + T t f urvr (r, X t,x r , (v kl (r, X t,x r )) (k,l)∈Γ 1 ×Γ 2 )dr -A δ T + B ν T |F t ]. (1.4.15)
Chapter 2

Systems of Integro-PDEs with Interconnected Obstacles and Multi-Modes Switching Problem

Driven by Lévy Process.

This chapter is a joint work with Said Hamadène.

Preliminaries

A Lévy process is an IR-valued RCLL (for right continuous with left limits) stochastic process L = {L t , t ≥ 0} defined on a probability space (Ω, F, P) with stationary and independent increments (L 0 = 0) and stochastically continuous.

For t ≤ T let us set F t = G t ∨ N where G t := σ{L s , 0 ≤ s ≤ t} and N is the P-null sets of F, therefore {F t } t≤T is complete and right continuous. Next by P we denote the σ-algebra of predictable processes on [0, T ] × Ω and finally for any RCLL process (Γ t ) t≤ we denote by Γ t-:= lim sրt Γ s and ∆Γ t := Γ t -Γ t-its jump at t, t ∈ (0, T ].

We now introduce the following spaces:

(a) S 2 := {ϕ := {ϕ t , 0 ≤ t ≤ T } is an IR-valued, F t -adapted RCLL process s.t. E( sup 0≤t≤T |ϕ t | 2 ) < ∞} ; A 2 is the subspace of S 2 of non-decreasing continuous processes null at t = 0 ; (b) H 2 := {ϕ := {ϕ t , 0 ≤ t ≤ T } is an IR-valued, F t -progressively measurable process s.t. E( T 0 |ϕ t | 2 dt) < ∞}; (c) ℓ 2 := {x = (x n ) n≥1 is an IR-valued sequence s.t. x 2 := ∞ i=1 x 2 i < ∞}; (d) H 2 (ℓ 2 ) := {ϕ = (ϕ t ) t≤T = ((ϕ n t ) n≥1 ) t≤T such that ∀n ≥ 1, ϕ n is a P-measurable process and E( T 0 ϕ t 2 dt) = ∞ i=1 E( T 0 |ϕ i t | 2 dt) < ∞}; L 2 := {ξ, an IR-valued and F T -measurable random variable such that E[|ξ| 2 ] < ∞} ; (e) Π g is the space of deterministic functions u(t, x) from [0, T ] × IR k into IR of polynomial growth, i.e.
, such that for some nonnegative constants p and C one has,

|u(t, x)| ≤ C(1 + |x| p ), ∀(t, x) ∈ [0, T ] × IR k .
Let us now recall the Lévy-Khintchine formula of a Lévy process (L t ) t≤T whose characteristic exponent is Ψ, i.e., ∀t ≤ T and θ ∈ IR, E(e iθLt ) = e tΨ(θ)

with

Ψ(θ) = iaθ - 1 2 ̟ 2 θ 2 + I R (e iθx -1 -iθx (|x|<1) )Π(dx) = iaθ - 1 2 ̟ 2 θ 2 + |x|≥1 (e iθx -1)Π(dx) + 0<|x|<1 (e iθx -1 -iθx)Π(dx)
where a ∈ IR, ̟ ≥ 0 and Π is a σ-finite measure on IR * := IR -{0} (we Π({0}) = 0 and then the domain of integration is the whole space), called the Lévy measure of L, satisfying

I R (1 ∧ x 2 )Π(dx) < ∞ (2.1.1)
and ∃ǫ > 0, λ > 0 s.t.

(-ǫ,ǫ) c e λ|x| Π(dx) < +∞. (2.1.2) Conditions (3.1.1)-(2.1.2) imply that for any i ≥ 2, I R |x| i Π(dx) < ∞
and then the process (L t ) t≤T have moments of any order.

Next following Nualart-Schoutens [START_REF] Nualart | Backward stochastic differential equations and Feynman-Kac formula for Lévy processes, with applications in finance[END_REF] we define, for every i ≥ 1, the so-called power-jump processes L (i) and their compensated version Y (i) , also called Teugels martingales, as follows: ∀t ≤ T , L

t = L t and for i ≥ 2, L (i) t = s≤t (∆L s ) i , Y (i) t = L (i) t -tE(L (i) 1 ). Note that for any t ≤ T , E(L (i) t ) = t I R x i Π(dx) < ∞ for any i ≥ 2 ([46], pp.29). (1)
An orthonormalization procedure can be applied to the martingales Y (i) in order to obtain a set of pairwise strongly orthonormal martingales (H (i) ) i≥1 such that each H (i) is a linear combination of (Y (j) ) j=1,i , i.e., 1) .

H (i) = c i,i Y (i) + ... + c i,1 Y (
It has been shown in [START_REF] Nualart | Backward stochastic differential equations and Feynman-Kac formula for Lévy processes, with applications in finance[END_REF] that the coefficients c i,k correspond to the orthonormalization of the polynomials 1, x, x 2 , ... with respect to the measure ν(dx) = x 2 Π(dx) + ̟ 2 δ 0 (dx) (δ 0 is the Dirac measure at 0). Specifically the polynomials (q i ) i≥0 defined by

q i-1 (x) = c i,i x i-1 + c i,i-1 x i-2 + ... + c i,1 , i ≥ 1 satisfy I R q n (x)q m (x)ν(dx) = δ nm , ∀n, m ≥ 0. Next let us set p i (x) = xq i-1 (x) = c i,i x i + c i,i-1 x i-1 + ... + c i,1 x and pi (x) = x(q i-1 (x) -q i-1 (0)) = c i,i x i + c i,i-1 x i-1 + ... + c i,2 x 2 .
Then for any i ≥ 1 and t ≤ T we have:

H (i) t = 0<s≤t {c i,i (∆L s ) i + ... + c i,2 (∆L s ) 2 } + c i,1 L t -tE[c i,i (L 1 ) (i) + ... + c i,2 (L 1 ) (2) ] -tc i,1 E(L 1 ) = q i-1 (0)L t + 0<s≤t pi (∆L s ) -tE[ 0<s≤1 pi (∆L s )] -tq i-1 (0)E(L 1 ).
As a consequence, for any t ≤ T and i ≥ 1, ∆H

= p i (∆L t ) for each i ≥ 1. In the particular case of i = 1, we obtain

H (1) t = c 1,1 (L t -tE(L 1 )) where c 1,1 = [ I R x 2 Π(dx) + ̟ 2 ] -1 2 and E[L 1 ] = a + |x|≥1 xΠ(dx). (2.1.3)
Finally note that for any i, j ≥ 1 the predictable quadratic variation process of H (i) and H (j) is

H (i) , H (j) t = δ ij t, ∀t ≤ T .
Remark 2.1.1. If Π = 0, we are in the classical Brownian case and all non-zero degree polynomials q i (x) will vanish, giving H (i) = 0, i ≥ 2. On the other hand, if Π only has mass at 1, we are in the Poisson case and once more H (i) = 0, i ≥ 2. Both cases are degenerate ones in this Lévy process framework.

The main result in the paper by Nualart-Schoutens [START_REF] Nualart | Chaotic and predictable representations for Lévy processes[END_REF] is the following representation property which allows for developping the BSDE theory in this Lévy framework.

Theorem 2.1.1. ( [START_REF] Nualart | Chaotic and predictable representations for Lévy processes[END_REF], pp.118). Let ζ be a random variable of L 2 , then there exists a process Z = (Z i ) i≥1 that belongs to H 2 (ℓ 2 ) such that:

ζ = E(ζ) + i≥1 T 0 Z i s dH (i) s .

Systems of Reflected BSDEs with Oblique Reflection driven by a Lévy process 2.2.1 Reflected BSDE driven by a Lévy process and their relationship with IPDEs

As a consequence of Theorem 2.1.1, and as in the framework of Brownian noise only, one can study standard BSDEs or reflected ones. The result below related to existence and uniqueness of a solution for a reflected BSDE driven by a Lévy process, is proved in [START_REF] Ren | Generalized reflected BSDE driven by a Levy Processes and an obstacle problem for PDIE with a nonlinear Neumann boundary condition[END_REF]. Indeed let us introduce a triplet (f, ξ, S) that satisfies:

Assumptions (A1): (i) ξ a random variable of L 2 which stands for the terminal value ; (ii) f : [0, T ] × Ω × IR × ℓ 2 -→
IR is a function such that the process (f (t, 0, 0)) t≤T belongs to H 2 and there exists a constant κ > 0 verifying

|f (t, y, z) -f (t, y ′ , z ′ )| ≤ κ(|y -y ′ | + z -z ′ ℓ 2 )
, for every t, y, y ′ , z and z ′ .

(iii) S := (S t ) 0≤t≤T is a process of S 2 such that S T ≤ ξ, Pa.s., and whose jump times are inaccessible stopping times. This in particular implies that for any t ≤ T , S p t = S t-, where S p is the predictable projection of S (see e.g. [START_REF] Dellacherie | Capacités et processus stochastiques[END_REF], pp.58 ) for more details on those notions.

In [START_REF] Ren | Generalized reflected BSDE driven by a Levy Processes and an obstacle problem for PDIE with a nonlinear Neumann boundary condition[END_REF], the authors have proved the following result related to existence and uniqueness of the solution of one barrier reflected BSDEs whose noise is driven by a Lévy process. Theorem 2.2.1. Assume that the triple (f, ξ, S) satisfies Assumptions (A1). Then there exists a unique triplet of processes (Y, U, K)

:= ((Y t , U t , K t )) t≤T with values in IR × ℓ 2 × IR + such that:        (Y, U, K) ∈ S 2 × H(ℓ 2 ) × A 2 ; Y t = ξ + T t f (s, Y s , U s )ds + K T -K t - ∞ i=1 T t U i s dH (i) s , ∀t ≤ T ; Y t ≥ S t , ∀ 0 ≤ t ≤ T and T 0 (Y t -S t )dK t = 0, P -a.s. (2.2.1)
The triple (Y, U, K) is called the solution of the reflected BSDE associated with (f, ξ, S).

To proceed we need to compare solutions of reflected BSDEs of types (2.2.1). So let us consider a stochastic process

V = (V t ) t≤T = (V i ) i≥1 = ((V i t ) t≤T ) i≥1 which belongs to H 2 (ℓ 2
) and let M := (M t ) t≤T be the stochastic integral defined by:

∀t ≤ T, M t := ∞ i=1 t 0 V i s dH (i) s .
We next denote by ε(M ) := (ε(M ) t ) t≤T the process that satisfies: ∀t ≤ T ,

ε(M ) t = 1 + t 0 ε(M ) s-dM s .
By Doléans-Dade's formula we have (see e.g. [START_REF] Protter | Stochastic integration and differential equations[END_REF]):

∀t ≤ T, ε(M ) t = exp M t - 1 2 [M, M ] c t - 0≤s≤t △M s 0≤s≤t {1 + △M s }.
Let us now introduce the following assumption on the process V .

Assumptions (A2):

The process

V = (V i ) i≥1 = ((V i t ) t≤T ) i≥1 verifies: ∞ i=1 V i t p i (△L t ) > -1, dP ⊗ dt -a.e (2.2.2)
and there exists a constant C such that:

∞ i=1 |V i t | 2 ≤ C, dP ⊗ dt -a.e. ( 2

.2.3)

We then have:

Proposition 2.2.1. Assume that Assumption (A2) is fulfilled. Then, P-a.s., for any t ∈ [0, T ], ε(M ) t > 0 and ε(M ) ∈ S 2 .
Proof. First note that for any t ≤ T ,

△M t = ∞ i=1 V i t △ H (i) t = ∞ i=1 V i t p i (△L t ) > -1, therefore for any t ≤ T , ε(M t ) > 0.
Next by using Doléans-Dade's formula and since

d H (i) , H (j) s = δ ij ds, we have: ∀t ≤ T , ε(M ) 2 t = ε(2M + [M, M ]) t = ε(2 ∞ i=1 . 0 V i s dH (i) s + ∞ i=1 ∞ j=1 . 0 V i s V j s d[H (i) , H (j) ] s ) t = ε(2 ∞ i=1 . 0 V i s dH (i) s + ∞ i=1 . 0 |V i s | 2 ds + ∞ i=1 ∞ j=1 . 0 V i s V j s d([H (i) , H (j) ] s -H (i) , H (j) s )) t = ε(N ) t exp{ ∞ i=1 t 0 |V i s | 2 ds}
where for t ≤ T ,

N t = 2 ∞ i=1 t 0 V i s dH (i) s + ∞ i=1 ∞ j=1 t 0 V i s V j s d([H (i) , H (j) ] s -H (i) , H (j) s )
is a local martingale. On the other hand, the quantity

∞ i=1 T 0 |V i s | 2 ds is bounded and ε(N ) ≥ 0, then for any t ≤ T , E[(ε(M ) t ) 2 ] ≤ CE[ε(N ) 0 ] ≤ C since ε(N ) is a supermartingale. It follows that ε(M ) is
∞ i=1 T 0 |V i s | 2 ds ≤ C, P -a.s. (2.2.4) Next for two processes U i = (U i k ) k≥1 , i = 1, 2, of H 2 (ℓ 2
) we define their scalar product in H 2 (ℓ 2 ) which we denote by U 1 , U 2 p := ( U 1 , U 2 p t ) t≤T as:

∀t ≤ T, U 1 , U 2 p t = k≥1 U 1 k (t)U 2 k (t). Proposition 2.2.2. : Let ξ ∈ L 2 , ϕ := (ϕ s ) s≤T ∈ H 2 , δ := (δ s )
s≤T a uniformly bounded process, and

finally let V = (V i ) i≥1 ∈ H 2 (ℓ 2 ) satisfying (A2). Let (Y, U ) := (Y t , U t ) t≤T ∈ S 2 × H 2 (ℓ 2
) be the solution of the following BSDE:

∀t ≤ T, Y t = ξ + T t (ϕ s + δ s Y s + V, U p s )ds - ∞ i=1 T t U i s dH (i) s .
(2.2.5)

For t ≤ T , let (X t s ) s∈[t,T ] be the process defined as follows:

∀s ∈ [t, T ], X t s = e s t δrdr ε(M ) s ε(M ) t . (2.2.6)
Then for any t ≤ T , Y t satisfies:

Y t = E[X t T ξ + T t X t s ϕ s ds|F t ], P -a.s.. On the other hand, if (Y ′ , U ′ ) ∈ S 2 × H 2 (ℓ 2
) is the solution of the BSDE:

Y ′ t = ξ + T t f (s, Y ′ s , U ′ s )ds - ∞ i=1 T t U ′i s dH (i) s , ∀t ≤ T (2.2.7) where f (t, Y ′ t , U ′ t ) ≥ ϕ t + δ t Y ′ t + V, U ′ p t , dP ⊗ dt -a.s.
then for any t ≤ T ,

Y ′ t ≥ E[X t T ξ + T t X t
-d(Y s X t s ) = -Y s-dX t s -X t s-dY s -d[Y, X t ] s = -X t s-Y s-δ s ds -Y s-X t s-dM s + X t s-ϕ s ds + X t s-δ s Y s ds -X t s-( i≥1 U i s dH (i) ) -X t s-{ ∞ i=1 ∞ j=1 V i s U j s d([H (i) , H (j) ] s -H (i) , H (j) s )} =X t s ϕ s ds -dN s where for any s ∈ [t, T ] dN s = Y s-X t s- ∞ i=1 V i s dH (i) s + X t s- i≥1 U i s dH (i) s + X t s- ∞ i=1 ∞ j=1 V i s U j s d([H (i) , H (j) ] s -H (i) , H (j) s ).
Note that since X t is uniformly square integrable, Y ∈ S 2 , U ∈ H 2 (ℓ 2 ) and finally taking into account Assumption (A2) on V , we get that N is a uniformly integrable martingale on [t, T ]. Therefore taking conditional expectation to obtain:

Y t = E[X t T ξ + T t X t s ϕ s ds|F t ], P -a.s.
which is the desired result.

We now focus on the second part of the claim. By Itô's formula we have:

∀s ∈ [t, T ], -d(Y ′ s X t s ) = -Y ′ s-dX t s -X t s-dY ′ s -d[Y ′ , X t ] s = -X t s-Y ′ s-δ s ds -Y ′ s-X t s- ∞ i=1 V i s dH (i) s + X t s-f (s, Y ′ s , U ′ s )ds -X t s- ∞ i=1 U ′i s dH (i) s -X t s- ∞ i=1 ∞ j=1 V i s U ′j s d[H (i) , H (j) ] s .
Next since X t ≥ 0 and taking into account the inequality satified by f to obtain:

-d(Y ′ s X t s ) ≥ X t s ϕ s ds -dN ′ s P -a.s.
, where for any s ∈ [t, T ],

dN ′ s = Y ′ s-X t s- ∞ i=1 V i s dH (i) s -X t s- U ′i s dH (i) s -X t s- ∞ i=1 ∞ j=1 V i s U ′j s d([H (i) , H (j) ] s -H (i) , H (j) s ).
But once more N ′ is a uniformly integrable martingale then by taking the conditional expectation we obtain:

Y ′ t ≥ E[X t T ξ + T t X t s ϕ s ds|F t ], P -a.s.
which completes the proof.

We are now ready to give a comparison result of solutions of two BSDEs of type (2.2.1).

Proposition 2.2.3.

For i = 1, 2, let (f i , ξ i ) be a pair that satisfies Assumption (A1)-(i),(ii) and let (Y i , U i ) ∈ S 2 × H 2 (ℓ 2
) be the solution of the following BSDE: ∀t ≤ T ,

Y i t = ξ i + T t f i (s, Y i s , U i s )ds - ∞ j=1 T t U i,j s dH (j) s .
Assume that: (i) For any U 1 , U 2 ∈ H 2 (l 2 ), there exists a process

V U 1 ,U 2 = (V U 1 ,U 2 j
) j≥1 (which may depend on U 1 and U 2 ) satisfying (A2) such that f 1 verifies:

f 1 (t, Y 2 t , U 1 t ) -f 1 (t, Y 2 t , U 2 t ) ≥ V U 1 ,U 2 , (U 1 -U 2 ) p t , dP ⊗ dt -a.e.;
(2.2.8)

(ii) P -a.s., ξ 1 ≥ ξ 2 and f 1 (t, Y 2 t , U 2 t ) ≥ f 2 (t, Y 2 t , U 2 t ), dP ⊗ dt -a.e.. ( 2

.2.9)

Then P-a.s.,

Y 1 t ≥ Y 2 t , ∀t ∈ [0, T ]. Proof. Let us set Ȳ = Y 1 -Y 2 , Ū = U 1 -U 2 and ξ = ξ 1 -ξ 2 , then ∀t ∈ [0, T ], Ȳt = ξ + T t {f 1 (s, Y 1 s , U 1 s ) -f 2 (s, Y 2 s , U 2 s )}ds - ∞ j=1 T t Ū j s dH (j) s .
Next let us set:

∀s ≤ T, δ s = (f 1 (s, Y 1 s , U 1 s ) -f 1 (s, Y 2 s , U 1 s )) × ( Ȳs ) -1 { Ȳs =0} and ϕ s = f 1 (s, Y 2 s , U 2 s ) -f 2 (s, Y 2 s , U 2
s ). Then by (2.2.9) we have, ϕ s ≥ 0, dP ⊗ dta.e.. On the other hand (δ s ) s∈[0,T ] is bounded since f 1 is uniformly Lipschitz. Finally we have

f 1 (s, Y 1 s , U 1 s ) -f 2 (s, Y 2 s , U 2 s ) ≥ ϕ s + δ s Ȳs + V U 1 ,U 2 , Ū p s , dP ⊗ ds -a.e.
. Therefore thanks to Proposition 3.2 we get,

∀t ≤ T, Ȳt ≥ E[X t T ξ + T t X t s ϕ s ds|F s ] ≥ 0, P -a.s.
where (X t s ) s∈[t,T ] is defined in the same way as in (2.2.6) with the new processes δ and ϕ. As X t , ξ and ϕ are non-negative then for any t ≤ T , Ȳt ≥ 0 which implies that Pa.s., ∀t ≤ T, Y

f 2 (t, Y 2 t , U 1 t ) -f 2 (t, Y 2 t , U 2 t ) ≥ V U 1 ,U 2 , (U 1 -U 2 ) p t , dP ⊗ dt -a.e.
(2.2.10)

and f 1 (t, Y 1 t , U 1 t ) ≥ f 2 (t, Y 1 t , U 1 t ), dP ⊗ dt -a.e.. ( 2

.2.11)

In this case, with the other properties, one can show in the same way that we actually have P-a.s.,

Y 1 ≥ Y 2 .
Remark 2.2.3. Point (i) of Proposition 2.2.3 is satisfied in the following cases:

(i) f does not depend on the component u ;

(ii) If L reduces to a Poisson process, we have

H (i) ≡ 0 for all i ≥ 2, then Assumption (A2) reads: (a) V = (V t ) t∈[0,T ] is bounded ; (b) for any stopping time τ , such that △L τ = 0, V τ > -1, P -a.s.. (iii) The generator f satisfies f (t, y, u) = h(t, y, i≥1 θ i t u i ), ∀(t, y, u) ∈ [0, T ] × IR k × ℓ 2
where the mapping η ∈ IR → h(t, y, η) is non decreasing and uniformly Lipschitz and, on the other hand,

((θ i t ) i≥1 ) t≤T satisfies i≥1 |θ i t | 2 ≤ C and i≥1 θ i t p i (∆L t ) ≥ 0, dt ⊗ dP -a.e.
We finally provide a comparison result of solutions of reflected BSDEs of type (2.2.1) which will be useful in the sequel. Proposition 2.2.4. For i = 1, 2, let (ξ i , S i , f i ) be a triple which satisfies Assumption (A1) and let

(Y i t , K i t , U i t ) t≤T be the solution of the RBSDE associated with (ξ i , S i , f i ). Assume that: (i) P -a.s, ξ 1 ≥ ξ 2 and ∀t ∈ [0, T ], f 1 (t, y, u) ≥ f 2 (t, y, u) and S 1 t ≥ S 2 t ; (ii) f 1 verifies condition (2.2.8). Then P-a.s. for any t ≤ T , Y 1 t ≥ Y 2 t .
Proof. For i = 1, 2, let us consider the following sequence of processes (Y i,n , U i,n ) ∈ S 2 × H 2 (ℓ 2 ), n ≥ 0, that satisfy:

Y i,n t = ξ i + T t f i (s, Y i,n s , U i,n s )ds + n T t (Y i,n s -S i s ) -ds - ∞ j=1 T t U i,n,j s dH (j)
s , ∀t ≤ T and let us denote by

f n i (s, y, u) := f i (s, y, u) + n(y -S i s ) -. For any n ≥ 0, f n 1 satisfies (2.2.8) and f n 1 ≥ f n 2 .
Therefore using the comparison result of Proposition 2.2.3, we deduce that: ∀n ≥ 0,

P -a.s., ∀t ≤ T, Y 1,n t ≥ Y 2,n t .
(2.2.12)

But since f 1 verifies (2.2.8) then we can show, as in [START_REF] Hamadène | Reflected backward stochastic differential equation with jumps and random obstacle[END_REF], Theorem 1.2.a, pp. 5, since the processes S i do not have predictable jumps, that for i = 1, 2 Y i,n ր Y i in S 2 . Thus, inequality (2.2.12) implies that

P-a.s., Y 1 ≥ Y 2 .
We are now going to make a connection between reflected BSDEs and their associated IPDEs with obstacle. So let (t, x) ∈ [0, T ] × IR k and let (X t,x s ) s≤T be the solution of the following standard SDE driven by the Lévy process L, i.e.,

X t,x s = x + t∨s t b(r, X t,x r )dr + t∨s t σ(r, X t,x r-)dL r , ∀s ≤ T, (2.2.13)
where we assume that the functions b and σ are jointly continuous, Lipschitz continuous w.r.t. x uniformly in t, i.e., there exists a constant C ≥ 0 such that for any t ∈ [0, T ], x,x ′ ∈ IR,

|σ(t, x) -σ(t, x ′ )| + |b(t, x) -b(t, x ′ )| ≤ C|x -x ′ |. (2.2.14)
As a consequence, the functions b(t, x) and σ(t, x) are of linear growth. We additionally assume that σ is bounded, i.e., there exists a constant C σ such that

∀(t, x) ∈ [0, T ] × IR, |σ(t, x)| ≤ C σ . (2.2.15)
Under the above conditions on b and σ, the process X t,x exists and is unique (see e.g. [START_REF] Protter | Stochastic integration and differential equations[END_REF], pp.249), and satisfies:

∀p ≥ 1, E[sup s≤T |X t,x s | p ] ≤ C(1 + |x| p ). (2.2.16)
Next let us consider the following functions:

h : x ∈ IR → h(x) ∈ IR; f : (t, x, y, u) ∈ [0, T ] × IR 1+1 × l 2 → f (t, x, y, u) ∈ IR; Ψ : (t, x) ∈ [0, T ] × IR → Ψ(t, x) ∈ IR,
which we assume satisfying:

Assumptions (A3): (i) h, Ψ and f (t, x, 0, 0) are jointly continuous and belong to Π g ; (ii) the mapping (y, z) → f (t, x, y, z) is Lipschitz continuous uniformly in (t, x) ; (iii) For any x ∈ IR, h(x) ≥ Ψ(T, x).
(iv) The generator f has the following form,

f (t, x, y, u) = h(t, x, y, i≥1 θ i t u i ), ∀(t, x, y, u) ∈ [0, T ] × IR 1+1 × ℓ 2
where the mapping η ∈ IR -→ h(t, x, y, η) is non decreasing, and there exists a constant

C > 0, such that ∀t ∈ [0, T ], x, y, z, z ′ ∈ IR, |h(t, x, y, z) -h(t, x, y, z ′ ) ≤ C|z -z ′ |. Moreover (θ i t ) i≥1 satisfies i≥1 |θ i t | 2 ≤ C and i≥1 θ i t p i (∆L t ) > 0, dt ⊗ dP -a.e..
Next let (t, x) ∈ [0, T ] × IR k be fixed and let us consider the following reflected BSDE:

       (Y t,x , U t,x , K t,x ) ∈ S 2 × H(ℓ 2 ) × A 2 ; Y t,x s = h(X t,x T ) + T s f (r, X t,x r , Y t,x r , U t,x r )dr + K t,x T -K t,x s - ∞ i=1 T s U t,x,i r dH (i) r ∀s ≤ T, Y t,x s ≥ Ψ(s, X t,x s ) and T 0 (Y t,x s -Ψ(s, X t,x s ))dK t,x s = 0, P -a.s.
(2.2.17)

Under assumptions (A3)-(i), (ii), (iii), the reflected BSDE (2.2.17) is well-posed and has a unique solution (Y t,x , U t,x , K t,x ), thanks to Theorem 3.1. Moreover the following estimate holds true:

E sup 0≤s≤T |Y t,x s | 2 + T 0 { i≥1 |U i,tx s | 2 }ds ≤ CE |h(X t,x T )| 2 + T 0 |f (s, X t,x s , 0, 0)| 2 ds + sup 0≤s≤T |Ψ(s, X t,x s )| 2 .
(2.2.18)

On the other hand, the quantity

u(t, x) = Y t,x t , (2.2.19)
is deterministic, continuous and satisfies

∀(t, x) ∈ [0, T ] × IR k , ∀s ∈ [t, T ], Y t,x s = u(s, X t,x s ).
Fore more details, one can see e.g. [START_REF] Ren | Reflected backward stochastic differential equations driven by Lévy processes[END_REF]. Finally note that under Assumptions (A3) and by (2.2.18) the function u belongs also to Π g .

Next let us introduce the following IPDE with obstacle:

min u(t, x) -Ψ(t, x); -∂ t u(t, x) -Lu(t, x) -f (t, x, u(t, x), Φ(u)(t, x)) = 0, (t, x) ∈ [0, T ) × IR, u(T, x) = h(x), (2.2
.20) where L is the generator associated with the process X t,x of (2.2.13) and which has the following expression:

Lu(t, x) = (E[L 1 ]σ(t, x) + b(t, x))∂ x u(t, x) + 1 2 σ(t, x) 2 ̟ 2 ∂ 2 xx u(t, x) + I R [u(t, x + σ(t, x)y) -u(t, x) -∂ x u(t, x)σ(t, x)y]Π(dy) and Φ(u)(t, x) = 1 c 1,1 ∂ x u(t, x)σ(t, x) k=1 + I R (u(t, x + σ(t, x)y) -u(t, x) -∂ x u(t,
min{u(t, x) -Ψ(t, x); -∂ t ϕ(t, x) -Lϕ(t, x) -f (t, x, ϕ(t, x), Φ(ϕ)(t, x))} ≤ 0, u(T, x) ≤ h(x); (resp. min{u(t, x) -Ψ(t, x); -∂ t ϕ(t, x) -Lϕ(t, x) -f (t, x, ϕ(t, x), Φ(ϕ)(t, x))} ≥ 0, u(T, x) ≥ h(x); ).
The function u is said to be a viscosity solution of (2.2.20) if it is both its viscosity subsolution and supersolution.

In [START_REF] Ren | Generalized reflected BSDE driven by a Levy Processes and an obstacle problem for PDIE with a nonlinear Neumann boundary condition[END_REF], Y.

Systems of reflected

BSDEs with inter-connected obstacles driven by a Lévy process and multi-modes switching problem.

We now introduce the following functions f i , h i and g ij , i, j ∈ A:

f i : (t, x, (y i ) i=1,m , u) ∈ [0, T ] × IR k × IR m × ℓ 2 -→ f i (t, x, (y i ) i=1,m , u) ∈ IR g ij : (t, x) ∈ [0, T ] × IR k -→ g ij (t, x) ∈ IR h i : x ∈ IR -→ h i (x) ∈ IR
which we assume satisfying:

Assumption (A4) (I) For any i ∈ A: (i) The mapping (t, x) → f i (t, x, -→ y , u) is continuous uniformly with respect to ( -→ y , u) where -→ y = (y i ) i=1,m ; (ii) The mapping ( -→ y , u) → f i (t, x, -→ y , u) is Lipschiz continuous uniformly w.r.t. (t, x) ; (iii) f i (t, x, 0, 0) is measurable and of polynomial growth w.r.t. (t, x). (iv) For any U 1 , U 2 ∈ H 2 (l 2 ), X, Y ∈ S 2 , there exists V U 1 ,U 2 ,i = (V U 1 ,U 2 ,i j ) j≥1 , i = 1, 2
, which may depend on U 1 and U 2 , that satisfies (A2) and such that :

f i (t, X t , Y t , U 1 t ) -f i (t, X t , Y t , U 2 t ) ≥ V U 1 ,U 2 ,i , (U 1 -U 2 ) p t , dP ⊗ dt -a.e., i = 1, 2; (2.2.21) (v) For any i ∈ A and k ∈ A i := A -{i}, the mapping y k → f i (t, x, y 1 , • • • , y k-1 , y k , y k+1 , • • • , y m , u) is nondecreasing whenever the other components (t, x, y 1 , • • • , y k-1 , y k+1 , • • • , y m , u) are fixed.
(II) ∀i, j ∈ A, g ii ≡ 0 and for k = j, g jk (t, x) is non-negative, continuous with polynomial growth and satisfy the following non-free loop property:

For any (t, x) ∈ [0, T ] × IR and for any sequence of indices i

1 , • • • , i k such that i 1 = i k and card{i 1 , • • • , i k } = k -1 we have g i1i2 (t, x) + g i2i3 (t, x) + • • • + g i k i1 (t, x) > 0.
(III) ∀i ∈ A, h i is continuous with polynomial growth and satisfies the following coherence condition:

h i (x) ≥ max j∈Ai (h j (x) -g ij (T, x)), ∀x ∈ IR.
We now describe precisely the switching problem. Let Υ = ((θ j ) j≥0 , (α j ) j≥0 ) be an admissible strategy and let a = (a s ) s∈[0,T ] be the process defined by

∀s ≤ T, a s := α 0 {θ0} (s) + ∞ j=1 α j-1 ]θj-1θj ] (s),
where {θ j } j≥0 is an increasing sequence of F t -stopping times with values in [0,T] and for j ≥ 0, α j is a random variable F θj -measurable with values in A = {1, ..., m}. If P[lim n θ n < T ] = 0, then the pair {θ j , α j } j≥0 (or the process a) is called an admissible strategy of switching. Next we denote by (A a s ) s≤T the switching cost process associated with an admissible strategy a, which is defined as following:

∀s < T, A a s = j≥1 g αj-1,αj (θ j , X t,x θj ) [θj ≤s] and A a T = lim s→T A a s (2.2.22)
where X t,x is the process given in (2.2.13). Next, for η ≤ T and i ∈ A, we denote by

A i η := {a admissible strategy such that α 0 = i, θ 0 = η and E[(A a T ) 2 ] < ∞}.
Assume momentarily that for i ∈ A, the function f i introduced previously does not depend on -→ y and u. For t ≤ T and a given admissible strategy a ∈ A i t , we define the payoff J a i (t, x) by:

J a i (t, x) := E[ T t f a(s) (s, X t,x s )ds + h a(T ) -A a T ]
where

f a(s) (. . . ) = f k (. . . ) (resp. h a(T ) (.) = h k (.)) if at time s (resp. T ) a(s) = k (resp. a(T ) = k) (k ∈ A). Finally let us define J i (t, x) := sup a∈A i t J a i (t, x), i = 1, ..., m. (2.2.23)
As a by-product of our main result which is given in Theorem 4.3 below, we get that the functions (J i (t, x)) i=1,...,m is the unique continuous viscosity solution of the Hamilton-Jacobi-Bellman system associated with this switching problem.

Let (t, x) ∈ [0, T ] × IR k and let us consider the following system of reflected BSDEs with oblique reflection: ∀j = 1, ..., m

         Y j ∈ S 2 , U j ∈ H 2 (ℓ 2 ), K j ∈ A 2 Y j s = h j (X t,x T ) + T s f j (r, X t,x r , Y 1 r , Y 2 r , • • • , Y m r , U j r )dr - ∞ i=1 T s U j,i r dH (i) r + K j T -K j s , ∀s ≤ T ; ∀s ≤ T, Y j s ≥ max k∈Aj {Y k s -g jk (s, X t,x s )} and T 0 {Y j s -max k∈Aj {Y k s -g jk (s, X t,x s )}}dK j s = 0.
(2.2.24) Note that the solution of this BSDE depends actually on (t, x) which we will omit for sake of simplicity, as far as there is no confusion. We then have the following result related to existence and uniqueness of the solution of (2.2.24). Theorem 2.2.3. Assume that Assumption (A4)(I)(ii)-(iv), (A4)(II) and (A4)(III) are fulfilled. Then system of reflected BSDE with oblique reflection (2.2.24) has a unique solution.

Proof. The proof follows the same lines as in [START_REF] Chassagneux | Discrete-time Approximation of Multidimensional BSDEs with oblique reflections[END_REF] and [START_REF] Hamadène | Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems[END_REF]. It will be given in two steps.

Step 1: We will first assume that the functions f i , i ∈ A, verify (A4)(I)(ii)-(v). The other assumptions remain fixed.

Let us introduce the following standard BSDEs : ∀s ≤ T ,

   Ȳ ∈ S 2 , Ū ∈ H 2 (ℓ 2 ); Ȳs = max j=1,m h j (X t,x T ) + T s max j=1,m f j (r, X t,x r , Ȳr , • • • , Ȳr , Ūr )dr - ∞ i=1 T s Ū i r dH (i) r (2.2.25) and    Y ∈ S 2 , U ∈ H 2 (ℓ 2 ); Y s = min j=1,m h j (X t,x T ) + T s min j=1,m f j (r, X t,x r , Y r , • • • , Y r , U r )dr - ∞ i=1 T s U i r dH (i)
r .

(2.2.26)

Note that thanks to Theorem 1 in [START_REF] Nualart | Backward stochastic differential equations and Feynman-Kac formula for Lévy processes, with applications in finance[END_REF], each one of the above BSDEs has a unique solutions. Next for j = 1, • • • , m and n ≥ 1, let us define (Y j,n , U j,n , K j,n ) by:

                 Y j,n ∈ S 2 , U j,n ∈ H 2 (ℓ 2 ), K j,n ∈ A 2 ; Y j,0 = Y Y j,n s = h j (X t,x T ) + T s f j (r, X t,x r , Y 1,n-1 r , • • • , Y j-1,n-1 r , Y j,n r , Y j+1,n-1 r , • • • , Y m,n-1 , U j,n r )dr - ∞ i=1 T s U i,j,n r dH (i) r + K j,n T -K j,n s , ∀s ≤ T ; Y j,n s ≥ max k∈Aj (Y k,n-1 s -g jk (s, X t,x s )), ∀s ≤ T ; T 0 [Y j,n r -max k∈Aj (Y k,n-1 r -g jk (r, X t,x r ))]dK j r = 0.
(2.2.27) By induction we can show that system (2.2.27) has a unique solution for any fixed n ≥ 1 since when n is fixed, (2.2.27) reduces to m decoupled reflected BSDEs. On the other hand it is easily seen that ( Ȳ , Ū , 0) is also a solution of :

     Ȳs = max j=1,m h j (X t,x T ) + T s max j=1,m f j (r, X t,x r , Ȳr , • • • , Ȳr , Ūr )dr - ∞ i=1 T s Ū i r dH (i) r + KT -Ks , ∀s ≤ T ; Ȳs ≥ max k∈Aj ( Ȳs -g jk (s, X t,x s )), ∀s ≤ T ; T 0 [ Ȳr -max k∈Aj ( Ȳs -g jk (s, X t,x s ))]d Kr = 0.
Next since for any i ∈ A, f i verifies Assumption A4(I), by Proposition 2.2.4 and an induction argument, we get that P-a.s. for any j, n and s ≤ T , Y j,n-1 s ≤ Y j,n s ≤ Ȳs . Then the sequence (Y j,n ) n≥0 , has a limit which we denote by Y j , for any j ∈ A. By the monotonic limit theorem in [START_REF] Fan | Reflected backward stochastic differential equations driven by a Levy process[END_REF], Y j ∈ S 2 and there exist

U j ∈ H 2 (ℓ 2 ), K j ∈ A 2 , such that: ∀s ≤ T ,      Y j s = h j (X t,x T ) + T s f j (r, X t,x r , -→ Y r , U j r )dr - ∞ i=1 T s U i,j r dH (i) r + K j T -K j s , Y j s ≥ max k∈Aj (Y k s -g jk (s, X t,x s )), (2.2.28)
where for any j ∈ A, U j is the weak limit of (U j,n ) n≥1 in H 2 (ℓ 2 ) and for any stopping time τ , K j τ is the weak limit of K j,n τ in L 2 (Ω, F τ , P). Finally note that K j is predictable since the processes K n,j are so, for any n ≥ 1.

Let us now consider the following RBSE:

             Ŷ j ∈ S 2 , Û j ∈ H 2 (ℓ 2 ), Kj ∈ S 2 , non-decreasing and Kj 0 = 0; Ŷ j s = h j (X t,x T ) + T s f j (r, X t,x r , Y 1 r , • • • , Y j-1 r , Ŷ j r , Y j+1 r , • • • , Y m r , Û j r )dr - ∞ i=1 T s Û i,j r dH (i) r + Kj T -Kj s , ∀s ≤ T ; Ŷ j s ≥ max k∈Aj (Y k s -g jk (s, X t,x s )), ∀s ≤ T ; T 0 [ Ŷ j r--max k∈Aj (Y k r--g jk (r, X t,x r-))]d Kj r = 0.
(2.2.29)

According to Theorem 3.3 in [START_REF] Aman | Reflected generalized backward doubly SDEs driven by Lévy processes and applications[END_REF], this equation has a unique solution. By Tanaka-Meyer's formula (see e.g. [START_REF] Protter | Stochastic integration and differential equations[END_REF], Theorem 68, pp. 216), for all j ∈ A:

( Ŷ j T -Y j T ) + =( Ŷ j s -Y j s ) + + T s { Ŷ j r--Y j r->0} d( Ŷ j r -Y j r ) + s<r≤T [ { Ŷ j r--Y j r->0} ( Ŷ j r -Y j r ) -+ { Ŷ j r--Y j r-≤0} ( Ŷ j r -Y j r ) + ] + 1 2 L 0 t ( Ŷ j -Y j )
where the process (L 0 t ( Ŷ j -Y j )) t≤T is the local time of the semi martingale ( Ŷ j s -Y j s ) 0≤s≤T at 0 which is a nonnegative process. Then we have

( Ŷ j T -Y j T ) + ≥( Ŷ j s -Y j s ) + + T s { Ŷ j r--Y j r->0} d( Ŷ j r -Y j r ) =( Ŷ j s -Y j s ) + - T s { Ŷ j r--Y j r->0} [f j (r, X t,x r , Y 1 r , • • • , Ŷ j r , • • • , Y m r , Û j r ) -f j (r, X t,x r , Y 1 r , • • • , Y j r , • • • , Y m r , U j r )]dr - T s { Ŷ j r--Y j r->0} d( Kj r -K j r ) + ∞ i=1 T s { Ŷ j r--Y j r->0} ( Û j,i r -U j,i r )dH (i) r .
First note that by (2.2.29),

T s { Ŷ j r--Y j r->0} d( Kj r -K j r ) ≤ 0. Now by Assumption (A4)(I)(iv)
, we obtain:

( Ŷ j s -Y j s ) + ≤ T s { Ŷ j r--Y j r->0} [f j (r, X t,x r , Y 1 r , • • • , Ŷ j r , • • • , Y m r , Û j r ) -f j (r, X t,x r , Y 1 r , • • • , Y j r , • • • , Y m r , Û j r ) + f j (r, X t,x r , Y 1 r , • • • , Y j r , • • • , Y m r , Û j r ) -f j (r, X t,x r , Y 1 r , • • • , Y j r , • • • , Y m r , U j r )]dr - ∞ i=1 T s { Ŷ j r--Y j r->0} ( Û j,i r -U j,i r )dH (i) r ≤ T s { Ŷ j r--Y j r->0} C( Ŷ j r--Y j r-) + dr + ∞ i=1 T s { Ŷ j r--Y j r->0} V j,U, Û ,i r ( Û j,i r -U j,i r )dr - ∞ i=1 T s { Ŷ j r--Y j r->0} ( Û j,i r -U j,i r )dH (i) r . Next for t ≤ T , let us set M t = ∞ i=1 t 0 V j,U 2 ,U 1 ,i r dH (i) r and Z t = ∞ i=1 t 0 { Ŷ j r--Y j r->0} ( Û j,i r -U j,i r )dH (i) r .
By Proposition 2.2.1, ε(M ) ∈ S 2 , ε(M ) > 0 and E[ε(M ) T ] = 1. Then using Girsanov's Theorem ( [START_REF] Protter | Stochastic integration and differential equations[END_REF], pp.136), under the probability measure d P := ε(M ) T dP, we obtain that the process

Zt = Z t -< M, Z > t , t ≤ T,
is a martingale and then

E P [ ∞ i=1 T s { Ŷ j r--Y j r->0} V j,U 2 ,U 1 ,i r ( Û j,i r -U j,i r )dr- ∞ i=1 T s { Ŷ j r--Y j r->0} ( Û j,i r -U j,i r )dH (i) r ] = -E P ( ZT -Zs ) = 0.
Thus for any s ≤ T ,

E P ( Ŷ j s -Y j s ) + ≤ E P [ T s C( Ŷ j r -Y j r ) + dr]
and finally by Gronwall's Lemma, ∀j ∈ A, ∀s ≤ T , ( Ŷ j s -Y j s ) + = 0 Pa.s. and then also Pa.s. since those probabilities are equivalent. It implies that P-a.s., Ŷ j ≤ Y j for any j ∈ A. On the other hand, since ∀n ≥ 1, ∀j ∈ A, Y j,n-1 ≤ Y j , then we have

∀s ≤ T, max k∈Aj (Y k,n-1 s -g jk (s, X t,x s )) ≤ max k∈Aj (Y k s -g jk (s, X t,x s )).
Therefore by comparison, we obtain Y j,n ≤ Ŷ j , and then Y j ≤ Ŷ j which implies Y j = Ŷ j , ∀j ∈ A.

Next by Itô's formula applied to (Y j -Ŷ j ) 2 we obtain: ∀s ∈ [0, T ],

(Y j s -Ŷ j s ) 2 = (Y j 0 -Ŷ j 0 ) 2 + 2 s 0 (Y j r--Ŷ j r-)d(Y j r -Ŷ j r ) + ∞ i=1 ∞ k=1 s 0 (U j,i r -Û j,i r )(U j,k r -Û j,k r )d[H i , H k ] r .
As Y j = Ŷ j and taking expectation in both-hand sides of the previous equality to obtain

E[ T 0 i≥1 (U j,i r -Û j,i r ) 2 dr] = 0.
It implies that U j = Û j , dt ⊗ dP and finally K j = Kj for any j ∈ A, i.e. (Y j , U j , K j ) j∈A verify (2.2.29).

Next we will show that the predictable process K j does not have jumps. So assume there exists j 1 ∈ A and a predictable stopping time τ such that △Y j1 τ = -△ K j1 τ = -△ Kj1 τ < 0 (note that the process K j is predictable). Then by the second equality in (2.2.29) we have

Y j1 τ -= max k∈Aj 1 (Y k τ --g j1k (τ, X t,x τ -)). (2.2.30)
Now let j 2 ∈ A j1 be the optimal index in (2.2.30), i.e.,

Y j2 τ --g j1,j2 (τ, X t,x τ ) = Y j1 τ -> Y j1 τ ≥ Y j2 τ -g j1,j2 (τ, X t,x τ ).
Note that g j1,j2 (τ, X t,x τ -) = g j1,j2 (τ, X t,x τ ) since the stopping time τ is predictable, and the process (X t,x s ) t≤s≤T does not have predictable jump times. Thus △Y j2 τ < 0 and once more we have,

Y j2 τ -= max k∈Aj 2 (Y k τ --g j2k (τ, X t,x τ -)). ( 2

.2.31)

We can now repeat the same argument as many times as necessary, to deduce the existence of a loop ℓ 1 , ..., ℓ p-1 , ℓ p = ℓ 1 (p ≥ 2) and l 2 = l 1 such that

Y ℓ1 τ -= Y ℓ2 τ --g ℓ1,ℓ2 (τ, X t,x τ -), • • • , Y ℓp-1 τ - = Y ℓp τ --g ℓp-1,ℓp (τ, X t,x τ -) which implies that g ℓ1,ℓ2 (τ, X t,x τ -) + • • • + g ℓp-1
,ℓp (τ, X t,x τ -) = 0 which is contradictory with Assumption (A4)(II). It implies that ∆K j1 τ = 0 and then K j1 is continuous since it is predictable. As j is arbitrary in A, then the processes K j are continuous and taking into account (2.2.29), we deduce that the triples (Y j , U j , K j ) j∈A , is a solution for system (2.2.24).

Step 2: We now deal with the general case i.e. we assume that f i , i ∈ A, do no longer satisfy the monotonicity assumption (A4)(I)(v) but (A4)(I)(ii)-(iv) solely.

Let i ∈ A and t 0 ∈ [0, T ] be fixed. For a ∈ A i t0 and Γ := ((

Γ l s ) s∈[0,T ] ) l∈A ∈ [H 2 ] m := H 2 × • • • × H 2 (m
times), we introduce the unique solution of the switched BSDE which is defined by: ∀s ∈ [t 0 , T ],

V a s = h a(T ) (X t,x T ) + T s f a(r) (r, X t,x r , -→ Γ r , N a r )dr - ∞ i=1 T s N a,i r dH (i) r -A a T + A a s (2.2.32)
where

V a ∈ S 2 and N a ∈ H 2 (ℓ 2 ) ( -→ Γ r = (Γ i r ) i∈A ).
First note that the solution of this equation exists and is unique since in setting, for s

∈ [t 0 , T ], Ṽ a s = V a s -A a s and ha T = h a(T ) (X t,x T ) -A a
T this equation becomes standard and has a unique solution by Nualart et al.'s result (see [START_REF] Nualart | Backward stochastic differential equations and Feynman-Kac formula for Lévy processes, with applications in finance[END_REF], Theorem 1, pp.765). Moreover as in [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] (see Appendix 5.1, Theorem 5.1.1) we have the following link between the BSDEs (2.2.24) and (2.2.32),

Y i t0 = esssup a∈A i t 0 (V a t0 -A a t0 ) = V a * t0 -A a * t0 (2.2.33)
for some a * ∈ A i t0 .

Next let us introduce the following mapping Θ defined on [H

2 ] m by Θ : [H 2 ] m → [H 2 ] m Γ = (Γ j ) j∈A → (Y j ) j∈A (2.2.34)
where (Y j , U j , K j ) j∈A is the unique solution of the following system of RBSDEs: ] m and for k = 1, 2, let (Y k,j , U k,j , K k,j ) j∈A = Θ(Γ k ), i.e., that satisfy: ∀s ≤ T ,

     Y j s = h j (X t,x T ) + T s f j (r, X t,x r , -→ Γ r , U j r )dr - ∞ i=1 T s U j,i r dH (i) r + K j T -K j s , ∀s ≤ T. Y j s ≥ max k∈Aj {Y k s -g jk (s, Y j s )}, ∀s ≤ T ; T 0 [Y j s -max k∈Aj {Y k s -g jk (s, Y j s )}]dK j s = 0 (2.
     Y k,j s = h j (X t,x T ) + T s f j (r, X t,x r , -→ Γ k r , U k,j r )dr - ∞ i=1 T s U k,j,i r dH (i) r + K k,j T -K k,j s Y k,j s ≥ max q∈Aj {Y k,q s -g jq (s, X t,x s )}; T 0 [Y k,j s -max q∈Aj {Y k,q s -g jq (s, X t,x s )}]dK k,j s = 0.
Next let us define ( Ŷ j ) j∈A through the following system of reflected BSDEs with oblique reflection:

∀s ≤ T ,      Ŷ j s = h j (X t,x T ) + T s f j (r, X t,x r , -→ Γ 1 r , Û j r ) ∨ f j (r, X t,x r , -→ Γ 2 r , Û j r )dr - ∞ i=1 T s Û j,i r dH (i) r + Kj T -Kj s Ŷ j s ≥ max q∈Aj { Ŷ q s -g jq (s, Ŷ j s )}; T 0 [ Ŷ j s -max q∈Aj { Ŷ q s -g jq (s, Ŷ j s )}]d Kj s = 0,
Now let t 0 ∈ [0, T ] and a an admissible strategy such that θ 0 = t 0 and E[(A a T ) 2 ] < ∞. Next let us define V k,a , k = 1, 2, and V a , via BSDEs, by: ∀s ∈ [t 0 , T ],

V a s = h a(T ) (X t,x T ) + T s f a(r) (r, X t,x r , -→ Γ 1 r , N a r ) ∨ f a(r) (r, X t,x r , -→ Γ 2 r , N a r )dr - ∞ i=1 T s N a,i r dH (i) r -A a T + A a s and for k = 1, 2, V k,a s = h a(T ) (X t,x T ) + T s f a(r) (r, X t,x r , -→ Γ k r , N k,a r )dr -A a T + A a s - ∞ i=1 T s N k,a,i r dH (i) r .
By Proposition 5.1 in Appendix, we have:

Y k,j t0 = esssup a∈A j t 0 (V k,a t0 -A a t0 ), k = 1, 2 and Ŷ j t0 = esssup a∈A j t 0 ( V a t0 -A a t0 ) := V a * t0 -A a * t0 . ( 2
= α 0 {θ0} (s) + ∞ j=1 α j-1 ]θj-1θj ] (s), s ∈ [t 0 , T ], and let U 1 , U 2 ∈ H 2 (l 2 ), X, Y ∈ S 2 .
For any s ∈ [t 0 , T ] we have:

f a(s) (s, X s , Y s , U 1 s ) -f a(s) (s, X s , Y s , U 2 s ) = [f α0 (s, X s , Y s , U 1 s ) -f α0 (s, X s , Y s , U 2 s )] {θ0≤s≤θ1} + j≥2 [f αj-1 (s, X s , Y s , U 1 s ) -f αj-1 (s, X s , Y s , U 2 s )] ]θj-1,θj ] (s) ≥ V U 1 ,U 2 ,α0 , (U 1 -U 2 ) p s {θ0≤s≤θ1} + j≥2 V U 1 ,U 2 ,αj-1 , (U 1 -U 2 ) p s ]θj-1,θj ] (s) =: V U 1 ,U 2 ,a , (U 1 -U 2 ) p s .
where for any s ∈ [t 0 , T ],

V U 1 ,U 2 ,a s = V U 1 ,U 2 ,α0 s {θ0≤s≤θ1} + j≥2 V U 1 ,U 2 ,αj-1 s ]θj-1,θj ] (s). But on [t 0 , T ] × Ω, ds ⊗ dP{ ∞ i=1 V i,U 1 ,U 2 ,a s (ω)p i (∆L s (ω)) ≤ -1} ≤ j∈A ds ⊗ dP{ ∞ i=1 V i,U 1 ,U 2 ,j s (ω)p i (∆L s (ω)) ≤ -1} = 0 which implies that on [t 0 , T ] × Ω it holds ∞ i=1 V i,U 1 ,U 2 ,a(ω) s (ω)p i (∆L s (ω)) > -1, ds ⊗ dP -a.e.
On the other hand, on [t 0 , T ] × Ω,

∞ i=1 |V i,U 1 ,U 2 ,a s | 2 ≤ ℓ∈A ∞ i=1 |V i,U 1 ,U 2 ,ℓ s | 2 ≤ C, ds ⊗ dP -a.e.
Thus the process V U 1 ,U 2 ,a verifies Assumption (A2) and f a(s) satisfies Assumption (A4)(I)(iv) on [t 0 , T ].

Consequently, by the comparison result (Proposition 2.2.3), for any strategy a ∈ A j t0 , P-a.s. for any

s ∈ [t 0 , T ], V a s ≥ V 1,a s ∨ V 2,a s . This combined with (2.2.36) leads to Y 1,j t0 ∨ Y 2,j t0 ≤ Ŷ j t0 = V a * t0 -A a * t0 . We then deduce V 1,a * t0 -A a * t0 ≤ Y 1,j t0 ≤ V a * t0 -A a * t0 and V 2,a * t0 -A a * t0 ≤ Y 2,j t0 ≤ V a * t0 -A a * t0 which implies |Y 1,j t0 -Y 2,j t0 | ≤ | V a * t0 -V 1,a * t0 | + | V a * t0 -V 2,a * t0 |. (2.2.37)
Next we first estimate the quantity

| V a * t0 -V 1,a * t0 |. For s ∈ [t 0 , T ] let us set △V a * s := V a * s -V 1,a * s and △N a * s := N a * s -N 1,a * s
. Applying Itô's Formula to the process e βs | △ V a * s | 2 we obtain: ∀s ∈ [t 0 , T ],

e βs | △ V a * s | 2 + T s e βr △ N a * r 2 dr = - T s βe βr | △ V a * r-| 2 dr -2 ∞ i=1 T s e βr △ V a * r-△ N i,a * r dH (i) r +2 T s e βr △ V a * r-[f a * (r) (r, X t,x r , Γ 1 r , N a * r ) ∨ f a * (r) (r, X t,x r , Γ 2 r , N a * r ) -f a * (r) (r, X t,x r , Γ 1 r , N 1,a * r )]dr - ∞ i=1 ∞ l=1 T s e βr △ N i,a * r △ N l,a * r d([H (i) , H (l) ] r -H (i) , H (l) r ).
By the Lipschitz property of f j , j ∈ A, and then of f a * and the fact that for any x, y ∈ IR, |x ∨ y -y| ≤ |x -y| we have: ∀s ∈ [t 0 , T ],

|f a * (r) (r, X t,x r , Γ 1 r , N a * r ) ∨ f a * (r) (r, X t,x r , Γ 2 r , N a * r ) -f a * (r) (r, X t,x r , Γ 1 r , N 1,a * r )| ≤ |f a * (r) (r, X t,x r , Γ 1 r , N a * r ) ∨ f a * (r) (r, X t,x r , Γ 2 r , N a * r ) -f a * (r) (r, X t,x r , Γ 1 r , N a * r )| +|f a * (r) (r, X t,x r , Γ 1 r , N a * r ) -f a * (r) (r, X t,x r , Γ 1 r , N 1,a * r )| ≤ L(|Γ 1 r -Γ 2 r | + N a * r -N 1,a * r ) (2.2.38)
The inequality 2xy ≤ 1 β x 2 + βy 2 (β > 0 and x, y ∈ IR) and (2.2.38) yield: ∀s ∈ [t 0 , T ],

e βs | △ V a * s | 2 ≤ - T s e βr △ N a * r 2 dr - T s βe βr | △ V a * r-| 2 ds -2 ∞ i=1 T s e βr △ V a * r-△ N i,a * r dH (i) r +2L T s e βr | △ V a * r-|(|Γ 1 r -Γ 2 r | + | N a * r -N 1,a * r |)dr - ∞ i=1 ∞ l=1 T s e βr △ N i,a * r △ N l,a * r d([H (i) , H (l) ] r -H (i) , H (l) r ) ≤ - T s e βr △ N a * r 2 dr - T s βe βr | △ V a * r-| 2 ds -2 ∞ i=1 T s e βr △ V a * r-△ N i,a * r dH (i) r + T s βe βr | △ V a * r-| 2 ds + L 2 β T s e βr (|Γ 1 r -Γ 2 r | + | N a * r -N 1,a * r |) 2 dr - ∞ i=1 ∞ l=1 T s e βr △ N i,a * r △ N l,a * r d([H (i) , H (l) ] r -H (i) , H (l) r ) ≤ 2L 2 β T s e βr |Γ 1 r -Γ 2 r | 2 dr -2 ∞ i=1 T s e βr △ V a * r-N i,a * r dH (i) r - ∞ i=1 ∞ l=1 T s e βr △ N i,a * r △ N j,a * r d([H (i) , H (l) ] r -H (i) , H (l) r ),
for β ≥ 2L 2 . We deduce, in taking expectation,

∀s ∈ [t 0 , T ], E[e βs | △ V a * s | 2 ] ≤ 2L 2 β E[ T s e βr |Γ 1 r -Γ 2 r | 2 dr].
Similarly, we get also ∀s ∈ [t 0 , T ],

E[e βs | V a * s -V 2,a * s | 2 ] ≤ 2L 2 β E[ T s e βr |Γ 1 r -Γ 2 r | 2 dr].
Therefore by (2.2.37) we obtain:

E[e βt0 |Y 1,j t0 -Y 2,j t0 | 2 ] ≤ 8L 2 β Γ 1 -Γ 2 2 2,β . (2.2.39)
As t 0 is arbitrary in [0, T ] then by integration w.r.t. t 0 we get

Θ(Γ 1 ) -Θ(Γ 2 ) 2,β ≤ 8L 2 T m β Γ 1 -Γ 2 2,β . ( 2
E[|Y 1,j s -Y 2,j s | 2 ] ≤ C (Y 1,j ) j∈A -(Y 2,j ) j∈A 2 2,β . (2.2.41)
This estimate will be useful later.

Corollary 2.2.1. Under Assumptions (A4)(I)(ii)-(iv), (A4)(II) and (A4)(III), there exist deterministic lower semi-continuous functions (u j (t, x)) j∈A of polynomial growth such that

∀(t, x) ∈ [0, T ] × IR k , ∀s ∈ [t, T ], Y j s = u j (s, X t,x s ), ∀j ∈ A.
Proof. This is a direct consequence of the construction by induction of the solution (Y j , U j , K j ) j∈A given in Step 1. Actually by Ren et al.'s result [START_REF] Ren | Generalized reflected BSDE driven by a Levy Processes and an obstacle problem for PDIE with a nonlinear Neumann boundary condition[END_REF] there exist deterministic continuous functions of polynomial growth ū(t, x), u(t, x) and u j,n (t, x), n ≥ 0 and j ∈ A, such that

∀(t, x) ∈ [0, T ] × IR k , ∀s ∈ [t, T ] (a) Ȳs = ū(s, X t,x s ) and Y s = u(s, X t,x s ). (b) Y j,n s = u j,n (s, X t,x s ), ∀j ∈ A, and Y ≤ Y j,n ≤ Y j,n+1 ≤ Ȳ .
This yields for any n ≥ 0 and (t,

x) ∈ [0, T ] × IR k , u(t, x) ≤ u n (t, x) ≤ u n+1 (t, x) ≤ ū(t, x).
Thus u j (t, x) := lim n→∞ u j,n (t, x), j ∈ A, verify the required properties since (Y j,n ) n converges to Y j , j ∈ A, in S 2 .

We now give a comparison result for solutions of systems (2.2.24). The induction argument allows to compare the solution of the approximating schemes, by Proposition 2.2.3, and then to deduce the same property for the limiting processes.

Remark 2.2.5. Let ( Ȳ j , Ū j , Kj ) j∈A be a solution of the system of RBSDEs (2.2.24) associated with (( fj ) j∈A , (ḡ jk ) j,k∈A , ( hj ) j∈A ) which satisfy [A4]. If for any j, k ∈ A, f j ≤ fj , h j ≤ hj , g jk ≥ ḡjk then for any j ∈ A, Y j ≤ Ȳ j .

Existence and uniqueness of the solution for the system of IPDEs with inter-connected obstacles

This section focuses on the main result of this paper which is the proof of existence and uniqueness of the solution for a system of IPDEs. For this objective we use its link with the system of RBSDEs (2.2.24). However we are led to make, hereafter, the following additional assumption.

Assumption (A5): For any i ∈ A, f i does not depend on the variable u ∈ ℓ 2 .

So we are going to consider the following system of IPDEs: ∀i ∈ A,

     min{u i (t, x) -max j∈Ai (u j (t, x) -g ij (t, x)); -∂ t u i (t, x) -Lu i (t, x) -f i (t, x, u 1 (t, x), • • • , u m (t, x))} = 0, (t, x) ∈ [0, T ] × [0, T ] × IR k ; u i (T, x) = h i (x) (2.3.1) where Lu(t, x) = L 1 u(t, x) + I(t, x, u) with L 1 u(t, x) := (E[L 1 ]σ(t, x) + b(t, x))∂ x u(t, x) + 1 2 σ(t, x) 2 ̟ 2 D 2 xx u(t, x) (2.3.2)
and I(t, x, u)

:= I R [u(t, x + σ(t, x)y) -u(t, x) -∂ x u(t, x)σ(t, x)y]Π(dy).
Note that for any

φ ∈ C 1,2 ([0, T ] × IR k ) ∩ Π g and (t, x) ∈ [0, T ] × IR k , the non-local term I(t, x, φ) := I R [φ(t, x + σ(t, x)y) -φ(t, x) -∂ x φ(t, x)σ(t, x)y]Π(dy) (2.3.3)
is well-defined. Actually let δ > 0 and let us define, for any q ∈ IR,

I 1,δ (t, x, φ) := |y|≤δ [φ(t, x + σ(t, x)y) -φ(t, x) -∂ x φ(t, x)σ(t, x)y]Π(dy), (2.3.4)
I 2,δ (t, x, q, u) := |y|>δ [u(t, x + σ(t, x)y)u(t, x)qσ(t, x)y]Π(dy).

(2.3.5)

By application of Taylor's expansion we have

φ(t, x + σ(t, x)y) -φ(t, x) -∂ x φ(t, x)σ(t, x)y = y 0 σ(t, x) 2 D 2 xx φ(t, x + σ(t, x)r)(y -r)dr.
But there exists a constant C tx such that for any |r| ≤ δ,

|D 2 xx φ(t, x + σ(t, x)r)| ≤ C tx since φ belongs to C 1,2 . Therefore |φ(t, x + σ(t, x)y) -φ(t, x) -∂ x φ(t, x)σ(t, x)y| ≤ C tx |y| 2
which implies that I 1,δ (t, x, φ) ∈ IR. Next for any (t, x), I 2,δ (t, x, D x φ(t, x), φ) ∈ IR since Π integrates any power function outside [-ǫ, ǫ]. Therefore I(t, x, φ) is well defined.

We are now going to give the definition of a viscosity solution of (2.3.1). First for a locally bounded function u: (t, x) ∈ [0, T ] × IR → u(t, x) ∈ IR, we define its lower semi-continuous (lsc for short) envelope u * and upper semi-continuous (usc for short) envelope u * as following:

u * (t, x) = lim (t ′ ,x ′ )→(t,x), t ′ <T u(t ′ , x ′ ), u * (t, x) = lim (t ′ ,x ′ )→(t,x), t ′ <T u(t ′ , x ′ ) Definition 2.3.1. A function (u 1 , • • • , u m ) : [0, T ] × IR → IR m
which belongs to Π g such that for any i ∈ A, u i is lsc (resp. usc), is said to be a viscosity subsolution (resp. supersolution) of (2.3.1) if for any

i ∈ A, ϕ ∈ Π g C 1,2 ([0, T ] × IR), u i (T, x) ≤ h i (x) (resp. u i (T, x) ≥ h i (x)) and if (t 0 , x 0 ) ∈ (0, T ) × IR is a global maximum (resp. minimum) point of u i -ϕ, min u i (t 0 , x 0 ) -max j∈Ai {u j (t 0 , x 0 ) -g ij (t 0 , x 0 )} ; -∂ t ϕ(t 0 , x 0 ) -Lϕ(t 0 , x 0 ) -f i (t 0 , x 0 , u 1 (t 0 , x 0 ), • • • , u i-1 (t 0 , x 0 ), u i (t 0 , x 0 ), • • • , u m (t 0 , x 0 )) ≤ 0 (resp. ≥ 0),
The function (u i ) m i=1 is called a viscosity solution of (2.3.1) if (u i * ) m i=1 and (u i * ) m i=1 ) are respectively viscosity supersolution and subsolution of (2.3.1).

The following result is needed later.

Lemma 2.3.1. Let (u i ) m
i=1 be a supersolution of (2.3.1) which belongs to Π g , i.e. for some γ > 0 and

C > 0, |u i (t, x)| ≤ C(1 + |x| g amma), ∀(t, x) ∈ [0, T ] × IR k and i ∈ A.
Then there exists λ 0 > 0 such that for any λ ≥ λ 0 and θ > 0,

-→ v (t, x) = (u i (t, x) + θe -λt (1 + |x| 2γ+2 )) m i=1
is supersolution of (4.1).

Proof. We use Definition 2.3.1. Let i ∈ A be fixed and

ϕ i ∈ C 1,2 ∩ Π g such that ϕ i (s, y) -(u i (s, y) + θe -λs (1+|y| 2γ+2
)) has a global maximum in (t, x) ∈ (0, T )×IR and ϕ i (t, x) = u i (t, x)+θe -λt (1+|x| 2γ+2 ).

We then have:

min u i (t, x) + θe -λt (1 + |x| 2γ+2 ) -max j∈Ai (-g ij (t, x) + (u j (t, x) + θe -λt (1 + |x| 2γ+2 ))); -∂ t (ϕ i (t, x) -θe -λt (1 + |x| 2γ+2 )) - 1 2 σ(t, x) 2 ̟ 2 D 2 xx (ϕ i (t, x) -θe -λt (1 + |x| 2γ+2 )) -(σ(t, x)E(L 1 ) + b(t, x))D x (ϕ i (t, x) -θe -λt (1 + |x| 2γ+2 )) - I R [ϕ i (t, x + σ(t, x)y) -θe -λt |x + σ(t, x)y| 2γ+2 -(ϕ i (t, x) -θe -λt |x| 2γ+2 ) -D x (ϕ i (t, x) -θe -λt |x| 2γ+2 )σ(t, x)y]Π(dy) -f i (t, x, -→ u ) ≥ 0. Then -∂ t ϕ i (t, x) -Lϕ i (t, x) -f i (t, x, -→ v (t, x)) ≥ θλe -λt (1 + |x| 2γ+2 ) -1 2 θe -λt σ(t, x) 2 ̟ 2 D 2 xx |x| 2γ+2 -(σ(t, x)E(L 1 ) + b(t, x))D x (θe -λt |x| 2γ+2 ) -I R (θe -λt |x + σ(t, x)y| 2γ+2 -θe -λt |x| 2γ+2 -θe -λt D x |x| 2γ+2 σ(t, x)y)Π(dy) + f i (t, x, -→ u (t, x)) -f i (t, x, -→ v (t, x)) ≥ θe -λt λ(1 + |x| 2γ+2 ) -1 2 σ(t, x) 2 ̟ 2 D 2 xx |x| 2γ+2 -(σ(t, x)E(L 1 ) + b(t, x))D x |x| 2γ+2 -I R (|x + σ(t, x)y| 2γ+2 -|x| 2γ+2 -D x |x| 2γ+2 σ(t, x)y)Π(dy) + m k=1 C k,i t,x,θ,λ |x| 2γ+2
(2.3.6) where C k,i t,x,θ,λ is bounded by the Lipschiz constant of f i with respect to (y i ) i=1,••• ,m which is independent of θ. But, since φ(y) = |y| 2γ+2 ∈ C 1,2 ∩ Π g , then the non-local term is well defined. Now let us set ψ(ρ) := φ(x + ρσ(t, x)y), for ρ, x, y ∈ IR. First note that for any t, x, y we have

|x + σ(t, x)y| 2γ+2 -|x| 2γ+2 -D x |x| 2γ+2 σ(t, x)y| = |ψ(1) -ψ(0) -D x ψ(0)| = | 1 0 (1 -ρ)ψ (2) (ρ)dρ| ≤ C|y| 2 (|x| 2γ + |y| 2γ ). Therefore I R (|x + σ(t, x)y| 2γ+2 -|x| 2γ+2 -D x |x| 2γ+2 σ(t, x)y)Π(dy) = |y|≤δ |x + σ(t, x)y| 2γ+2 -|x| 2γ+2 -D x |x| 2γ+2 σ(t, x)yΠ(dy) + |y|≥δ |x + σ(t, x)y| 2γ+2 -|x| 2γ+2 -D x |x| 2γ+2 σ(t, x)yΠ(dy)| ≤ C(1 + |x| 2γ )
since the measure Π integrates any power function away from 0. Therefore there exists a constant λ 0 ∈ IR + which does not depend on θ such that if λ ≥ λ 0 then the right-hand side of (2.3.6) is nonnegative. Thus v is a viscosity supersolution of (2.3.1), which is the desired result.

Remark 2.3.1. In the same way one can show that if (u i ) m i=1 is a viscosity subsolution of (2.3.1) which belongs to Π g , i.e. for some γ > 0 and C > 0,

|u i (t, x)| ≤ C(1 + |x| γ ), ∀(t, x) ∈ [0, T ] × IR k and i ∈ A.
Then there exists λ 0 > 0 such that for any λ ≥ λ 0 and θ > 0, -→ v (t, x) = (u i (t, x)θe -λt (1 + |x| 2γ+2 )) m i=1 is subsolution of (2.3.1).

Existence of the viscosity solution

In this section we deal with the issue of existence of the viscosity solution of (2.3.1). Recall that (Y j , U j , K j ) j∈A is the unique solution of (2.2.24) and let (u j (t, x)) j∈A be the functions defined in Corollary 2. Proof. The proof will be divided into two steps.

Step 1: We first show that (u j ) m j=1 is a supersolution of (2.3.1). Note that for all j ∈ A, as u j is lsc, we then have

u j * = u j . Next let us set u n j (t, x) = Y j,n,t,x t
, where (Y j,n,t,x ; U j,n,t,x , K j,n,t,x ) j∈A is the unique solution of (2.2.27). As pointed out in Corollary 2.2.1, for any n ≥ 0, (t,

x) ∈ [0, T ] × IR k and s ∈ [t, T ], Y j,n,t,x s = u n j (s, X t,x s ) and u n j (t, x) ր u j (t, x).
Additionally by induction, (u n j ) j∈A , n ≥ 0, are continuous, belong to Π g and by Ren et al.'s result (Theorem 2.2.2) verify in viscosity sense the following system (n ≥ 1): ∀j ∈ A,

       min u j,n (t, x) -max k∈Aj (u j,n-1 (t, x) -g jk (t, x)); -∂ t u j,n (t, x) -Lu j,n (t, x) -f j (t, x, (u 1,n-1 , • • • , u j-1,n-1 , u j,n , u j+1,n-1 , • • • , u m,n-1 )(t, x)) = 0; u j,n (T, x) = h j (x).
(2.3.7) First note that for any j ∈ A, u j verifies

u j (T, x) = h j (x) and u j (t, x) ≥ max k∈Aj {u k (t, x) -g jk (t, x)}, ∀(t, x) ∈ [0, T ] × IR k .
Next let (t, x) ∈ (0, T )×IR and let j ∈ A be fixed. Let φ be a function which belongs to C 1,2 ([0, T ]×IR)∩Π g such that u jφ has a strict global minimum in (t, x) on [0, T ] × IR k and wlog we assume that u j (t, x) = φ(t, x). Now let δ > 0 be fixed. Then (t, x) is a global strict minimum of u jφ in [0, T ] × B(x, C σ δ). Next let (t n , x n ) be the global minimum of u n jφ on [0, T ] × B(x, C σ δ). Therefore lim n (t n , x n ) = (t, x) and u n j (t n , x n ) → u(t, x).

Actually let us consider a convergent subsequence of (t n , x n ), which we still denote by (t n , x n ), and let set (t * , x * ) its limit. Then

u n j (t n , x n ) -φ(t n , x n ) ≤ u n j (t, x) -φ(t, x).
(2.3.8)

Taking the limit wrt n and since u j * = u j is lsc to obtain

u j (t * , x * ) -φ(t * , x * ) ≤ u j (t, x) -φ(t, x).
As the minimum (t, x) of u jφ on [0, T ] × IR k is strict then (t * , x * ) = (t, x). It follows that the sequence ((t n , x n )) n converges to (t, x). Going back now to (2.3.8) and in sending n to infinite we obtain

u j * (t, x) = u j (t, x) ≤ lim inf n u n j (t n , x n ) ≤ lim sup n u n j (t n , x n ) ≤ u j (t, x) which implies that u n j (t n , x n ) → u j (t, x) as n → ∞. Now for n large enough (t n , x n ) ∈ (0, T ) × B(x, C σ δ) and it is the global minimum of u n j -φ in [0, T ] × B(x, C σ δ). As u n
j is a supersolution of (2.3.7), then by Definition 5.2.1 (see Appendix 5.2) we have

-∂ t φ(t n , x n ) -L 1 φ(t n , x n ) -I 1,δ (t n , x n , φ) -I 2,δ (t n , x n , D x φ(t n , x n ), u j,n ) ≥ f j (t n , x n , u 1,n-1 (t n , x n ), • • • , u j-1,n-1 (t n , x n ), u j,n (t n , x n ), u j+1,n-1 (t n , x n ), • • • , u m,n-1 (t n , x n )).
(2.3.9) But there exists a subsequence of {n} such that:

(i) for any k ∈ A j , (u n-1 k (t n , x n )) n is convergent and then lim n u n-1 k (t n , x n ) ≥ u k (t, x) = u k * (t, x) ; (ii) (I 1,δ (t n , x n , φ)) n → I 1,δ (t, x, φ) as n → ∞ ;
Sending now n to infinite (through the previous subsequence) in (2.3.9), using the fact that f j is continuous and verifies (A4)(I)(v) and finally by Fatou's Lemma to obtain:

-∂ t φ(t, x) -L 1 φ(t, x) -I 1,δ (t, x, φ) ≥ I 2,δ (t, x, D x φ(t, x), u j ) + f j (t, x, u 1 (t, x), • • • , u j-1 (t, x), u j (t, x), u j+1 (t, x), • • • , u m (t, x)).
But u j (t, x) = φ(t, x) and u j ≥ φ, then I 2,δ (t, x, D x φ(t, x), u j ) ≥ I 2,δ (t, x, D x φ(t, x), φ). Plugging now this inequality in the previous one to obtain

-∂ t φ(t, x) -L 1 φ(t, x) -I(t, x, φ) -f j (t, x, u 1 (t, x), • • • , u j-1 (t, x), u j (t, x), u j+1 (t, x), • • • , u m (t, x)) ≥ 0. Therefore u j is a viscosity supersolution of      min{u j (t, x) -max k∈Aj (u k (t, x) -g jk (t, x)); -∂ t u j (t, x) -Lu j (t, x) -f j (t, x, u 1 (t, x), • • • , u m (t, x))} = 0; u j (T, x) = h j (x).
As j is arbitrary then (u j ) j∈A is a viscosity supersolution of (2.3.1).

Step 2: We will now show that (u * j ) j∈A is a subsolution of (2.3.1). As a first step we are going to show that ∀j ∈ A, min{u * j (T, x) -

h j (x); u * j (T, x) -max k∈Aj (u * k (T, x) -g jk (T, x))} = 0.
By definition of u * j and since u n j ր u j , we have

min{u * j (T, x) -h j (x); u * j (T, x) -max k∈Aj (u * k (T, x) -g jk (T, x))} ≥ 0
Next suppose that for some x 0 ∈ IR, ∃j > 0, s.t.

min{u * j (T, x 0 ) -h j (x 0 ); u * j (T, x 0 ) -max k∈Aj (u * k (T, x 0 ) -g jk (T, x 0 ))} = 2ǫ.
We will show that leads to a contradiction. Let (t k , x k ) k≥1 → (T, x 0 ) and u j (t k , x k ) → u * j (T, x 0 ). We can find a sequence of functions (v n ) n≥0 ∈ C 1,2 ([0, T ] × IR) of compact support such that v n → u * j , since u * j is usc. On some neighborhood B n of (T, x 0 ) we have,

∀(t, x) ∈ B n , min{v n (t, x) -h j (x); v n (t, x) -max k∈Aj (u * k (t, x) -g jk (t, x))} ≥ ǫ. (2.3.10)
Let us denote by

B n k := [t k , T ] × B(x k , δ k n ), for some δ k n ∈]0, 1] small enough such that B n k ⊂ B n . Since u *
j is of polynomial growth, there exists c > 0, such that |u * j | ≤ c on B n . We can then assume v n ≥ -2c on B n . Define

V n k (t, x) := v n (t, x) + 4c|x -x k | 2 δ n k 2 + √ T -t Note that V n k (t, x) ≥ v n (t, x) and (u * j -V n k )(t, x) ≤ -c ∀(t, x) ∈ [t k , T ] × ∂B(x k , δ n k ). (2.3.11)
On the other hand, by Itô's formula we have

-{∂ t V n k (t, x) + LV n k (t, x)} = -∂ t v n (t, x) + ∂ t ((T -t) 1 2 ) + {E(L 1 )σ(t, x) + b(t, x)}(∂ x v n (t, x) + 8c(x-x k ) (δ n k ) 2 ) + 1 2 σ(t, x) 2 ̟ 2 (D 2 xx v n (t, x) + 8c (δ n k ) 2 ) + I R [v n (t, x + σ(t, x)y) + 4c|x-x k +σ(t,x)y| 2 (δ n k ) 2 +(T -t) 1 2 -v n (t, x) -4c|x-x k | 2 (δ n k ) 2 -(T -t) 1 2 -∂ x v n (t, x)σ(t, x)y -8c(x-x k ) (δ n k ) 2 σ(t, x)y]Π(dy) = -∂ t v n (t, x) + ∂ t ((T -t) 1 2 ) + {E(L 1 )σ(t, x) + b(t, x)}{∂ x v n (t, x) + 8c(x-x k ) (δ n k ) 2 } + 1 2 σ(t, x) 2 ̟ 2 (D 2 xx v n (t, x) + 8c (δ n k ) 2 ) + I R [v n (t, x + σ(t, x)y) -v n (t, x) -∂ x v n (t, x)σ(t, x)y]Π(dy) + I R [ 4c|x-x k +σ(t,x)y| 2 (δ n k ) 2 -4c|x-x k | 2 (δ n k ) 2 -8c(x-x k ) (δ n k ) 2 σ(t, x)y]Π(dy) . Note that Φ(x) := 4c|x-x k | 2 (δ n k ) 2 ∈ C 2 ∩ Π g and v n ∈ C 1,
-(∂ t V n k (t, x) + LV n k )(t, x) ≥ 0, ∀(t, x) ∈ B k n .
(2.3.12)

Consider now the stopping time

θ k n := inf {s ≥ t k , (s, X t k ,x k s ) ∈ B k n c } ∧ T , where B k n c is the complement of B k n and θ k := inf {s ≥ t k , u j (s, X t k ,x k s ) = max l∈Aj (u l (s, X t k ,x k s ) -g jl (s, X t k ,x k s ))} ∧ T . Applying Itô's formula with V n k (t, x) on [t k , θ k n ∧ θ k ]
and taking into account (2.3.11), (2.3.10), (2.3.12) and the fact that

V n k ∈ C 1,2
, to obtain:

V n k (t k , x k ) = V n k (θ k n ∧ θ k , X t k ,x k θ k n ∧θ k ) - θ k n ∧θ k t k [b(r, X t k ,x k r )∂ x V n k (r, X t k ,x k r ) + ∂ t V n k (t, x)(r, X t k ,x k r )]dr - θ k n ∧θ k t k σ(r, X t k ,x k r- )∂ x V n k (r, X t k ,x k r- )dL r -1 2 θ k n ∧θ k t k σ 2 (r, X t k ,x k r )̟ 2 ∂ 2 xx V n k (r, X t k ,x k r )dr - t k <r≤θ k n ∧θ k {V n k (r, X t k ,x k r ) -V n k (r, X t k ,x k r- ) -σ(r, X t k ,x k r- )∂ x V n k (r, X t k ,x k r- ) △ L r }
(2.3.13) Next let us deal with the last term of the last equality and let us set

h(s, y) = V n k (s, X t k ,x k s- + σ(s, X t k .x k s-)y) -V n k (s, X t k ,x k s-) -∂ x V n k (s, X t k .x k s-)σ(s, X t k .
x k s-)y. By the mean value theorem we have

h(s, y) = 1 2 ∂ 2 xx v n (s, X t k ,x k s- + Xσ(s, X t k .x k s-)y)(σ(s, X t k .x k s-)y) 2 + 4c δ n k 2 (σ(s, X t k .x k s-)y) 2
where X is a stochastic processes which is valued in (0, 1). As v n is of compact support and σ is bounded then

E[ T 0 I R |h(s, y)|Π(dy)ds] < ∞.
It follows that

E[ t k ≤r≤θ k n ∧θ k {V n k (r, X t k ,x k r ) -V n k (r, X t k ,x k r- ) -σ(r, X t k ,x k r- )∂ x V n k (r, X t k ,x k r- ) △ L r }] =E[ θ k n ∧θ k t k I R h(s, y)Π(dy)ds] < ∞.
Next going back to (2.3.13) and taking expectation to obtain

V n k (t k , x k ) = E[V n k (θ k n ∧ θ k , X t k ,x k θ k n ∧θ k ) - θ k n ∧θ k t k (∂ t V n k (r, X t k ,x k r ) + LV n k (r, X t k ,x k r ))dr] ≥ E[V n k (θ k n , X t k ,x k θ k n ) {θ k n ≤θ k } + V n k (θ k , X t k ,x k θ k ) {θ k n >θ k } ] = E[{V n k (θ k n , X t k ,x k θ k n ) {θ k n <T } + V n k (T, X t k ,x k T ) {θ k n =T } } {θ k n ≤θ k } + V n k (θ k , X t k ,x k θ k ) {θ k n >θ k } ] ≥ E[{(u * j (θ k n , X t k ,x k θ k n ) + c) {θ k n <T } + (ǫ + h j (X t k ,x k T )) {θ k n =T } } {θ k n ≤θ k } + {ǫ + max k∈Aj (u * k (θ k , X t k ,x k θ k ) -g jk (θ k , X t k ,x k θ k ))} {θ k n >θ k } ] ≥ E[u j (θ k n ∧ θ k , X t k ,x k θ k n ∧θ k )] + c ∧ ǫ = E[u j (t k , x k ) - θ k n ∧θ k t k f j (s, X t k ,x k s , (u l (s, X t k ,x k s )) l=1,m ds] + c ∧ ǫ
since the processes (Y j = u j (., X.)) j∈A stopped at time θ k n ∧ θ k solves an explicit RBSDE system with triple of data given by ((f j ) j∈A , (h j ) j∈A , (g i,j ) i,j∈A ). In addition, dK j,t,x = 0 on [t k , θ k ]. On the other hand, (u j ) j∈A ∈ Π g and then taking into account (2.2.16) and Assumption (A4)(1)(iii), we deduce that

lim k→∞ E[ θ k n ∧θ k t k f j (s, X t k ,x k s , (u l (s, X t k ,x k s )) l=1,m )ds] = 0.
Taking the limit in the previous inequalities yields:

lim k→∞ V n k (t k , x k ) = lim k→∞ v n (t k , x k ) + T -t k = v n (T, x 0 ) ≥ lim k→∞ u j (t k , x k ) + c ∧ ǫ = u * j (T, x 0 ) + c ∧ ǫ.
As v n → u * j pointwisely, then we get a contradiction, when taking the limit in the previous inequalities, and the result follows, i.e., ∀x ∈ IR, ∀j ∈ A,

min{u * j (T, x) -h j (x); u * j (T, x) -max l∈Aj (u * l (T, x) -g jl (T, x))} = 0.
Finally the proof of u * j (T, x) = h j (x), ∀j ∈ A is obtained in the same way as in ( [START_REF] Hamadène | Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems[END_REF], pp.180) since the function g ij , i, j ∈ A verify the non-free loop property (A4)(II). Now let us show (u * j ) j∈A is a subsolution of (2.3.1). First note that since u n j ր u j and u n j is continuous, we have

u * j (t, x) = lim sup n→∞ * u n j (t, x) = lim n→∞,t ′ →t,x ′ →x u n j (t ′ , x ′ ).
Besides ∀j ∈ A and n ≥ 0 we deduce from the construction of u n j that

u n j (t, x) ≥ max l∈Aj (u n-1 l (t, x) -g jl (t, x))
and by taking the limit in n we obtain:

∀j ∈ A, ∀x ∈ IR, u * j (t, x) ≥ max l∈Aj (u * l (t, x) -g jl (t, x)). Next fix j ∈ A. Let (t, x) ∈ (0, T ) × IR be such that u * j (t, x) -max l∈Aj (u * l (t, x) -g jl (t, x)) > 0. (2.3.14)
Let φ be a C 1,2 ([0, T ] × IR) ∩ Π g function such that u * jφ has a global maximum at (t, x) in [0, T ] × IR k which wlog we suppose strict and u j (t, x) = φ(t, x). Therefore (t, x) is a global strict maximum of u jφ in [0, T ] × B(x, C σ δ). On the other hand there exist subsequences {n k } and ((t

′ n k , x ′ n k )) k such that ((t ′ n k , x ′ n k )) k → k (t, x) and u n k j (t ′ n k , x ′ n k ) → k u * j (t, x). Let now (t n k , x n k ) be the global maximum of u n k j -φ on [0, T ] × B(x, C σ δ). Therefore (t n k , x n k ) → k (t, x) and u n k j (t n k , x n k ) → k u * j (t, x).
Actually let us consider a convergent subsequent of (t n k , x n k ), which we still denote by (t n k , x n k ), and let ( t, x) be its limit. Then for some k 0 and for k ≥ k 0 we have

u n k j (t n k , x n k ) -φ(t n k , x n k ) ≥ u n k j (t ′ n k , x ′ n k ) -φ(t ′ n k , x ′ n k ).
(2.3.15)

Taking the limit wrt k to obtain

u * j ( t, x) -φ( t, x) ≥ u * j (t, x) -φ(t, x).
As the maximum (t, x) of u jφ on [0, T ] × IR k is strict then ( t, x) = (t, x). It follows that the sequence ((t n k , x n k )) k converges to (t, x). Going back now to (3.3.24) and in sending k to infinite we obtain

u * j (t, x) ≥ lim sup k u n k j (t n k , x n k ) ≥ lim inf k u n k j (t n k , x n k ) ≥ lim inf k u n k j (t ′ n k , x ′ n k ) = u * j (t, x) which implies that u n k j (t n k , x n k ) → u * j (t, x) as k → ∞. Now for k large enough, (i) (t n k , x n k ) ∈ (0, T ) × B(x, C σ δ) and is the global maximum of u n k j -φ in (0, T ) × B(x, C σ δ) ; (ii) u n k j (t n k , x n k ) > max l∈Aj (u n k -1 l (t n k , x n k ) -g jl (t n k , x n k )).
As u n k j is a subsolution of (2.3.7), then by Definition 5.2.1 (see Appendix 5.2) we have

-∂ t φ(t n k , x n k ) -L 1 φ(t n k , x n k ) -I 1,δ (t n k , x n k , φ) -I 2,δ (t n k , x n k , D x φ(t n k , x n k ), u j,n k ) ≤ f j (t n k , x n k , u 1,n k -1 (t n k , x n k ), • • • , u j-1,n k -1 (t n k , x n k ), u j,n k (t n k , x n k ), u j+1,n k -1 (t n k , x n k ), • • • , u m,n k -1 (t n k , x n k )).
(2.3.16) But there exists a subsequence of {n k } (which we still denote by {n k }) such that:

(i) for any l ∈ A j , (u

n k -1 l (t n k , x n k )) k is convergent and then lim k u n k -1 l (t n k , x n k ) ≤ u * l (t, x) ; (ii) (I 1,δ (t n k , x n k , φ)) n k → I 1,δ (t, x, φ) as k → ∞.
Sending now k to infinite (through the previous subsequence) in (3.3.25), using the fact that f j is continuous and verifies (A4)(I)(v) and finally by Lebesgue's Theorem to obtain

-∂ t φ(t, x) -L 1 φ(t, x) -I 1,δ (t, x, φ) ≤ I 2,δ (t, x, D x φ(t, x), u j ) + f j (t, x, u * 1 (t, x), • • • , u * j-1 (t, x), u * j (t, x), u * j+1 (t, x), • • • , u * m (t, x)).
But u j (t, x) = φ(t, x) and u j ≤ φ, then I 2,δ (t, x, D x φ(t, x), u j ) ≤ I 2,δ (t, x, D x φ(t, x), φ). Plugging now this inequality in the previous one to obtain

-∂ t φ(t, x) -L 1 φ(t, x) -I(t, x, φ) -f j (t, x, u * 1 (t, x), • • • , u * j-1 (t, x), u * j (t, x), u * j+1 (t, x), • • • , u * m (t, x)) ≤ 0.
Therefore u j is a viscosity subsolution of

     min{u j (t, x) -max k∈Aj (u k (t, x) -g jk (t, x)); -∂ t u j (t, x) -Lu j (t, x) -f j (t, x, u 1 (t, x), • • • , u m (t, x))} = 0; u j (T, x) = h j (x).
As j is arbitrary then (u j ) j∈A is a viscosity subsolution of (2.3.1).

Uniqueness of the viscosity solution

We now give a comparison result of subsolution and supersolution of system (2.3.1), from which we get the continuity and uniqueness of its solution.

Proposition 2.3.1. Assume Assumptions (A4) fulfilled. Let (u j ) j∈A (resp. (w j ) j∈A ) be a subsolution (resp. supersolution) of (2.3.1) which belongs to Π g . Then for any j ∈ A,

∀(t, x) ∈ [0, T ] × IR, u j (t, x) ≤ w j (t, x)
Proof. Let γ be a real constant such that for any j ∈ A and (t,

x) ∈ [0, T ] × IR k , |u j (t, x)| + |w j (t, x)| ≤ C(1 + |x| γ ).
To begin with we additionally assume the existence of a constant λ such that λ < -m. max j∈A

{C j } (C j is the Lipschitz constant of f j w.r.t -→ y ) and ∀j ∈ A, ∀t, x, y 1 , • • • , y j-1 , y j+2 , • • • , y m , y ≥ y ′ , f j (t, x, y 1 , • • • , y j-1 , y, • • • , y m ) -f j (t, x, y 1 , • • • , y j-1 , y ′ , • • • , y m ) ≤ λ(y -y ′ ) (2.3.17)
Thanks to Lemma 4.1 and Remark 2.3.1, we know there exists ν large enough such that for any θ > 0,

w j,θ,ν (t, x) = w j (t, x) + θe -νt |x| 2γ+2 (resp. u j,θ,ν (t, x) = u j (t, x) -θe -νt |x| 2γ+2 ). So it is enough to show that ∀j ∈ A, ∀(t, x) ∈ [0, T ] × IR, u j,θ,ν (t, x) ≤ w j,θ,ν (t, x),
then taking limits as θ → 0, the result follows. On the other hand by the growth condition there exists a constant C > 0 such that

∀j ∈ A, ∀(t, x) ∈ [0, T ] × IR, s.t. |x| ≥ C, u j,θ,ν (t, x) < 0 < w j,θ,ν (t, x). (2.3.18)
Now for the sake of simplicity we merely denote u j,θ,ν (resp. w j,θ,ν ) by u j (resp. w j ).

To get the comparison result, we proceed by contradiction assuming that

∃(t 1 , x 1 ) ∈ [0, T ] × IR, such that max j∈A (u j (t 1 , x 1 ) -w j (t 1 , x 1 )) > 0.
Taking into account the values of the subsolution and the supersolution at T , there exist ( t, x) ∈ [0, T [×B(0, C) (wlog we assume that t > 0), such that :

0 < max (t,x)∈[0,T ]×I R max j∈A (u j (t, x) -w j (t, x)) = max (t,x)∈[0,T [×B(0,C) max j∈A (u j (t, x) -w j (t, x))
= max j∈A (u j ( t, x)w j ( t, x)).

We now define à as follows:

à := {j ∈ A, u j ( t, x) -w j ( t, x) = max k∈A (u k ( t, x) -w k ( t, x))}. (2.3.19)
By the assumption (A4)(2), using the same argument as in ( [START_REF] Hamadène | Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems[END_REF], pp. 171), we can prove that for some j ∈ Ã,

u j ( t, x) > max k∈Aj (u k ( t, x) -g jk ( t, x)). ( 2

.3.20)

Let us now take such a j ∈ Ã. For ε > 0 and ρ > 0, let us define

Φ j ε,ρ (t, x, y) := u j (t, x) -w j (t, y) - |x -y| 2 ε -|t -t| 2 -ρ|x -x| 4 .
For any ε > 0 and ρ > 0, let (t 0 , x 0 , y 0 ) be such that

Φ j ε,ρ (t 0 , x 0 , y 0 ) = max (t,x,y)∈[0,T ]× B(0,C) 2 Φ j ε,ρ (t, x, y) = max (t,x,y)∈[0,T ]×I R 2 Φ j ε,ρ (t, x, y).
Note that the maximum exists since Φ j ε,ρ is usc and B(0, C) 2 is the closure of B(0, C) 2 and by (2.3.18) the maximum on [0, T ] × IR 2 can only be reached in B(0, C) 2 . Finally let us point out that (t 0 , x 0 , y 0 ) depends actually on ε and ρ which we omit for sake of simplicity. We then have, Φ j ε,ρ ( t, x, x) = u j ( t, x)w j ( t, x)

≤ u j ( t, x) -w j ( t, x) + |x 0 -y 0 | 2 ε + |t 0 -t| 2 + ρ|x 0 -x| 4
≤ u j (t 0 , x 0 )w j (t 0 , y 0 ).

(2.3.21)

The growth condition of u j and w j implies that |x0-y0| 2 ε + |t 0 -t| 2 + ρ|x 0 -x| 4 is bounded and hence lim ε→0 (x 0y 0 ) = 0. Next by (3.3.6), for any subsequence (t 0 l , x 0 l , y 0 l ) l which converges to ( t, x, x), u j ( t, x)w j ( t, x) ≤ u j ( t, x)w j ( t, x), since u j is usc and w j is lsc. By the definition of ( t, x) this last inequality is an equality. Using both the definiton of Φ j ε,ρ and (3.3.6), it implies that the sequence lim

ε→0 (t 0 , x 0 , y 0 ) = ( t, x, x) (2.3.22)
and once more from (3.3.6) we deduce

lim ε→0 |x 0 -y 0 | 2 ε = 0. (2.3.23)
Finally classically (see e.g. [START_REF] Hamadène | Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems[END_REF], pp. 173) we have also lim ε→0 (u j (t 0 , x 0 ), w j (t 0 , y 0 )) = (u j ( t, x), w j ( t, x)).

(2.3.24)

Next as the functions (u k ) k∈A are usc and (g ij ) i,j∈A are continuous, and since the index j satisfies (2.3.19), there exists r > 0 such that for (t, x) ∈ B(( t, x), r) we have u j (t, x) > max k∈Aj (u k (t, x)g jk (t, x)).

But by (3.3.9), (3.3.7) and once more since u j is usc then there exists ε 0 such that for any 0 < ε < ε 0 , we have:

u j (t 0 , x 0 ) > max k∈Aj (u k (t 0 , x 0 ) -g ij (t 0 , x 0 )).
Now for ε small enough, we are able to apply Jensen-Ishii's Lemma for non local operators established by Barles and Imbert ([6], pp.583) (one can see also [START_REF] Biswas | Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion processes[END_REF], Lemma 4.1, pp.64) with u j , w j and φ(t, x, y) := |x-y| 2 ε + |t -t| 2 + ρ|x 0 -x| 4 at the point (t 0 , x 0 , y 0 ). For any δ ∈ (0, 1) there are p 0 u , q 0 u , p 0 w , q 0 w , M 0 u and M 0 w real constants such that: (i) p 0 up 0 w = ∂ t φ(t 0 , x 0 , y 0 ), q 0 u = ∂ x φ(t 0 , x 0 , y 0 ), q 0 w = -∂ y φ(t 0 , x 0 , y 0 ) (2.3.25)

and

M 0 u 0 0 -M 0 w ≤ 4 ε 1 -1 -1 1 + 12ρ|x 0 -x| 2 0 0 0 ; (2.3.26) (ii) -p 0 u -{σ(t 0 , x 0 )E(L 1 ) + b(t 0 , x 0 )}q 0 u - 1 2 σ(t 0 , x 0 ) 2 ̟ 2 M 0 u -f j (t 0 , x 0 , (u k (t 0 , x 0 )) m k=1 ) -I 1,δ (t 0 , x 0 , φ(t 0 , ., y 0 )) -I 2,δ (t 0 , x 0 , q 0 u , u j ) ≤ 0 ; (2.3.27) (iii) -p 0 w -{σ(t 0 , y 0 )E(L 1 ) + b(t 0 , y 0 )}q 0 w - 1 2 σ(t 0 , y 0 ) 2 ̟ 2 M 0 w -f j (t 0 , y 0 , (w k (t 0 , y 0 )) m k=1
) -I 1,δ (t 0 , y 0 , -φ(t 0 , x 0 , .)) -I 2,δ (t 0 , y 0 , q 0 w , w j ) ≥ 0.

(2.3.28)

We are now going to provide estimates for the non-local terms. First let us define ψ ρ (t, x) := ρ|x -x| 4 + |t -t| 2 . By definition of (t 0 , x 0 , y 0 ), for any d, d ′ ∈ IR,

u j (t 0 , x 0 + d ′ ) -w j (t 0 , y 0 + d) - |x 0 + d ′ -y 0 -d| 2 ε -ψ ρ (t 0 , x 0 + d ′ ) ≤u j (t 0 , x 0 ) -w j (t 0 , y 0 ) - |x 0 -y 0 | 2 ε -ψ ρ (t 0 , x 0 ).
Therefore for z ∈ IR, in taking d ′ = σ(t 0 , x 0 )z and d = σ(t 0 , y 0 )z, we obtain u j (t 0 , x 0 + σ(t 0 , x 0 )z)u j (t 0 , x 0 )q 0 u σ(t 0 , x 0 )z ≤ w j (t 0 , y 0 + σ(t 0 , y 0 )z)w j (t 0 , y 0 )q 0 w σ(t 0 , y 0 )z + |σ(t0,x0)-σ(t0,y0)| 2 z 2 ε +ψ ρ (t 0 , x 0 + σ(t 0 , x 0 )z)ψ ρ (t 0 , x 0 ) -D x ψ ρ (t 0 , x 0 )σ(t 0 , x 0 )z.

It implies that for any δ > 0,

I 2,δ (t 0 , x 0 , q 0 u , u j ) -I 2,δ (t 0 , y 0 , q 0 w , w j ) ≤ C |x 0 -y 0 | 2 ε + I 2,δ (t 0 , x 0 , D x ψ ρ (t 0 , x 0 ), ψ ρ ) (2.3.29)
since σ(t, x) is uniformly Lipschitz w.r.t. x. But it easy to check that

|I 2,δ (t 0 , x 0 , D x ψ ρ (t 0 , x 0 ), ψ ρ )| ≤ ρ |z|≥δ {|z| 2 + |z| 4 }Π(dz).
On the other hand, since φ ∈ C 2 I 1,δ (t 0 , x 0 , φ(t 0 , ., y 0 )) = |z|≤δ {φ(t 0 , x 0 + σ(t 0 , x 0 )z, y 0 )φ(t 0 , x 0 , y 0 ) -D x φ(t 0 , x 0 , y 0 )σ(t 0 , x 0 )z}Π(dz) ≤ σ(t 0 , x 0 ) 2 |z|≤δ {ε -1 + 6ρ(1 + |z| 2 )}|z| 2 Π(dz), and I 1,δ (t 0 , y 0 , -φ(t 0 , x 0 , .)) = |z|≤δ {-φ(t 0 , x 0 , y 0 + σ(t 0 , y 0 )z) + φ(t 0 , x 0 , y 0 ) + D y φ(t 0 , x 0 , y 0 )σ(t 0 , y 0 )z}Π(dz) = -ε -1 σ(t 0 , y 0 ) 2 |z|≤δ |z| 2 dΠ(z).

Therefore we have

-I 1,δ (t 0 , x 0 , φ(t 0 , ., y 0 )) + I 1,δ (t 0 , y 0 , -φ(t 0 , x 0 , .))

≥ -σ(t 0 , x 0 ) 2 |z|≤δ {ε -1 + 6ρ(1 + |z| 2 )}|z| 2 Π(dz) -ε -1 σ(t 0 , y 0 ) 2 |z|≤δ |z| 2 dΠ(z).
(2.3.30)

Making now the difference between (3.3.14) and (3.3.15) yields

-(p 0 u -p 0 w ) -[(σ(t 0 , x 0 )E(L 1 ) + b(t 0 , x 0 ))q 0 u -(σ(t 0 , y 0 )E(L 1 ) + b(t 0 , y 0 ))q 0 w ] - 1 2 ̟ 2 [σ(t 0 , x 0 ) 2 M 0 u -σ(t 0 , y 0 ) 2 M 0 w ] -[f j (t 0 , x 0 , (u k (t 0 , x 0 )) m k=1
)f j (t 0 , y 0 , (w k (t 0 , y 0 )) m k=1 )] -I 1,δ (t 0 , x 0 , φ(t 0 , ., y 0 )) + I 1,δ (t 0 , y 0 , -φ(t 0 , x 0 , .)) -I 2,δ (t 0 , x 0 , q 0 u , u j ) + I 2,δ (t 0 , y 0 , q 0 w , w j ) ≤ 0.

Taking now into account (3.3.16) and (3.3.17) we get

-(p 0 u -p 0 w ) -[(σ(t 0 , x 0 )E(L 1 ) + b(t 0 , x 0 ))q 0 u -(σ(t 0 , y 0 )E(L 1 ) + b(t 0 , y 0 ))q 0 w ] - 1 2 ̟ 2 [σ(t 0 , x 0 ) 2 M 0 u -σ(t 0 , y 0 ) 2 M 0 w ] -[f j (t 0 , x 0 , (u k (t 0 , x 0 )) m k=1 ) -f j (t 0 , y 0 , (w k (t 0 , y 0 )) m k=1 )] -σ(t 0 , x 0 ) 2 |z|≤δ {ε -1 + 6ρ(1 + |z| 2 )}|z| 2 Π(dz) -ε -1 σ(t 0 , y 0 ) 2 |z|≤δ |z| 2 dΠ(z) -C |x 0 -y 0 | 2 ε -I 2,δ (t 0 , x 0 , D x ψ ρ (t 0 , x 0 ), ψ ρ ) ≤ 0.
Next by using the properties satisfied by p 0 u , q 0 u , p 0 w , q 0 w , M 0 u and M 0 w and sending δ to 0 to obtain the existence of a constant C ε,ρ such that for any fixed ρ we have lim sup ε→0 C ε,ρ ≤ 0 and

-{f j (t 0 , x 0 , u k (t 0 , x 0 )) m k=1 ) -f j (t 0 , x 0 , (w k (t 0 , y 0 )) m k=1 , )} ≤ C ε,ρ + ρ I R {|z| 2 + |z| 4 }Π(dz). (2.3.31)
Next since f j is Lipschitz w.r.t. (y k ) m k=1 and by condition (2.3.17) we have

-λ(u j (t 0 , x 0 ) -w j (t 0 , y 0 )) - k∈Aj Υ j,k ε,ρ (u k (t 0 , x 0 ) -w k (t 0 , y 0 )) ≤ C ε,ρ + ρ I R {|z| 2 + |z| 4 }Π(dz),
where Υ j,k ε,ρ stands for the increment rate of f j with respect to y k (k = j), which, by monotonicity condition (A4)(1)(v) on f j , is non-negative and bounded by C j . Thus -λ(u j (t 0 , x 0 )w j (t 0 , y 0 ))

≤ k∈Aj Υ j,k ε,ρ (u k (t 0 , x 0 ) -w k (t 0 , y 0 )) + + C ε,ρ + ρ I R {|z| 2 + |z| 4 }Π(dz) ≤ C j k∈Aj (u k (t 0 , x 0 ) -w k (t 0 , y 0 )) + + C ε,ρ + ρ I R
{|z| 2 + |z| 4 }Π(dz).

Taking the limit superior in both hand-sides as ε → 0, once again u k (resp. w k ) is usc (resp. lsc) and j ∈ Ã, we get

-λ(u j ( t, x) -w j ( t, x)) ≤ C j k∈Aj (u k ( t, x) -w k ( t, x)) + + ρ I R {|z| 2 + |z| 4 }Π(dz),
finally take ρ → 0 to obtain,

-λ(u j ( t, x) -w j ( t, x)) ≤ C j k∈Aj (u k ( t, x) -w k ( t, x)) + ≤ (m -1)C j (u j ( t, x) -w j ( t, x)).
But this is contradictory since u j ( t, x)w j ( t, x) > 0 and -λ > (m -1)C j . Henceforth for any j ∈ A, u j ≤ w j .

We now consider the general case. Let (u j ) j∈A (resp. (w j ) j∈A ) be a subsolution (resp. supersolution) of (2.3.1). Denote ũj (t, x) = e λt u j (t, x) and wj (t, x) = e λt w j (t, x). Then it is easy to show that (ũ j ) j∈A (resp. ( wj ) j∈A ) is a subsolution (resp. supersolution) of the following system of variational inequalities which is similar to (2.2.20):

             min{ũ j (t, x) -max k∈Aj (ũ k (t, x
)e λt g jk (t, x)); -∂ t ũj (t, x) -Lũ j (t, x) + λũ j (t, x)e λt f j (t, x, (e -λt ũk ) m k=1 )} = 0 ; ũj (T, x) = e λT h j (x).

(2.3.32)

Next let us set F j (t, x, -→ y ) := -λy + e λt f j (t, x, (e -λt y) m i=1 ) with λ is chosen such that λ = m(1 + max k∈A C k ) where C k is the Lipschitz constant of f k w.r.t. to (y l ) l∈A .

Then the functions F k , k ∈ A, verify condition (2.3.17). It follows, from Step 1, that ∀j ∈ A, ũj ≤ wj and then u j ≤ w j . The proof is now complete.

As a by-product we have: Theorem 2.3.2. Under Assumptions [A4], [A5], and (2.2.14), (2.2.15) as well, the system of variational inequalities with inter-connected obstacles (2.3.1) has a unique continuous viscosity solution with polynomial growth.

In the case when f j , j ∈ A, do not depend on y, by the characterization (2.2.32)-(2.2.33) (see also Remark 5.1.1), we deduce that the functions (u j (t, x)) j∈A are nothing but (J j (t, x)) j∈A . Thus, as a by product of Theorem 2.3.2, we have: Corollary 2.3.1. The value functions (J j (t, x)) j∈A defined in (2.2.23) are continuous, belong to Π g and is the unique viscosity solution of the Hamilton-Jacobi-Bellman system associated with the stochastic optimal switching problem.

Second existence and uniqueness result

In this section we consider the issue of existence and uniqueness of a solution for the systems of IPDEs (2.3.1) when the functions (-f j ) j∈A verify [A4](I). This turns into assuming that (f j ) j∈A verify, instead of [A4](I)(v), the following:

[A4]( †): For any j ∈ A, for any k = j, the mapping

y k → f j (t, x, y 1 , • • • , y k-1 , y k , y k+1 , • • • , y m ) is nonincreasing whenever the other components (t, x, y 1 , • • • , y k-1 , y k+1 , • • • , y m ) are fixed.
The other assumptions on (-f j ) j∈A remain the same. Proof. : We first focus on the issue of existence.

For any j ∈ A and λ ∈ R let us define F j by:

F j (t, x, y 1 , • • • , y m ) = e λt f j (t, x, e -λt y 1 , • • • , e -λt y m ) -λy j .
Since f j is uniformly Lipschitz w.r.t. (y k ) k=1,m then F j is so and for λ large enough, F j satisfies:

For any k = 1, m, the mapping

y k → F j (t, x, y 1 , • • • , y k-1 , y k , y k+1 , • • • , y m ) is nonincreasing when- ever the other components (t, x, y 1 , • • • , y k-1 , y k+1 , • • • , y m ) are fixed.
Let us now consider the following iterative Picard sequence : ∀j ∈ A, Y j,0 = 0 and for n ≥ 1, define:

(Y 1,n , • • • , Y m,n ) = Θ((Y 1,n-1 , • • • , Y m,n-1 ))
where Θ is the mapping defined in (2.2.34)-(2.2.35) where f j is replaced with F j . By (2.2.40), the sequence (Y j,n ) j∈A converges in ([H 2 ] m , . 2,β ) to the unique solution (Y j ) j∈A of the system of RBSDEs associated with ((F j (s, X t,x s , y 1 , • • • , y m )) j∈A , (e λT h j (X t,x T )) j∈A , (e λt g jk (s, X t,x s )) j,k∈A ).

So using an induction argument on n and Theorem 2.3.2, there exist deterministic continuous functions with polynomial growth (u n j ) j∈A such that:

∀ n ≥ 0, j ∈ A, ∀(t, x) ∈ [0, t] × IR, ∀s ∈ [t, T ], Y j,n s = u n j (s, X t,x s ). (2.3.33)
By (2.2.41), take s = t we obtain ∀j, n, q, t ≤ T, x ∈ IR, |u n j (t, x)u q j (t,

x)| = E[|Y j,n t -Y j,q t | 2 ] ≤ C (Y j,n-1 ) j∈A -(Y j,q-1 ) j∈A 2 2,β .
Thus for any j ∈ A, (u n j ) n≥0 is of Cauchy type and converges pointwisely to a deterministic function u j . But (Y j ) j∈A = Θ((Y j ) j∈A ), then once more by (2.2.41), we also have: Next as Θ is a contraction then, by induction on n we have

∀s ∈ [0, T ], E[|Y j s -Y j,m s | 2 ] ≤ C (Y j ) j∈A -(Y j,m-
∀n, q ≥ 0, (Y j,n+q ) j∈A -Y j,n ) j∈A 2,β ≤ C n Θ 1 -C Θ (Y j,1 ) j∈A 2,β
where C Θ ∈]0, 1[ is the constant of contraction of Θ. Since the norms . and . 2,β are equivalent, then there exists a constan C 1 such that :

∀n, q ≥ 0, (Y j,n+q ) j∈A -Y j,n ) j∈A ≤ C 1 C n Θ (Y j,1
) j∈A .

Take now the limit as q goes to +∞ and in the view of (2.3.34) and (2.3.35), if we take s = t we deduce that :

∀(t, x) ∈ [0, T ] × IR, |u j (t, x) -u n j (t, x)| ≤ C 2 (Y j,1
) j∈A . But it is easy to check that (Y j,1 ) j∈A 2,β (t, x) is of polynomial growth (by (2.2.18) and since E[sup s≤T |X t,x s | γ ] is of polynomial growth for any γ ≥ 0) and since for any fixed n ≥ 0, u n j is so. Therefore for any j ∈ A, u j is of polynomial growth, i.e., belongs to Π g .

We will now show the continuity of u j . For any j ∈ A, let us set

Ȳ j,0 s = C(1 + |X t,x s | p ), s ≤ T,
where C and p are related to polynomial growth of (u j ) j∈A , i.e.,

∀j ∈ A, |u j (t, x)| ≤ C(1 + |x| p ), ∀(t, x) ∈ [0, T ] × IR.
Next for any n ≥ 1 and j ∈ A let us set

( Ȳ 1,n , • • • , Ȳ m,n ) = Θ(( Ȳ 1,n-1 , • • • , Ȳ m,n-1 )).
As Θ is a contraction then once more the sequence (( Ȳ j,n ) j∈A ) n≥0 converges in ([H 2 ] m , . 2,β ) to (Y j,t,x ) j∈A the unique solution of the system of RBSDEs associated with ((F j (s, X t,x s , y 1 , • • • , y m )) j∈A , (e λT h j (X t,x T )) j∈A , (e λt g jk (s, X t,x s )) j,k∈A ).

By the definition of Ȳ j,0 , we have

P -a.s., ∀j ∈ A, s ∈ [t, T ], Y j,t,x s ≤ Ȳ j,0 s and taking into account of [A4]( †) we obtain ∀j ∈ A, ∀s ∈ [t, T ], F j (s, X t,x s , Y 1,t,x s , • • • , Y m,t,x s ) ≥ F j (s, X t,x s , Ȳ 1,0 s , • • • , Ȳ m,0 s ).
Next by the comparison result of Remark 2.2.5 and since ( Ȳ j,1 ) j∈A = Θ(( Ȳ j,0 ) j∈A ) , (Y j,t,x ) j∈A = Θ((Y j,t,x ) j∈A ) we get

∀j ∈ A, s ∈ [t, T ], Ȳ j,1 s ≤ Y j,t,x s .
Now by an induction argument we obtain, for any n ≥ 0 and j ∈ A,

∀s ∈ [t, T ], Ȳ j,2n+1 s ≤ Y j,t,x s ≤ Ȳ j,2n s . ( 2

.3.36)

In the same way as previously there exist deterministic continuous functions ūn j with polynomial growth such that

∀(t, x) ∈ [0, T ] × IR, s ∈ [t, T ], Ȳ j,n s = ūn j (s, X t,x s ).
Moreover for any j ∈ A, the sequence (ū n j ) n converges pointwisely to u and by (2.3.36) we have

∀j ∈ A, ∀(t, x), u j (t, x) = lim n ր ū2n+1 j (t, x) = lim n ց ū2n j (t, x).
Therefore, u j , j ∈ A, is both lsc and usc and then continuous. Finally as (Y j,t,x ) j∈A = Θ((Y j,t,x ) j∈A ) and ∀j ∈ A, Y j,t,x s = u j (s, X t,x s ), s ∈ [t, T ], with u j a deterministic continuous function with polynomial growth, then (u j ) j∈A is a viscosity solution for the corresponding system of IPDEs, thus (e -λt u j ) j∈A is a viscosity solution the system of IPDEs (4.1) with polynomial growth.

Let us now deal with the issue of uniqueness. Let (ū j ) j∈A be another solution of (2.3.1) which belongs to Π g and ( Ȳ j ) j∈A ∈ [H 2 ] m such that for any j ∈ A, s ∈ [t, T ], Ȳ j,t,x s = ūj (s, X t,x s ).

Define ( Ỹ j,t,x ) j∈A as follow: ( Ỹ j,t,x ) j∈A = Θ(( Ȳ j,t,x ) j∈A )

Then there exist (ũ j ) j∈A deterministic continuous functions with polynomial growth (ũ j ) j∈A such that:

∀j ∈ A, s ∈ [t, T ], Ỹ j,t,x s = ũj (s, X t,x s ).
Moreover (ũ j ) j∈A is the unique viscosity solution of the following system of IPDEs :

∀j ∈ A          min{ũ j (t, x) -max k∈Aj (ũ k (t, x) -g jk (t, x)); -∂ t u j (t, x) -Lũ j (t, x) -f j (t, x, (ū k ) k∈A )} = 0 ; ũj (T, x) = h j (x).
(2.3.37)

As (ū j ) j∈A is also a solution of (2.3.37), then by uniqueness of Theorem 2.3.3 we obtain ũj = ūj , for any j ∈ A. Therefore ( Ȳ j,t,x ) j∈A = Θ(( Ȳ j,t,x ) j∈A ).

As (Y j ) j∈A is the unique fixed point of Θ in [H 2 ] m , we then have

∀j ∈ A, s ∈ [t, T ], Ȳ j,t,x s = Y j s .
It follows that ∀j ∈ A, ūj = u j . Finally (u j (t, x)) j∈A is the unique continuous with polynomial growth functions viscosity solution of the system of IPDEs (4.1).

Chapter 3

Viscosity solution of system of variational inequalities with interconnected bilateral obstacles and connections to multiple modes switching game of jump-diffusion processes

Preliminaries

Let (Ω, F, (F t ) t≥0 , P ) be a stochastic basis such that F 0 contains all P-null elements of F, and F t + ε>0 F t+ε = F t , t ≥ 0, and suppose that the filtration is generated by the following two mutually independent process: -a d-dimensional standard Brownian motion (W t ) t≥0 -a Poisson random measure N on R + × E, where E R l -{0} is equipped with its Borel field B E , with compensator ν(dtde) = dtn(de), such that n(E) < ∞, and { N ((0, t] × A) = (Nν)((0, t] × A)} 0≤t≤T is and F t -martingale for all A ∈ B E satisfying n(A) < ∞. n is assumed to be a σ-finite measure on (E, B E ) satisfying:

E (1 ∧ x 2 )n(dx) < ∞. (3.1.1)
Let T be a fixed positive constant and A 1 (resp. A 2 ) denote the set of switching modes for player 1 (resp. player 2). Let m 1 (resp. m 2 ) be the cardinal of the set A 1 (resp. A 2 ) and for (i, j)

∈ A 1 × A 2 , A 1 i := A 1 -{i} and A 2 j := A 2 -{j}. Next, for -→ y = (y kl ) (k,l)∈A 1 ×A 2 ∈ R m1×m2 .
For any y 1 ∈ R, denote by [ -→ y i,j , y 1 ] the matrix which is obtained from -→ y by replacing the element y ij with y 1 . A function Φ : (t, x) ∈ [0, T ] × R → Φ(t, x) ∈ R is called of polynomial growth if there exist two non-negative real constant C and γ such that

|Φ(t, x)| ≤ C(1 + |x| γ ).
Hereafter, this class of functions is denoted by Π g .

We now define the probabilistic tools and sets we need later. Let: In this paper, we investigate existence and uniqueness of viscosity solutions v(t, x) := (v ij (t, x)) (i,j)∈A 1 ×A 2 of the following system of variational inequalities with upper and lower interconnected obstacles:

(i) P be the σ-algebra of F t -predictable subsets of Ω × [0, T ]; (ii) L 2 := {ϕ is an IR-valued, F T -random variable such that ϕ 2 L 2 := E[|ϕ| 2 ] < ∞}; (iii) H 2 := {{ϕ t , 0 ≤ t ≤ T } is an IR-valued, F t -
∀(i, j) ∈ A 1 × A 2 ,            min{(v ij -L ij [ v])(t, x); max{(v ij -U ij [ v])(t, x); -∂ t v ij (t, x) -Lv ij (t, x) -g ij (t, x, (v kl (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x v ij (t, x), I B ij (t, x, v ij ))}} = 0 ; v ij (T, x) = h ij (x) (3.1.2) where for any (i, j) ∈ Γ 1 × Γ 2 and (t, x) ∈ [0, T ] × R and φ ∈ C 1,2 , (a) L ij [ v](t, x) := max k∈A 1 i {v kj (t, x) -g ik (t, x)} and U ij [ v](t, x) := min l∈A 2 j {v il (t, x) -g jl (t, x)}; (b) Lφ(t, x) := b(t, x)D x φ(t, x) + 1 2 σ 2 (t, x)D 2 xx φ(t, x) + E (φ(t, x + β(x, e)) -φ(t, x) -D x φ(t, x)β(x, e))n(de); (c) I B ij (t, x, φ) = E (φ(t, x + β(x, e)) -φ(t, x))γ ij (x, e)n(de). . Next for δ > 0, (t, x) ∈ [0, T ] × IR k , q ∈ R, φ a C 1,2 function and (i, j) ∈ Γ 1 × Γ 2 , let us set: (d) I 1 δ (t, x, φ) = |e|≤δ (φ(t, x + β(x, e)) -φ(t, x) -D x φ(t, x)β(x, e))n(de);
(e) I 2 δ (t, x, q, φ) = |e|≥δ (φ(t, x + β(x, e))φ(t, x)qβ(x, e))n(de);

(f ) I 1,B ij δ (t, x, φ) = |e|≤δ (φ(t, x + β(x, e)) -φ(t, x))γ ij (
x, e)n(de);

(g) I 2,B ij δ (t, x, φ) = |e|≥δ (φ(t, x + β(x, e))φ(t, x))γ ij (x, e)n(de);

(h) I(t, x, φ) = E (φ(t, x + β(x, e))φ(t, x) -D x φ(t, x)β(x, e))n(de) = I 1 δ (t, x, φ) + I 2 δ (t, x, D x φ, φ);

(i) I B ij (t, x, φ) = E (φ(t, x + β(x, e)) -φ(t, x))γ ij (x, e)n(de) = I 1,B ij δ (t, x, φ) + I 2,B ij δ (t, x, φ); (j) L φ u(t, x) := b(t, x)D x φ(t, x) + 1 2 σ 2 (t, x)D 2 xx φ(t, x) + I 1 δ (t, x, φ) + I 2 δ (t, x, D x φ, u).
The following assumptions will be in force throughout the rest of the paper.

(A0) The functions b(t, x) and σ(t, x): [0, T ] × R → R are jointly continuous in (t, x), of linear growth in (t, x) and Lipschitz continuous w.r.t. x, meaning that there exists a non-negative constant C such that for any (t, x, x ′ ) ∈ [0, T ] × R we have:

|b(t, x)| + |σ(t, x)| ≤ C(1 + |x|), |σ(t, x) -σ(t, x ′ )| + |b(t, x) -b(t, x ′ )| ≤ C|x -x ′ |.
The function β : R × E → R is measurable, continuous in x and such that for some real K and all e ∈ E, for any x, x ′ ∈ R,

|β(x, e)| ≤ K(1 ∧ |e|), |β(x, e) -β(x ′ , e)| ≤ K|x -x ′ |(1 ∧ |e|). (A1) For any (i, j) ∈ A 1 × A 2 , g ij (t, x, -→ y , z, q) : R × R × R m1×m2 × R d × R → R, (i)
is continuous in (t, x) uniformly w.r.t. the other variables ( -→ y , z, q) and for any (t, x) the mapping (t, x) → g i,j (t, x, 0, 0, 0) is of polynomial growth.

(ii) satisfies the standard hypothesis of Lipschitz continuity w.r.t. the variables ( -→ y , z, q), i.e. ∀(t, x) ∈

[0, T ] × R, ∀( -→ y 1 , -→ y 2 ) ∈ R m1×m2 × R m1×m2 , (z 1 , z 2 ) ∈ R × R, (q 1 , q 2 ) ∈ R × R, |g ij (t, x, -→ y 1 , z 1 , q 1 ) -g ij (t, x, -→ y 2 , z 2 , q 2 )| ≤ C(| -→ y 1 --→ y 2 | + |z 1 -z 2 | + |q 1 -q 2 |),
where, | -→ y | stands for the standard Euclidean norm of -→ y in R m1 × R m2 .

(iii) q → g ij (t, x, y, z, q) is non-decreasing, for all (t, x, y, z)

∈ [0, T ] × R × R m1×m1 × R. Futhermore, let γ ij : R × B E → R such that there exists C > 0, 0 ≤ γ ij (x, e) ≤ C(1 ∧ |e|), x ∈ R, e ∈ B E |γ ij (x, e) -γ ij (x ′ , e ′ )| < C|x -x ′ |(1 ∧ |e|), x, x ′ ∈ R, e ∈ E.
We set

f ij (t, x, y, z, u) = g ij (t, x, y, z, E u(e)γ ij (x, e)n(de), for (t, x, y, z, u) ∈ [0, t] × R × R m1×m2 × R × L 2 (R, B E , n).
(A2) Monotonicity: For any (i, j) ∈ A 1 × A 2 and any (k, l) = (i, j) the mapping y k,l → g i,j (t, x, -→ y , z, u) is non-decreasing.

(A3) The functions h ij (x) : R → R are continuous w.r.t. x, belong to class Π g and satisfy

∀(i, j) ∈ A 1 × A 2 and x ∈ R, max k∈A 1 i (h kj (x) -g ik (T, x)) ≤ h ij (x) ≤ min l∈A 2 j (h il (x) -g jl (T, x)),
where g ik and g jl are given in the next assumption.

(A4) The no free loop property: The switching costs g ik and ḡjl are non-negative, jointly continuous in (t, x), belong to Π g and satisfy the following condition: For any loop in A 1 ×A 2 , i.e., any sequence of pairs (i 1 , j 1 ), . . . , (i N , j N ) of A 1 ×A 2 such that (i N , j N ) = (i 1 , j 1 ), card{(i 1 , j 1 ), . . . , (i N , j N )} = N -1 and ∀ q = 1, . . . , N -1, either i q+1 = i q or j q+1 = j q , we have

∀(t, x) ∈ [0, T ] × IR k , q=1,N -1 ϕ iqiq+1 (t, x) = 0, (3.1.3)
where, ∀ q = 1, . . . , N -1, ϕ iqiq+1 (t, x) = -g iqiq+1 (t, x)1 1 iq =iq+1 + ḡjqiq+1 (t, x)1 1 jq =jq+1 .

To begin with let us point out that the non-local terms I(t, x, φ) and I B ij (t, x, φ) introduced previously are well defined under Assumptions (A0) and (A2) since for any function φ of class C 1,2 , by the mean value theorem, we have

|φ(t, x + β(x, e)) -φ(t, x) -∂ x φ(t, x)β(e, x)| ≤ C 1 t,x |β(x, e)| 2 ≤ C 1 t,x (1 ∧ |e|)
X t,x s = x + s t b(r, X t,x r )dr + s t σ(r, X t,x r )dW r + s t E β(X t,x r-, e) N (drde), s ∈ [t, T ], x ∈ R. (3.1.4)
The existence and uniqueness of the solution X t,x s follows from [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF].

Next, we give two definitions of the viscosity solution of (3.1.2), and according to [START_REF] Hamadène | Systems of integro-PDEs with interconnected obstacles and multimodes switching problem driven by Lévy process[END_REF](proposition 5.1 in appendix), they are equivalent. For locally bounded function u:

(t, x) ∈ [0, T ] × R → u(t, x) ∈ R,
we define its lower semi-continuous (lsc for short) envelope u * , and upper semi-continuous(usc for short) envelope u * as following:

u * (t, x) = lim (t ′ ,x ′ )→(t,x), t ′ <T u(t ′ , x ′ ), u * (t, x) = lim (t ′ ,x ′ )→(t,x), t ′ <T u(t ′ , x ′ ) Definition 3.1.1. A function -→ u = (u ij (t, x)) (i,j)∈A 1 ×A 2 : [0, T ] × R → R A 1 ×A 2 such that for any (i, j) ∈ A 1 × A 2 , u ij ∈ Π g is lsc (resp. usc)
, is said to be a viscosity subsolution (resp. supsolution) of (3.1.2) if for any test function

ϕ ∈ C 1,2 ([0, T ] × R) Π g , if (t 0 , x 0 ) ∈ [0, T ] × R is a global maximum (resp. minimum) point of u i,j -ϕ,        min{(u ij -L ij [ -→ u ])(t 0 , x 0 ), max{(u ij -U ij [ -→ u ])(t 0 , x 0 ), -∂ t φ(t 0 , x 0 ) -b(t 0 , x 0 )∂ x φ(t 0 , x 0 ) -1 2 σ 2 (t 0 , x 0 )∂ 2 xx φ(t 0 , x 0 ) -I 1 δ (t 0 , x 0 , φ) -I 2 δ (t 0 , x 0 , ∂ x φ, φ) -g ij (t 0 , x 0 , (u kl (t 0 , x 0 )) (k,l)∈A 1 ×A 2 , σ(t 0 , x 0 ))∂ x φ(t 0 , x 0 ), I 1,B ij δ (t 0 , x 0 , φ) + I 2,B ij δ (t 0 , x 0 , φ))}} ≤ 0 (resp. ≥ 0) v ij (T, x) ≤ h ij (x) (resp. ≥). Definition 3.1.2. A function -→ u = (u ij (t, x)) (i,j)∈A 1 ×A 2 : [0, T ] × R → R A 1 ×A 2 such that for any (i, j) ∈ A 1 × A 2 , u ij ∈ Π g is lsc (resp. usc)
, is said to be a viscosity subsolution (resp. supsolution) of (3.1.2) if for any (t 0 , x 0 ) ∈ (0, T ) × R, δ > 0 and a function φ ∈ C 1,2 ([0, T ] × R) Π g such that u ij (t 0 , x 0 ) = φ(t 0 , x 0 ) and u ij -φ has a global maximum (resp. minimum) at (t 0 , x 0 ) on [0, T ]×B(x 0 , Kδ) where K is the constant such that for any

x ∈ R, |β(x, e)| ≤ K(1 ∧ |e|), we have        min{(u ij -L ij [ -→ u ])(t 0 , x 0 ), max{(u ij -U ij [ -→ u ])(t 0 , x 0 ), -∂ t φ(t 0 , x 0 ) -b(t 0 , x 0 )∂ x φ(t 0 , x 0 ) -1 2 σ 2 (t 0 , x 0 )∂ 2 xx φ(t 0 , x 0 ) -I 1 δ (t 0 , x 0 , φ) -I 2 δ (t 0 , x 0 , ∂ x φ, u ij ) -g ij (t 0 , x 0 , (u kl (t 0 , x 0 )) (k,l)∈A 1 ×A 2 , σ(t 0 , x 0 ))∂ x φ(t 0 , x 0 ), I 1,B ij δ (t 0 , x 0 , φ) + I 2,B ij δ (t 0 , x 0 , u ij ))}} ≤ 0 (resp. ≥ 0) v ij (T, x) ≤ h ij (x) (resp. ≥). Definition 3.1.3. A function -→ u = (u ij (t, x)) (i,j)∈A 1 ×A 2 such that for any (i, j) ∈ A 1 × A 2 , u ij ∈ Π g , is called a viscosity solution of (3.1.2) if (u ij * (t, x)) (i,j)∈A 1 ×A 2 (resp.(u * ij (t, x)) (i,j)∈A 1 ×A 2 is a viscosity supersolution (resp. subsolution) of (3.1.2)

Approximation schemes of the solution of systems of reflected BSDEs

For n, m ≥ 0, let (Y i,j,n,m , Z i,j,n,m , U i,j,n,m ) (i,j)∈A 1 ×A 2 be the solution of the following system of BSDEs.

       (Y i,j,n,m , Z i,j,n,m , U i,j,n,m ) ∈ S 2 × H 2 × H 2 ( N ); dY i,j,n,m s = -f i,j,n,m (s, X t,x s , (Y k,l,n,m s ) (k,l)∈A 1 ×A 2 , Z i,j,n,m s , U i,j,n,m s )ds +Z i,j,n,m s dB s + E U i,j,n,m s (e) N (dsde) Y i,j,n,m T = h i,j (X t,x T ), (3.2.1)
where,

f i,j,n,m (s, X t,x s , (y ij ) (ij)∈A 1 ×A 2 , z s , u s ) :=g i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z s , E u s (e)λ ij (X t,x s , e)n(de)) =g i,j (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z s , E u s (e)λ ij (X t,x
s , e)n(de))

+ n(y ijmax

k∈A 1 i {y kj -g ik (s, X t,x s )}) - -m(y ij -min l∈A 2 j {y il -g jl (s, X t,x s )}) + .
Let's recall that under the assumption (A1), the solution (Y i,j,n,m , Z i,j,n,m , U i,j,n,m ) (i,j)∈A 1 ×A 2 of (3.2.1) exists and is unique(see [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]). By the assumption( A1)(3), we have the comparison theorem for BSDE with jumps(see [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF] Theorem 2.4). Next, let's show the monotonicity properties for the matrix (Y i,j,n,m ) n,m .

Proposition 3.2.1. For any (i, j) ∈ A 1 × A 2 and n, m ≥ 0 we have

P -a.s., Y i,j,n,m ≤ Y i,j,n+1,m and Y i,j,n,m+1 ≤ Y i,j,n,m , (i, j) ∈ A 1 × A 2 . (3.2.2)
Moreover, for any (i, j) ∈ A 1 × A 2 and n, m ≥ 0, there exists a deterministic continuous function v i,j,n,m ∈ Π g s.t., for any t ≤ T ,

Y i,j,n,m s = v i,j,n,m (s, X t,x s ), s ∈ [t, T ]. (3.2.3)
Finally, for any (i, j) ∈ A 1 × A 2 and n, m ≥ 0, v i,j,n,m (t, x) ≤ v i,j,n+1,m (t, x) and v i,j,n,m+1 (t, x) ≤ v i,j,n,m (t, x),

(t, x) ∈ [0, T ] × R (3.2.4)
Proof. First, we recall the result by Xuehong Zhu (2010)([62] Theorem 3.1) related to the comparison of solutions of multi-dimensional BSDEs. Actually, it is enough to show the for any t, (y ij

) (i,j)∈A 1 ×A 2 , (y ij ) (i,j)∈A 1 ×A 2 ∈ R m1×m2 , (z ij ) (i,j)∈A 1 ×A 2 , (z ij ) (i,j)∈A 1 ×A 2 ∈ R m1×m2 and (u ij ) (i,j)∈A 1 ×A 2 , (u ij ) (i,j)∈A 1 ×A 2 ∈ L 2 (R, B E , n) m1×m2 , there exists a constant C, - 4
(i,j)∈A 1 ×A 2 y - ij (f i,j,n+1,m (s, X t,x s , (y + kl + y kl ) (kl)∈A 1 ×A 2 , z ij , u ij ) -(f i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij )) ≤2 (i,j)∈A 1 ×A 2 {y ij <0} |z ij -z ij | 2 + C (i,j)∈A 1 ×A 2 (y - ij ) 2 + 2 (i,j)∈A 1 ×A 2 E {y ij ≥0} |(y ij + u ij (e) -u ij (e)) -| 2 n(de) + 2 (i,j)∈A 1 ×A 2 E {y ij <0} [|(y ij + u ij (e) -u ij (e)) -| 2 -|y - ij | 2 -2y ij (u ij (e) -u ij (e))]n(de)
This inequality follows from the fact that, for any (i, j)

∈ A 1 × A 2 , (i)f i,j,n,m (s, X t,x s , (y kl ) ∈A 1 ×A 2 , z ij , u ij ) ≤ f i,j,n+1,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij ) (ii)For any (u kl ) (kl)∈A 1 ×A 2 ≥ 0, f i,j,n,m (s, X t,x s , (y kl + u kl ) ∈A 1 ×A 2 , z ij , u ij ) ≤ f i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij ).
(iii)f ij depends only on z ij , u ij and not on the other components z kl , u kl , (kl) = (ij).

(iv)∀(i, j) ∈ A 1 × A 2 , -4y - ij (f i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij ) -f i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij )) ≤C(y - ij ) 2 + 2 E {yij ≥0} |(y ij + u ij (e) -u ij (e)) -| 2 n(de) + 2 E {yij <0} [|(y ij + u ij (e) -u ij (e)) -| 2 -|y - ij | 2 -2y ij (u ij (e) -u ij (e))
]n(de) (i),(ii) and (iii) are easy to check, now we proof (iv), in the case that y ij ≥ 0, (iv) is obvious, so we discuss only when y ij < 0. The right-hand side of the inequality is non-negative, in fact,

E [|(y ij + u ij (e) -u ij (e)) -| 2 -|y - ij | 2 -2y ij (u ij (e) -u ij (e))]n(de) = uij (e)-uij (e)<-yij (u ij (e) -u ij (e)) 2 n(de) + uij (e)-uij (e)≥-yij [-|y - ij | 2 -2y ij (u ij (e) -u ij (e))
]n(de)

≥0

So we consider only the case that

-4y - ij (f i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij ) -f i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij )) = -4y ij ((f i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij ) -f i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij )) >0 By the assumption (A1)(iii), E u ij (e)γ ij (X t,x
s , e)n(de) > E u ij (e)γ ij (X t,x s , e)n(de), otherwise the inequality above can not hold. Next, by the assumption (A1)(ii), there exists a constant C > 0 such that:

-4y ij ((f i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij ) -f i,j,n,m (s, X t,x s , (y kl ) (kl)∈A 1 ×A 2 , z ij , u ij )) ≤ -4y ij C E γ ij (X t,x s , e)(u ij (e) -u ij (e))n(de) = -4y ij C uij (e)-uij (e)<-yij γ ij (X t,x s , e)(u ij (e) -u ij (e))n(de) -4y ij C uij (e)-uij (e)≥-yij γ ij (X t,x s , e)(u ij (e) -u ij (e))n(de) ≤4C 2 y 2 ij E γ 2 ij (X t,x s , e)n(de) + uij (e)-uij (e)<-y ij (u ij (e) -u ij (e)) 2 n(de) ≤K(y - ij ) 2 + 2 E {y ij ≥0} |(y ij + u ij (e) -u ij (e)) -| 2 n(de) + 2 E {y ij <0} [|(y ij + u ij (e) -u ij (e)) -| 2 -|y - ij | 2 -2y ij (u ij (e) -u ij (e))
]n(de), finally, we have (iv). Consequently, we have

P -a.s., Y i,j,n,m ≤ Y i,j,n+1,m .
In the same way we can show that Pa.s., Y i,j,n,m+1 ≤ Y i,j,n,m . The second claim is just the representation of solutions of standard BSDEs with jumps by deterministic functions in the Markovian framework(see [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]). The inequalities of (3.2.4) are obtained by taking s = t in (3.2.2) in view of the representation (3.2.3) of Y i,j,n,m by v i,j,n,m and X t,x .

We will show two approximation schemes obtained from the sequence Y i,j,m,n , (i, j) ∈ A 1 × A 2 ) n,m of the solution of system (3.2.1). The first scheme is a sequence of decreasing reflected BSDEs with interconnected lower obstacles:

∀(i, j) ∈ A 1 × A 2 ,                  ( Ȳ i,j,m , Zi,j,m , Ū i,j,m , Ki,j,m ) ∈ S 2 × H 2 × H 2 ( N ) × A 2 ; Ȳ i,j,m s = h i,j (X t,x T ) + T s f i,j,m (r, X t,x r , ( Ȳ k,l,m r ) (k,l)∈A 1 ×A 2 , Zi,j,m r , Ū i,j,m r )dr - T s Zi,j,m r dB r - T s E Ū i,j,m r (e) N (drde) + Ki,j,m T -Ki,j,m s Ȳ i,j,m s ≥ max k∈A 1 i { Ȳ k,j,m s -g ik (s, X t,x s )}, s ≤ T, T 0 ( Ȳ i,j,m s -max k∈A 1 i { Ȳ k,j,m s -g ik (s, X t,x s )})d Ki,j,m s = 0, (3.2.5) where, ∀(i, j) ∈ A 1 × A 2 , m ≥ 0 and (s, -→ Y ij , Z ij , U ij ), f i,j,m (s, X t,x s , -→ Y , Z ij , U ij ) :=g ij,+,m (s, X t,x s , (Y kl ) (k,l)∈A 1 ×A 2 , Z ij , E U ij (e)γ ij (X t,x s , e)n(de)) =g ij (s, X t,x s , (Y kl ) (k,l)∈A 1 ×A 2 , Z ij , E U ij (e)γ ij (X t,x s , e)n(de)) -m(Y ij -min l∈A 2 j (Y il + g jl (s, X t,x s ))) + .
Thanks to the assumption (A1)-(A3) and non free loop assumption, by Theorem (5.2) in appendix, the solution of (3.2.5) exists and is unique. Moreover, we have the following properties.

Proposition 3.2.2. For any (i, j) ∈ A 1 × A 2 and m ≥ 0, we have:

(i) lim n→∞ E[ sup t≤s≤T |Y i,j,n,m s -Ȳ i,j,m s | 2 ] → 0 (3.2.6)
(ii) Pa.s., Ȳ i,j,m ≥ Ȳ i,j,m+1 .

(iii)There exsits a unique

A 1 × A 2 -uplet of deterministic continuous functions (ū k,l,m ) (k,l)∈A 1 ×A 2 in Π g such that, for every t ≤ T , Ȳ i,j,m s = ūi,j,m (s, X t,x s ), s ∈ [t, T ]. (3.2.7) Moreover, ∀(i, j) ∈ A 1 × A 2 and (t, x) ∈ [0, T ] × R k , ūi,j,m (t, x) ≥ ūi,j,m+1 (t, x).
Finally, (ū i,j,m ) (i,j)∈A 1 ×A 2 is a viscosity solution in the class Π g of the following system of variational inequalities with inter-connected obstacles.

∀(i, j) ∈ A 1 × A 2 ,      min{ū i,j,m (t, x) -max k∈A 1 i (ū k,j,m (t, x) -g ik (t, x)); -∂ t ūi,j,m (t, x) -Lū i,j,m (t, x) -g ij,+,m (t, x, (ū k,l,m (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x ūi,j,m (t, x), B ij ūi,j,m (t, x))} = 0 ūi,j,m (T, x) = h i,j (x).
(3.2.8)

Proof. (i)It is enough to consider the case m = 0, since for any i, j ∈ A 1 × A 2 , the function

(s, x, (y kl ) (kl)∈A 1 ×A 2 ) → -m(y ij -min l∈A 2 j (y il + g jl (s, x))) +
has the same properties as f ij displayed in (A1) and (A2). First, let us show that ∀(i, j) ∈ A 1 × A 2 and n ≥ 0, Pa.s., Y i,j,n,0 ≤ Ȳ i,j,0 .

(3.2.9)

First and w.l.o.g we may assume that f ij is non-decreasing w.r.t. (y kl ) (k,l)∈A1×A2 , since thanks to assumption (A2), it is enough to multiply the solution by e λt , where λ is appropriately chosen in order to fall in this latter case, since f ij is Lipschitz in y ij . Now, for fixed n, let us define recursively the sequence ( Ỹ k,ij,n ) k≥0 as follows: for k = 0 and any (i, j) ∈ A 1 × A 2 , we set Ỹ 0,ij,n := Ȳ ij,0 and, for any

k ≥ 1, let us define ( Ỹ k,ij,n , Z k,ij,n , U k,ij,n ) ∈ S 2 × H 2 × H 2 ( N )
as the solution of the following system of BSDEs:

∀(i, j) ∈ A 1 × A 2 ,      -d Ỹ k,ij,n s = f ij (s, X t,x s , ( Ỹ k-1,pq,n s ) (p,q)∈A 1 ×A 2 , Zk,ij,n s , Ũ k,ij,n s )ds +n( Ỹ k,ij,n s -max l∈A 1 i ( Ỹ k-1,lj,n s -g il (s, X t,x s ))) -ds -Zk,ij,n s dB s -E Ũ k,ij,n s (e) N (dsde) Ỹ k,ij,n T = h i,j (X t,x T ), (3.2.10)
The solution of (3.2.10) exists since it is a multi-dimensional standard BSDE with a Lipschiz coefficient, nothing that ( Ỹ k-1,pq,n s ) (p,q)∈A 1 ×A 2 is already given. Since, n is fixed and the coefficient

φ ij,n (s, ω, (y pq ) (p,q)∈A 1 ×A 2 , z ij , U ij ) := f ij (s, X t,x s , (y pq ) (p,q)∈A 1 ×A 2 , z ij , U ij )+n(y ij -max l∈A 1 i (y lj -g il (s, X t,x s ))) -
is Lipschitz w.r.t. ((y pq ) (p,q)∈A 1 ×A 2 , z ij , U ij ), the sequence ( Ỹ k,ij,n ) k≥0 converges in S 2 to Y ij,n,0 as k → ∞, for any i, j and n.

Using an induction argument w.r.t.k, we prove that for any i, j and n,

P -a.s., Ỹ k,ij,n ≤ Ȳ ij,0 , k ≥ 0.
Indeed, for k = 0 the inequalities hold true through the definition of Ỹ 0,ij,n . Assume not that these inequalities are valid for some k -1, i.e. for any i, j and n,

P -a.s., Ỹ k-1,ij,n ≤ Ȳ ij,0 . (3.2.11)
Thus, taking into account that Ȳ ij,0 satisfies (3.2.5) with m = 0 and the fact that f ij is non-decreasing w.r.t. (y pq ) (p,q)∈A 1 ×A 2 then for any i, j and n, it holds

f ij (s, X t,x s , ( Ỹ k-1,pq,n s ) (p,q)∈A 1 ×A 2 , z ij , U ij )ds + n( Ȳ ij,0 s -max l∈A 1 i ( Ỹ k-1,lj,n s -g il (s, X t,x s ))) - ≤ f ij (s, X t,x s , ( Ȳ pq,0 s ) (p,q)∈A 1 ×A 2 , z ij , U ij ).
Using comparison result of solution of one dimensional BSDEs we obtain that for any i, j and n, Ỹ k,ij,n ≤ Ȳ ij,0 , a.s.. Thus the property (3.2.11) is valid for any k. Taking the limit as k tends to ∞, we obtain (3.2.9).

(3.2.9) together with (3.2.2) imply that ∃ Ŷ ij ∈ S 2 s.t. Y i,j,n,0 ր Ŷ ij . Now by Essaky's monotonic limit theorem in [START_REF] Essaky | Reflected backward stochastic differential equation with jumps and RCLL obstacle[END_REF], there exist ( Ẑi,j , Û i,j , Ki,j ) ∈ H 2 × H 2 ( N ) × A 2 : (a) Ŷ ij is RCLL and uniformly P-square integrable, for any stopping time τ , lim

n→∞ ր Y i,j,n,0 τ = Ŷ ij τ .
(b) Kij is RCLL non-decreasing, Kij 0 = 0 and for any stopping time τ , lim

n→∞ K i,j,n,0 τ = Kij τ , P -a.s.. (c) Ẑij ∈ H 2 , Û ij ∈ H 2 ( N ) and for any p ∈ [1, 2), lim n→∞ E[ T 0 |Z i,j,n,0 s -Ẑij | p ds] = 0, lim n→∞ E[ T 0 E |U i,j,n,0 s -Û ij | p 2 n(de)ds] = 0. (d)For any i, j ∈ A 1 × A 2 , s ≤ T ,        Ŷ ij s = h ij (X t,x T ) + T s f ij (r, X t,x r , ( Ŷ kl r ) (k,l)∈A 1 ×A 2 , Ẑij r , Û ij r ) + Kij T -Kij s - T s Ẑij r dB r - T s E Û ij r (e)n(deds) Ŷ ij s ≥ max k∈A 1 i { Ŷ kj s -g ik (s, X t,x s )}.
(3.2.12)

The remaining of the proof is in the same way with Theorem (5.2) in appendix, that is Ŷ ij is regular(see [START_REF] Hamadène | Systems of integro-PDEs with interconnected obstacles and multimodes switching problem driven by Lévy process[END_REF] page 14 for the definition of regular) and Kij is continuous and the Skorohod condition is satisfied by using the no-free loop property (A4). So ( Ŷ ij , Ẑij , Û ij , Kij ) satisfy (3.2.5) with m = 0, by the uniqueness of the solution of (3.2.5), Ŷ ij = Ȳ ij,0 , a.s., which completes the proof of (i).

(ii)is an immediate consequence of (i) and Proposition 3.1.

(iii)One can see Theorem 5.3 in Appendix for the proof of existence of ū in (3.2.8). Also it is similar as theorem 4.1 in [START_REF] Hamadène | Systems of integro-PDEs with interconnected obstacles and multimodes switching problem driven by Lévy process[END_REF]. Actually, there is nonlocal-term I B ij in the generator g ij which is different from the generator in [START_REF] Hamadène | Systems of integro-PDEs with interconnected obstacles and multimodes switching problem driven by Lévy process[END_REF], but since I B ij has the same proposition with the one in [START_REF] Hamadène | Systems of integro-PDEs with interconnected obstacles and multimodes switching problem driven by Lévy process[END_REF] and g ij is non-decreasing and Lipschitz continuous in the nonlocal-term, uniformly with respect to all the other variables, we can use the same method to show the existence and uniqueness. Finally as ∀s ∈

[t, T ], Ȳ i,j,m s ≥ Ȳ i,j,m+1 s a.s., let s = t we get ∀(t, x) ∈ [0, T ] × R, ūi,j,m (t, x) ≥ ūi,j,m+1 (t, x).
We now consider the increasing approximating scheme:

∀(i, j) ∈ A 1 × A 2 ,                    (Y i,j,n , Z i,j,n , U i,j,n , K i,j,n ) ∈ S 2 × H 2 × H 2 ( N ) × A 2 ; Y i,j,n s = h i,j (X t,x T ) + T s f i,j,n (r, X t,x r , (Y k,l,n r ) (k,l)∈A 1 ×A 2 , Z i,j,n r , U i,j,n r )dr - T s Z i,j,n r dB r - T s E U i,j,n r (e) N (drde) + K i,j,n T -K i,j,n s Y i,j,n s ≤ min l∈A 2 j {Y i,l,n s + g jl (s, X t,x s )}, s ≤ T, T 0 (Y i,j,n s -min l∈A 2 j {Y k,j,n s + g jl (s, X t,x s )})dK i,j,n s = 0, (3.2.13)
where,

∀(i, j) ∈ A 1 × A 2 , n ≥ 0 and (s, -→ Y ij , Z ij , U ij ), f i,j,n (s, X t,x s , -→ Y , Z ij , U ij ) :=g ij,-,n (s, X t,x s , (Y kl ) (k,l)∈A 1 ×A 2 , Z ij , E U ij (e)γ ij (X t,x s , e)n(de)) =g ij (s, X t,x s , (Y kl ) (k,l)∈A 1 ×A 2 , Z ij , E U ij (e)γ ij (X t,x s , e)n(de)) + n(Y ij -max k∈A 1 i (Y kj -g ik (s, X t,x s ))) -.
Thanks to the assumption (A1)-(A3), by Theorem 5.4.1 in appendix, the solution of (3.2.13) exists and is unique.

Next we give a analogous of Proposition 3.2, we do not give its proof since it is deduced from this latter proposition in considering the equation satisfied by (-Y i,j,n , -Z i,j,n , -U i,j,n , -K i,j,n ) Proposition 3.2.3. For any (i, j) ∈ A 1 × A 2 and n ≥ 0, we have:

(i) lim m→∞ E[ sup t≤s≤T |Y i,j,n,m s -Y i,j,n s | 2 ] → 0 (3.2.14)
(ii) For any n ≥ 0, Pa.s., Y i,j,n ≤ Y i,j,n+1 .

Theorem 3.3.1. Let ( -→ u ) (i,j) ∈ A 1 × A 2 (resp. ( -→ w ) (i,j) ∈ A 1 × A 2
) is an usc subsolution (resp. lsc supersolution) of (3.1.2) which belongs to Π g . Then it holds that for any (i, j)

∈ A 1 × A 2 , u ij (t, x) ≤ w ij (t, x), (t, x) ∈ [0, T ] × R.
Proof. We peoceed by contradiction, let (t 0 , x 0 ) ∈ [0, T ] × R such that there exists ǫ > 0, max i,j

(u ij -w ij )(t 0 , x 0 ) ≥ 0. (3.3.2)
Next, w.l.o.g. assume that for any (i, j)

∈ A 1 × A 2 , lim x→∞ (u ij -w ij )(t, x) = -∞, (3.3.3)
otherwise one may replace w ij with w i,j,θ,ν defined by

w i,j,θ,ν (t, x) = w ij (t, x) + θe -νt |x| 2γ+2 (t, x), (t, x) ∈ [0, T ] × R,
which is still an usc supersolution of (3.1.2) for θ > 0 and ν ≥ ν 0 which satisdies (3.3.3). It suffices to show that u ij (t, x) ≤ w i,j,θ,ν (t, x), (t, x) ∈ [0, T ] × R, since by taking the limit as θ → 0, we deduce that

u ij (t, x) ≤ w ij (t, x), (t, x) ∈ [0, T ] × R.
max ij {(u ij -w ij )(t, x)} = max (t,x)∈[0,T ]×B(0,r) max ij {(u ij -w ij )(t, x)} = max ij {(u ij -w ij )(t * , x * )}, ( 3
.3.4) where, (t * , x * ) ∈ [0, T ] × B(0, r), where B(0, r) denotes the ball in R with center the origin and radius r, since by definition u ij (T, x) ≤ w ij (T, x), for all (i, j) ∈ A 1 × A 2 . Next we will finish the proof in two steps: Step 1. First we make the following assumption on the functions (g ij

) (i,j)∈A 1 ×A 2 . For all λ > c ij (m 1 × m 2 -1), (i, j) ∈ A 1 × A 2 , (t, x, -→ y , z, U ) ∈ [0, T ] × R × R m1×m2 × R × R, and (u, v) ∈ R 2 s.t.u ≥ v, g ij (t, x, [ -→ y ij , u], z, U ) -g ij (t, x, [ -→ y ij , v], z, U ) ≤ -λ(u -v), (3.3.5)
where c ij is the Lipschitz constant of g ij w.r.t. -→ y . Next, let (i 0 , j 0 ) ∈ Γ(t * , x * ) that satisfies (3.3.1). For ǫ > 0, ρ > 0, let Φ i0,j0 ǫ,ρ be the function defined as follows:

Φ i0,j0 ǫ,ρ (t, x, y) := (u i0,j0 (t, x)w i0,j0 (t, y))

- |x -y| ǫ -|t -t * | 2 -ρ|x -x * | 4 .
Let (t 0 , x 0 , y 0 ) be such that Φ i0,j0 ǫ,ρ (t 0 , x 0 , y 0 ) = max

(t,x,y)∈[0,T ]× B(0,r) 2 Φ i0,j0 ǫ,ρ (t, x, y) = max (t,x,y)∈[0,T ]×I R 2 Φ i0,j0
ǫ,ρ (t, x, y) which exists since which exists since Φ i0,j0 ǫ,ρ is usc and B(0, r) 2 is the closure of B(0, r) 2 . Then we have

Φ i0,j0 ǫ,ρ (t * , x * , x * ) = u i0,j0 (t * , x * ) -w i0,j0 (t * , x * ) ≤ u i0,j0 (t * , x * ) -w i0,j0 (t * , x * ) + |x 0 -y 0 | 2 ǫ + |t 0 -t * | 2 + ρ|x 0 -x * | 4 ≤ u i0,j0 (t 0 , x 0 ) -w i0,j0 (t 0 , y 0 ) (3.3.6)
The growth condition of u i0,j0 and w i0,j0 implies that |x0-y0|

2 ǫ + |t 0 -t * | 2 + ρ|x 0 -x * | 4
is bounded and hence lim ǫ→0 (x 0y 0 ) = 0. Next by (3.3.6), for any subsequence (t 0 l , x 0 l , y 0 l ) l which converges to ( t, x, x), u i0,j0 (t * , x * )w i0,j0 (t * , x * ) ≤ u i0,j0 ( t, x)w i0,j0 ( t, x), since u i0,j0 is usc and w i0,j0 is lsc. By the definition of (t * , x * ) this last inequality is an equality. Using both the definiton of Φ i0,j0 Finally classically (see e.g. [START_REF] Djehiche | Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles[END_REF], pp. 14) we have also lim ǫ→0 (u j (t 0 , x 0 ), w j (t 0 , y 0 )) = (u j (t * , x * ), w j (t * , x * )).

(3.3.9)

Next recalling that u i0,j0 (resp. w i0,j0 ) is usc (resp. lsc) and satisfies (3.3.1), then for ǫ small enough and there exists a subsequence which we still index by ǫ, we obtain u i0,j0 (t 0 , x 0 ) > max (w i0,l (t 0 , y 0 )g j0l (t 0 , y 0 )). (3.3.11) We are able to apply Jensen-Ishii's Lemma for non local operators established by Barles and Imbert ([5], pp.583) (one can see also [START_REF] Biswas | Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion processes[END_REF], Lemma 4.1, pp.64) with u i0,j0 , w i0,j0 and φ(t, x, y)

k∈A 1 i 0 (u k,j0 (t 0 , x 0 ) -g i0k (t 0 , x 0 )), ( 3
:= |x-y| 2 ǫ + |t - t * | 2 + ρ|x -x * | 4 at the point (t 0 , x 0 , y 0 ).
For any δ ∈ (0, 1) there are p ǫ u , q ǫ u , p ǫ w and q ǫ w elements of IR and M ǫ u , M ǫ w two non-negative constants such that:

(i) p ǫ u -p ǫ w = ∂ t φ(t 0 , x 0 , y 0 ), q ǫ u = ∂ x φ(t 0 , x 0 , y 0 ), q ǫ w = -∂ y φ(t 0 , x 0 , y 0 ) (3.3.12) and M 0 u 0 0 -M 0 w ≤ D 2 xx φ(t 0 , x 0 , y 0 ) = 12|x 0 -x * | 2 0 0 0 + 2 ǫ 1 -1 -1 1 ; (3.3.13) (ii) -p ǫ u -b(t 0 , x 0 )q ǫ u - 1 2 σ(t 0 , x 0 ) 2 M ǫ u -g i0,j0 (t 0 , x 0 , (u ij (t 0 , x 0 )) (i,j)∈A 1 ×A 2 , σ(t 0 , x 0 )q ǫ u , I 1,B i 0 ,j 0 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) + I 2,B i 0 ,j 0 δ (t 0 , x 0 , u i0,j0 ))
-I 1 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) -I 2 δ (t 0 , x 0 , q ǫ u , u i0,j0 ) ≤ 0 ;

(3.3.14) (iii)p ǫ wb(t 0 , y 0 )q ǫ w -1 2 σ(t 0 , y 0 ) 2 M ǫ wg i0,j0 (t 0 , y 0 , (w ij (t 0 , y 0 )) (i,j)∈A 1 ×A 2 , σ(t 0 , y 0 )q ǫ w , I 1,B i 0 ,j 0 δ (t 0 , x 0 , -φ(t 0 , x 0 , .)) + I 2,B i 0 ,j 0 δ (t 0 , x 0 , w i0,j0 ))

-I 1 δ (t 0 , y 0 , -φ(t 0 , x 0 , .)) -I 2 δ (t 0 , y 0 , q ǫ w , w i0,j0 ) ≥ 0.

(3.3.15)

Next we are going to provide estimates for the non-local terms. Define ψ ρ (t, x)

:= ρ|x -x * | 4 + |t -t * | 2 .
By definition of (t 0 , x 0 , y 0 ), for any d, d ′ ∈ IR,

u i0,j0 (t 0 , x 0 + d ′ ) -w i0,j0 (t 0 , y 0 + d) -|x0+d ′ -y0-d| 2 ǫ -ψ ρ (t 0 , x 0 + d ′ ) ≤ u i0,j0 (t 0 , x 0 ) -w i0,j0 (t 0 , y 0 ) -|x0-y0| 2 ǫ -ψ ρ (t 0 , x 0 ).
Therefore for z ∈ E, in taking d = β(x 0 , z) and d ′ = β(y 0 , z), we obtain

u i0,j0 (t 0 , x 0 + β(x 0 , z)) -u i0,j0 (t 0 , x 0 ) -q ǫ u β(x 0 , z) ≤ w i0,j0 (t 0 , y 0 + β(y 0 , z)) -w i0,j0 (t 0 , y 0 ) -q ǫ w β(y 0 , z) + |β(x0,z)-β(y0,z)| 2 ǫ +ψ ρ (t 0 , x 0 + β(x 0 , z)) -ψ ρ (t 0 , x 0 ) -D x ψ ρ (t 0 , x 0 )β(x 0 , z).
By assumption (A0), for any δ > 0,

I 2 δ (t 0 , x 0 , q ǫ u , u i0,j0 ) -I 2 δ (t 0 , y 0 , q ǫ w , w i0,j0 ) ≤ K|x0-y0| 2 ǫ |e|≥δ (1 ∧ |e|) 2 n(de) + I 2 δ (t 0 , x 0 , D x ψ ρ (t 0 , x 0 ), ψ ρ ) ≤ K |x0-y0| 2 ǫ + I 2 δ (t 0 , x 0 , D x ψ ρ (t 0 , x 0 ), ψ ρ ). (3.3.16) It is easy to check that |I 2 δ (t 0 , x 0 , D x ψ ρ (t 0 , x 0 ), ψ ρ )| ≤ Cρ |z|≥δ {|z| 2 + |z| 4 }n(dz).
On the other hand, since φ ∈ C 2 and β is bounded, I 1 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) = |e|≤δ {φ(t 0 , x 0 + β(x 0 , e), y 0 )φ(t 0 , x 0 , y 0 ) -D x φ(t 0 , x 0 , y 0 )β(x 0 , e)}n(de)

= C 1 ǫ,ρ |e|≤δ β(x0,e) 2 ǫ dn(e) ≤ K 1 ǫ,ρ ǫ |e|≤δ (1 ∧ |e|)
2 n(de) and I 1 δ (t 0 , y 0 , -φ(t 0 , x 0 , .)) = |e|≤δ {-φ(t 0 , x 0 , y 0 + β(y 0 , e)) + φ(t 0 , x 0 , y 0 ) + D y φ(t 0 , x 0 , y 0 )β(y 0 , e)}n(de)

= -C 2 ǫ,ρ |e|≤δ β(y0,e) 2 ǫ
dn(e) ≥ -

K 2 ǫ,ρ ǫ |e|≤δ (1 ∧ |e|) 2 n(de). Therefore we have -I 1 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) + I 1 δ (t 0 , y 0 , -φ(t 0 , x 0 , .)) ≥ - Kǫ,ρ ǫ |e|≤δ (1 ∧ |e|) 2 n(de). (3.3.17)
Now by the definition of (t * , x * ) and (3.3.9), for ǫ small enough,

d, d ′ ∈ IR, u i0,j0 (t 0 , x 0 + d ′ ) -w i0,j0 (t 0 , y 0 + d) ≤ u i0,j0 (t 0 , x 0 ) -w i0,j0 (t 0 , y 0 ).
Therefore for z ∈ E, in taking d = β(x 0 , z) and d ′ = β(y 0 , z), we obtain

u i0,j0 (t 0 , x 0 + β(x 0 , z)) -u i0,j0 (t 0 , x 0 ) ≤ w i0,j0 (t 0 , y 0 + β(y 0 , z)) -w i0,j0 (t 0 , y 0 ) + |β(x0,z)-β(y0,z)| 2 -2(x0-y0)(β(x0,z)-β(y0,z)) ǫ +ψ ρ (t 0 , x 0 + β(x 0 , z)) -ψ ρ (t 0 , x 0 ).
Since γ i0,j0 is nonnegative, and by the assumption on β,for any δ > 0,

I 2,B i 0 ,j 0 δ (t 0 , x 0 , u i0,j0 ) -I 2,B i 0 ,j 0 δ (t 0 , y 0 , w i0,j0 ) ≤ I 2,B i 0 ,j 0 δ (t 0 , x 0 , φ ρ ) + O( |x 0 -y 0 | 2 ǫ ), (3.3.18)
and it is easy to check that

|I 2,B i 0 ,j 0 δ (t 0 , x 0 , ψ ρ )| ≤ Cρ |z|≥δ {|z| 2 + |z| 4 }n(dz).
On the other hand, since φ ∈ C 2 , I 1,B i 0 ,j 0 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) = |e|≤δ {φ(t 0 , x 0 + β(x 0 , e), y 0 )φ(t 0 , x 0 , y 0 )}γ i0,j0 (e, x 0 )n(de)

≤ C 1,B ǫ,ρ |e|≤δ |β(x 0 , e)|γ i0,j0 (e, x 0 )dn(e) and I 1,B i 0 ,j 0 δ (t 0 , y 0 , -φ(t 0 , x 0 , .)) = |e|≤δ {-φ(t 0 , x 0 , y 0 + β(y 0 , e)) + φ(t 0 , x 0 , y 0 )}γ i0,j0 (e, y 0 )n(de) ≥ -C 2,B ǫ,ρ |e|≤δ |β(y 0 , e)|γ i0,j0 (e, y 0 )dn(e).

By the assumption of β and for ǫ small enough,

I 1,B i 0 ,j 0 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) -I 1,B i 0 ,j 0 δ (t 0 , y 0 , -φ(t 0 , x 0 , .)) ≤ K B ǫ,ρ |e|≤δ (1 ∧ |e|) 2 n(de). (3.3.19)
Making now the difference between (3.3.14) and (3.3.15) yields

-(p ǫ u -p ǫ w ) -[b(t 0 , x 0 )q ǫ u -b(t 0 , y 0 )q ǫ w ] - 1 2 [σ(t 0 , x 0 ) 2 M ǫ u -σ(t 0 , y 0 ) 2 M ǫ w ] -[g i0,j0 (t 0 , x 0 , (u ij (t 0 , x 0 )) (i,j)∈A 1 ×A 2 , σ(t 0 , x 0 )q ǫ u , I 1,B i 0 ,j 0 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) + I 2,B i 0 ,j 0 δ (t 0 , x 0 , u i0,j0 ))
g i0,j0 (t 0 , y 0 , (w ij (t 0 , y 0 )) (i,j)∈A 1 ×A 2 , σ(t 0 , y 0 )q ǫ w ), I 1,B i 0 ,j 0 δ (t 0 , y 0 , -φ(t 0 , x 0 , .)) + I 2,B i 0 ,j 0 δ (t 0 , y 0 , w i0,j0 ))] -I 1,δ (t 0 , x 0 , φ(t 0 , ., y 0 )) + I 1,δ (t 0 , y 0 , -φ(t 0 , x 0 , .)) -I 2,δ (t 0 , x 0 , q ǫ u , u i0,j0 ) + I 2,δ (t 0 , y 0 , q ǫ w , w i0,j0 ) ≤ 0.

Taking now into account (3.3.16) and (3.3.17), let δ → 0, we get

-(p ǫ u -p ǫ w ) -[b(t 0 , x 0 )q ǫ u -b(t 0 , y 0 )q ǫ w ] - 1 2 [σ(t 0 , x 0 ) 2 M ǫ u -σ(t 0 , y 0 ) 2 M ǫ w ]
-[g i0,j0 (t 0 , x 0 , (u ij (t 0 , x 0 )) (i,j)∈A 1 ×A 2 , σ(t 0 , x 0 )q ǫ u , I 1,B i 0 ,j 0 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) + I 2,B i 0 ,j 0 δ (t 0 , x 0 , u i0,j0 ))

g i0,j0 (t 0 , y 0 , (w ij (t 0 , y 0 )) (i,j)∈A 1 ×A 2 , σ(t 0 , y 0 )q ǫ w , I 1,B i 0 ,j 0 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) + I 2,B i 0 ,j 0 δ (t 0 , x 0 , u i0,j0 ))]

- K|x 0 -y 0 | 2 ǫ |e|≥δ (1 ∧ |e|) 2 n(de) ≤ Cρ I R {|z| 2 + |z| 4 }n(dz).
and finally as usual, by (3.3.18) and (3.3.19) and the assumption on g i0,j0 , the exists a constant C ǫ,ρ such that for any fixed ρ > 0, lim sup ǫ→0,ρ→0 C ǫ,ρ ≤ 0 and

-{g i0,j0 (t 0 , x 0 , (u ij (t 0 , x 0 )) (i,j)∈A 1 ×A 2 , σ(t 0 , x 0 )q ǫ u , I 1,B i 0 ,j 0 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) + I 2,B i 0 ,j 0 δ (t 0 , x 0 , u i0,j0 )) -g i0,j0 (t 0 , x 0 , (w ij (t 0 , y 0 )) (i,j)∈A 1 ×A 2 , σ(t 0 , y 0 )q ǫ u , I 1,B i 0 ,j 0 δ (t 0 , x 0 , φ(t 0 , ., y 0 )) + I 2,B i 0 ,j 0 δ (t 0 , x 0 , u i0,j0 ))} ≤ C ǫ,ρ + Cρ I R {|z| 2 + |z| 4 }n(dz).
By the convergence to zero of (x 0 -y 0 ) and the Lipschitz condition of g ij with respect to (u kl (t 0 , x 0 )) (k,l)∈A 1 ×A 2 , and using condition (3.3.5) we obtain:

λ(u i0,j0 (t 0 , x 0 )-w i0,j0 (t 0 , y 0 ))- (i,j)∈A 1 ×A 2 =(i0,j0) β ij 0 (u i,j (t 0 , x 0 )-w i,j (t 0 , y 0 )) ≤ C 0 +Cρ I R {|z| 2 +|z| 4 }n(dz),
where β ij ǫ,ρ stands for the increment rate of f i0,j0 with respect to u ij which is uniformly bounded with respect to ǫ, ρ, and non-negative, thanks to the monotonicity assumption (A2). Therefore, λ(u i0,j0 (t 0 , x 0 )w i0,j0 (t 0 , y 0 ))

≤ (i,j)∈A 1 ×A 2 =(i0,j0) β ij ǫ,ρ (u i,j (t 0 , x 0 ) -w i,j (t 0 , y 0 )) + C ǫ,ρ + Cρ I R {|z| 2 + |z| 4 }n(dz) ≤ K ǫ,ρ (f i0,j0 ) (i,j)∈A 1 ×A 2 =(i0,j0) (u i,j (t 0 , x 0 ) -w i,j (t 0 , y 0 )) + + C ǫ,ρ + Cρ I R
{|z| 2 + |z| 4 }n(dz).

Taking the limit as ǫ → 0, ρ → 0,

λ(u i0,j0 (t * , x * ) -w i0,j0 (t * , x * )) ≤ lim ǫ→0,ρ→0 K ǫ,ρ (f i0,j0 ) (i,j)∈A 1 ×A 2 =(i0,j0) (u i,j (t 0 , x 0 ) -w i,j (t 0 , y 0 )) + ≤ K(f i0,j0 ) (i,j)∈A 1 ×A 2 =(i0,j0) lim n→∞ (u i,j (t 0 , x 0 ) -w i,j (t 0 , y 0 )) + ≤ K(f i0,j0 ) (i,j)∈A 1 ×A 2 =(i0,j0) (u i,j (t * , x * ) -w i,j (t * , x * )) + , since u ij (resp. w ij ) is usc (resp. lsc). As (i 0 , j 0 ) ∈ Γ(t * , x * ), we obtain λ(u i0,j0 (t * , x * ) -w i0,j0 (t * , x * )) ≤ C(f i0,j0 )(m 1 × m 2 -1)(u i0,j0 (t * , x * ) -w i0,j0 (t * , x * )).
But this is contradictory with (3.3.4) and (3.3.5), thus, for any (i, j)

∈ A 1 × A 2 , u ij ≤ w ij .
Step 2. Now we deal with the general case. For arbitarary λ ∈ R, if (u ij ) (i,j)∈A 1 ×A 2 (resp.(w ij ) (i,j)∈A 1 ×A 2 ) be a subsolution (resp. supersolution) of (3.1.2). Denote ũij = e -λt u ij (t, x) and wij = e -λt w ij (t, x). It is easy to show that ũij (t, x)(resp. wij (t, x)) is a subsoltion (resp. supersolution) of the following system of variational inequalities with oblique reflection :

∀(i, j) ∈ A 1 × A 2 ,          min{ṽ ij (t, x) -max l∈A 1 i {ṽ lj (t, x) -e -λt g il (t, x)}, max{ṽ ij (t, x) -min k∈A 2 j {ṽ ik (t, x) + e -λt g jk (t, x)}, -∂ t ṽij (t, x) -Lṽ ij (t, x) -λṽ ij (t, x) -e -λt g ij (t, x, (e λt ṽkl (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)e λt D x ṽij (t, x), e λt B ij ṽij (t, x))}} = 0 ṽij (T, x) = e -T λ h ij (x) (3.3.20)
But, by choosing λ large enough the functions

F ij (t, x, u kl (t, x)) (k,l)∈A 1 ×A 2 , z, U ) := λu ij + e -λt g ij (t, x, (e λt u kl (t, x)) (k,l)∈A 1 ×A 2 , e λt z, e λt U ), (i, j) ∈ A 1 × A 2 ,
satisfy condition (3.3.5). Hence thanks to the result stated in step 1, we have ũij ≤ wij , (i, j) ∈ A 1 × A 2 . Thus, u ij ≤ w ij , for any (i, j) ∈ A 1 × A 2 , which is the desired result.

Corollary 3.3.1. System (3.1.2) has at most one viscosity solution belongs to Π g , and it is necessary continuous.

Now we prove the existence, this will be done in 3 steps.

Proposition 3.3.1. The family (ū ij ) (i,j)∈A 1 ×A 2 is a viscosity subsolution of the system (3.1.2). Proof. Recall that ∀(i, j) ∈ A 1 × A 2 , ūij = lim m ց u ij,m , so ūij is usc, and ūij (T, x) = h ij (x), x ∈ R.
For a fixed (i, j) ∈ A 1 × A 2 , we suppose that there exists

ǫ 0 > 0 s.t. ūij (t, x) ≥ L ij [ū](t, x) + ǫ 0 ,
otherwise the subsolution property holds. Thanks to the decreasing convergence of (ū ij,m ) m≥0 to ūij , there exists m 0 such that for any m ≥ m 0 , we have

ūij,m (t, x) ≥ L ij [(ū pq,m ) (p,q)∈A 1 ×A 2 ](t, x) + ǫ 0 2 . (3.3.21)
Next, since

∀(i, j) ∈ A 1 × A 2 , m ≥ 0, ūij,m and L ij [(ū pq,m ) (p,q)∈A 1 ×A 2 ] are continuous, there exists a neighborhood Θ m of (t, x) such that ūij,m (t ′ , x ′ ) ≥ L ij [(ū pq,m ) (p,q)∈A 1 ×A 2 ](t ′ , x ′ ) + ǫ 0 4 , (t ′ , x ′ ) ∈ Θ m . (3.3.22)
By the same argument of [START_REF] Djehiche | Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles[END_REF](page 17), we have

ūij ≤ U ij [ū]. (3.3.23)
Let φ be a C 1,2 ([0, T ]×IR)∩Π g function such that ūij -φ has a global maximum at (t, x) in [0, T ]×IR k which wlog we suppose strict and ūij (t, x) = φ(t, x). Therefore (t, x) is a global strict maximum of ūij -φ in [0, T ] × B(x, Kδ). On the other hand there exist subsequences {m k } and ((t

′ m k , x ′ m k )) k such that ((t ′ m k , x ′ m k )) k → k (t, x) and ūij,m k (t ′ m k , x ′ m k ) → k ūij (t, x). Let now (t m k , x m k ) be the global maximum of ūij,m k -φ on [0, T ] × B(x, Kδ). Therefore (t m k , x m k ) → k (t, x) and u m k j (t m k , x m k ) → k ūij,m k (t, x).
Actually let us consider a convergent subsequent of (t m k , x m k ), which we still denote by (t m k , x m k ), and let ( t, x) be its limit. Then for some k 0 and for k

≥ k 0 we have ūij,m k (t m k , x m k ) -φ(t m k , x m k ) ≥ ūij,m k (t ′ m k , x ′ m k ) -φ(t ′ m k , x ′ m k ). (3.3.24)
Taking the limit wrt k to obtain ūij ( t, x)φ( t, x) ≥ ūij (t, x)φ(t, x).

As the maximum (t, x) of ūijφ on [0, T ] × IR k is strict then ( t, x) = (t, x). It follows that the sequence ((t m k , x m k )) k converges to (t, x). Going back now to (3.3.24) and in sending k to infinite we obtain

ūij,m k (t, x) ≥ lim sup k ūij,m k (t m k , x m k ) ≥ lim inf k ūij,m k (t m k , x m k ) ≥ lim inf k ūij,m k (t ′ m k , x ′ m k ) = ūij (t, x) which implies that ūij,m k (t m k , x m k ) → ūij (t, x) as k → ∞. Now for k large enough, (i) (t m k , x m k ) ∈ (0, T ) × B(x, Kδ) and is the global maximum of ūij,m k -φ in (0, T ) × B(x, Kδ) ; (ii) ūij,m k (t m k , x m k ) > L ij [(ū pq,m k ) (p,q)∈A 1 ×A 2 ](t m k , x m k ).
As ūij,m k is a subsolution of (3.2.8), then by Definition 2.2, we have

-∂ t φ(t m k , x m k ) -L 1 φ(t m k , x m k ) -I 1 δ (t m k , x m k , φ) -I 2 δ (t m k , x m k , D x φ(t n k , x n k ), ūij,m k ) ≤ g ij (t m k , x m k , (ū k,l,m k (t m k , x m k )) (k,l)∈A 1 ×A 2 , σ(t m k , x m k )D x ūi,j,m k (t m k , x m k ), I 1,B ij δ (t m k , x m k , φ) + I 2,B ij δ (t m k , x m k , ūi,j,m k )). (3.3.25)
But there exists a subsequence of {m k } (which we still denote by {m k }) such that:

(i) for any (k, l)

∈ A 1 i × A 2 j , (ū k,l,m k -1 (t m k , x m k )) k is convergent and then lim k ūk,l,m k -1 (t m k , x m k ) ≤ ūkl (t, x) ; (ii) (I 1 δ (t m k , x m k , φ)) m k → I 1 δ (t, x, φ) as k → ∞; (iii) (I 1,B ij δ (t m k , x m k , φ)) m k → I 1,B ij δ (t, x, φ) as k → ∞.
Sending now k to infinite, using the fact that g ij is continuous and verifies (A2) and finally by Lebesgue's Theorem to obtain ,x,φ) . Plugging now this inequality in the previous one to obtain

-∂ t φ(t, x) -L 1 φ(t, x) -I 1 δ (t, x, φ) ≤ I 2 δ (t, x, D x φ(t, x), ūij ) + g ij (t, x, (ū k,l (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x ūij (t, x), I 1,B ij δ (t, x, φ) + I 2,B ij δ (t, x, ūij )). But ūij (t, x) = φ(t, x) and ūij ≤ φ, then I 2 δ (t, x, D x φ(t, x), ūij ) ≤ I 2 δ (t, x, D x φ(t, x), φ) and I 2,B ij δ (t, x, ūij ) ≤ I 2,B ij δ (t
-∂ t φ(t, x) -L 1 φ(t, x) -I(t, x, φ) -g ij (t, x, (ū k,l (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x ūij (t, x), I B ij (t, x, φ)) ≤ 0.
Therefore ūij is a viscosity subsolution of

   min{(v ij -L ij [ -→ v ])(t, x), max{(v ij -U ij [ -→ v ])(t, x), -∂ t v ij (t, x) -Lv ij (t, x) -g ij (t, x, (v kl (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x v ij (t, x), I B ij (t, x, v ij ))}} = 0 v ij (T, x) = h ij (x)
As i, j is arbitrary then (ū ij ) (i,j)∈A 1 ×A 2 is a viscosity subsolution of (3.1.2).

Proposition 3.3.2. Let m 0 be fixed in N . Then the family (ū ij,m0 ) (i,j)∈A 1 ×A 2 is a viscosity supersolution of the system (3.1.2).

Proof. By the Proposition 3.2, the triples ( Ȳ ij,m0 , Zij,m0 , Ū ij,m0 , Kij,m0 ) (i,j)∈A 1 ×A 2 introduced in (3.2.5), solve the following system of reflected BSDEss:

∀(i, j) ∈ A 1 × A 2 ,                  ( Ȳ i,j,m0 , Zi,j,m0 , Ū i,j,m0 , Ki,j,m0 ) ∈ S 2 × H 2 × H 2 ( N ) × A 2 ; Ȳ i,j,m0 s = h i,j (X t,x T ) - T s f i,j,m0 (r, X t,x r , ( Ȳ k,l,m0 r ) (k,l)∈A 1 ×A 2 , Zi, j,m0 r , Ū i,j,m0 r )dr +

T s

Zi,j,m0

r dB r + T s E Ū i,j,m0 r (e) N (drde) + Ki,j,m0 T -Ki,j,m0 s Ȳ i,j,m0 s ≥ max k∈A 1 i { Ȳ k,j,m0 s -g ik (s, X t,x s )}, s ≤ T, T 0 ( Ȳ i,j,m0 s -max k∈A 1 i { Ȳ k,j,m0 s -g ik (s, X t,x s )})d Ki,j,m0 s = 0, (3.3.26) where, ∀(i, j) ∈ A 1 × A 2 , m ≥ 0 and (s, -→ Y ij , Z ij , U ij ), f i,j,m0 (s, X t,x s , -→ Y , Z ij , U ij ) :=f ij (s, X t,x s , (Y kl ) (k,l)∈A 1 ×A 2 , Z ij , U ij ) -m 0 (Y ij -min l∈A 2 j (Y il + g jl (s, X t,x s ))) + =g ij (s, X t,x s , (Y kl ) (k,l)∈A 1 ×A 2 , Z ij , E U ij (e)γ ij (X t,x s , e)n(de)) -m 0 (Y ij -min l∈A 2 j (Y il + g jl (s, X t,x s ))) + . :=g ij,+,m0 (s, X t,x s , (Y kl ) (k,l)∈A 1 ×A 2 , Z ij , E U ij (e)γ ij (X t,x s , e)n(de))
Furthermore, there exsits a unique A 1 ×A 2 -uplet of deterministic continuous functions (ū k,l,m0 ) (k,l)∈A 1 ×A 2 in Π g such that, for every t ≤ T , Ȳ i,j,m0 s = ūi,j,m0 (s, X t,x s ), s ∈ [t, T ].

By Theorem 5.1 in Appendix, it follows that ūi,j,m0 is a viscosity solution for the following PDE with two obstacles:

       min{θ(t, x) -max k∈A 1 i {ū kj,m0 (t, x) -g ik (t, x)}, max{θ(t, x) -ūij,m0 (t, x) ∨ min l∈A 2 j (ū il,m0 (t, x) -ḡjl (t, x)), -∂ t θ(t, x) -Lθ(t, x) -g ij (t, x, (ū kl,m0 (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x θ(t, x), I B ij (t, x, θ(t, x)))}} = 0 θ(T, x) = h ij (x) Next, let (t, x) ∈ [t, T ] × R and (p, q, M ) ∈ J-ūij,m0 (t, x).
As ūij,m0 is a solution in a viscosity sense of the previous PDE with two obstacles then it holds that ūij,m0 (t, x) ≥ max

k∈A 1 i {ū kj,m0 (t, x) -g ik (t, x)} (3.3.27)
and

max{ū ij,m0 (t, x) -ūij,m0 (t, x) ∨ min l∈A 2 j (ū il,m0 (t, x) -ḡjl (t, x)); -p -L (q,M ) ūij,m0 (t, x) -g ij (t, x, (ū kl,m0 (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)q, I B ij (t, x, ūij,m0 (t, x)))} ≥ 0. But, for any constants a, b ∈ R we have a -(a ∨ b) ≤ a -b and thus a -(a ∨ b) ≥ 0 ⇒ a -b ≥ 0.
Therefore, we have

max{ū ij,m0 (t, x) -min l∈A 2 j (ū il,m0 (t, x) -ḡjl (t, x)); -p -L (q,M ) ūij,m0 (t, x) -g ij (t, x, (ū kl,m0 (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)q, I B ij (t, x, ūij,m0 (t, x)))} ≥ 0.
Combining this inequality with (3.3.27) and since ūij,m0 (T, x) = h ij (x) it follows that ūij,m0 is a viscosity supersolution of the system

   min{θ(t, x) -max k∈A 1 i {ū kj,m0 (t, x) -g ik (t, x)}, max{θ(t, x) -min l∈A 2 j (ū il,m0 (t, x) -ḡjl (t, x)), -∂ t θ(t, x) -Lθ(t, x) -g ij (t, x, (ū kl,m0 (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x θ(t, x)I ,B ij (t, x, θ(t, x)))}} = 0 θ(T, x) = h ij (x) Since (i, j) is arbitrary in A 1 ×A 2 , the system of continuous functions (ū ij,m0 ) (i,j)∈A 1 ×A 2 is a supersolution of (3.1.2).
Consider the set U m0 defined as follows.

U m0 = { -→ u := (u ij ) (i,j)∈A 1 ×A 2 s.t. -→ u is subsolution of (3.1.2) and ∀(i, j) ∈ A 1 × A 2 , ūi,j ≤ u i,j ≤ ūij,m0 }. U m0 is not empty since it contains (¯u ij ) (i,j)∈A 1 ×A 2 . Next for (t, x) ∈ [0, T ] × R and (i, j) ∈ A 1 × A 2 , set: m0 u ij (t, x) = sup{u ij (t, x), (u kl ) (k,l)∈A 1 ×A 2 ∈ U m0 }.
Now we give the main result of this section. To begin with we give a third definition of viscosity solution, in the same way with [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]Proposition 1), we can prove that they are equivalent. Definition 3.3.1. (i) For a function u: [0, T ] × R → R, lsc (resp. usc), we denote J -u(t, x) the parabolic subjet (resp. J + u(t, x) the parabolic superjet ) of u at (t,x)∈ [0, T ] × R, as the set of triples (p,q,M)∈ R × R × S k ; where S k is the set of symmetric real matrices of dimension k

u(t ′ , x ′ ) ≥ u(t, x) + p(t ′ -t) -q, x ′ -x + 1 2 x ′ -x, M (x ′ -x) + o(|t ′ -t| + |x ′ -x|) 2 (resp. ≤)
(ii) We denote Ju(t, x) (resp. J+ u(t, x)) the parabolic limiting superjet (resp. superjet) of u at (t,x), as the set of triples (p,q,M)∈ R × R × S k s.t.

(p, q, M ) = lim n→∞ (p n , q n , M n ), (t, x) = lim n→∞ (t n , x n ) where (p n , q n , M n ) ∈ J -u(t n , x n )(resp.J + u(t n , x n )) and u(t, x) = lim n→∞ u(t n , x n ). (iii) A function -→ u = (u ij (t, x)) (i,j)∈A 1 ×A 2 : [0, T ] × R → R A 1 ×A 2 such that for any (i, j) ∈ A 1 × A 2 , u ij ∈ Π g is lsc (resp. usc)
, is said to be a viscosity subsolution (resp. supsolution) of (3.1.2) if for any test function

φ ∈ C 1,2 ([0, T ] × R), δ > 0, if (t 0 , x 0 ) ∈ [0, T ] × R is a maximum (resp. minimum) point of u i,j -φ on [0, T ] × B(x 0 , Kδ)
where K is the same as in definition 2.2, and if (p, q, M ) ∈ Ju i,j (t 0 , x 0 )(resp. J+ u i,j (t 0 , x 0 )) with q = D t φ(t 0 , x 0 ), p = D x φ(t 0 , x 0 ), and

M ≥ D 2 xx φ(t 0 , x 0 ) (resp. M ≤ D 2
xx φ(t 0 , x 0 )), then:

       min{(u ij -L ij [ -→ u ])(t 0 , x 0 ), max{(u ij -U ij [ -→ u ])(t 0 , x 0 ), -p -b(t 0 , x 0 )q -1 2 σ 2 (t 0 , x 0 )M -I 1 δ (t 0 , x 0 , φ) -I 2 δ (t 0 , x 0 , q, u ij ) -g ij (t 0 , x 0 , (u kl (t 0 , x 0 )) (k,l)∈A 1 ×A 2 , σ(t 0 , x 0 )q, I 1,B ij δ (t 0 , x 0 , φ) + I 2,B ij δ (t 0 , x 0 , u ij ))}} ≤ 0 (resp. ≥ 0) v ij (T, x) ≤ h ij (x) (resp. ≥) Theorem 3.3.2.
The family ( m0 u ij ) (i,j)∈A 1 ×A 2 does not depend on m 0 and is the unique continuous viscosity solution in the class Π g of the system (3.1.2).

Proof. Note that for any (i, j) ∈ A 1 × A 2 , ūi,j ≤ m0 u ij ≤ ūij,m0 . Since ūi,j and ūij,m0 are of polynomial growth, then ( m0 u ij ) (i,j)∈A 1 ×A 2 belongs to Π g . The remaining of the proof is divided into two steps, to easy notation, we denote ( m0 u ij ) (i,j)∈A 1 ×A 2 by (u ij ) (i,j)∈A 1 ×A 2 as no confusion is possible.

Step 1. First we show that (u ij ) (i,j)∈A 1 ×A 2 is a subsolution of (3.1.2). It is clear that for any (t, x) ∈ [0, T ] × R, ūi,j ≤ u ij ≤ ūij,m0 . This implies that ūi,j ≤ u ij, * ≤ ūij,m0 since ūij is usc and ūij,m0 is continuous. Therefore, for any

x ∈ R, since ūij (T, x) = ūij,m0 = h ij (x), we have u ij, * (T, x) = h ij (x). For the fixed (i, j) ∈ A 1 × A 2 , let (ũ ij ) (i,j)∈A 1 ×A 2 be an arbitrary element of U m0 . Then, for any (t, x) ∈ [0, T ] × R and any (p, q, M ) ∈ J+ ũij, * (t, x) we have min{(ũ ij, * -L ij [(ũ kl, * ) (k,l)∈A 1 ×A 2 ])(t, x), max{(ũ ij, * -U ij [(ũ kl, * ) (k,l)∈A 1 ×A 2 ])(t, x), -p -L (q,M ) ũij, * (t, x) -g ij (t, x, (ũ kl, * ) ij (k,l)∈A 1 ×A 2 , σ(t, x)q, I B ij (t, x, ũij, * (t, x)))}} ≥ 0
By definition we have ũij ≤ u ij and then ũij, * ≤ u ij, * . By the monotonicity property (A2), we have

min{(ũ ij, * -L ij [(u kl, * ) (k,l)∈A 1 ×A 2 ])(t, x), max{(ũ ij, * -U ij [(u kl, * ) (k,l)∈A 1 ×A 2 ])(t, x), -p -L (q,M ) ũij, * (t, x) -g ij (t, x, [(u kl, * ) ij (k,l) =(i,j)∈A 1 ×A 2 , ũij ], σ(t, x)q, I B ij (t, x, ũij, * (t, x)))}} ≥ 0
This means that ũij (t, x) is a subsotluion of the following equation.

min{(w -L ij [(u kl, * ) (k,l)∈A 1 ×A 2 ])(t, x), max{(w -U ij [(u kl, * ) (k,l)∈A 1 ×A 2 ])(t, x), -p -L (q,M ) w -g ij (t, x, [(u kl, * ) ij (k,l) =(i,j)∈A 1 ×A 2 , w], σ(t, x)q, I B ij (t, x, w))}} = 0
Relying on the lower semi continuity of the function

   (t, x, w, p, q, M ) → min{(w -L ij [(u kl, * ) (k,l)∈A 1 ×A 2 ])(t, x), max{(w -U ij [(u kl, * ) (k,l)∈A 1 ×A 2 ])(t, x), -p -L (q,M ) w -g ij (t, x, [(u kl, * ) ij (k,l) =(i,j)∈A 1 ×A 2 , w], σ(t, x)q, I B ij (t, x, w))}}
and using Lemma 4.2 in Crandall et al.(1992)( [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF],pp.23), related to suprema of subsolutions, combined with the above result, it holds that u ij is a subsolution of the following equation

   min{(w -L ij [(u kl, * ) (k,l)∈A 1 ×A 2 ])(t, x), max{(w -U ij [(u kl, * ) (k,l)∈A 1 ×A 2 ])(t, x), -p -L (q,M ) w -g ij (t, x, [(u kl, * ) ij (k,l) =(i,j)∈A 1 ×A 2 , w], σ(t, x)q, I B ij (t, x, w))}} = 0 u ij (T, x) = h ij (x). (3.3.28) Since (i, j) is arbitrary in A 1 × A 2 , (u ij ) (i,j)∈A 1 ×A 2 is a subsolution of (3.1.2).
Step 2. Now we will use Perron's method to show that (u ij ) (i,j)∈A 1 ×A 2 is a viscosity supersolution of (3.1.2). Note that for any (i, j)

∈ A 1 × A 2 , u ij = u ij * ≤ u ij * ≤ u ij * ≤ u ij,m0 * = u ij,m0 , (i, j) ∈ Γ 1 × Γ 2
, and y ∈ IR, we denote by [(y kl ) (k,l)∈Γ 1 ×Γ 2 -{i,j} , y] the matrix obtained from the matrix y = (y kl ) (k,l)∈Γ 1 ×Γ 2 by replacing the element y ij with y.

For any (i,

j) ∈ Γ 1 × Γ 2 , let b : (t, x) ∈ [0, T ] × IR k → b(t, x) ∈ IR k ; σ : (t, x) ∈ [0, T ] × IR k → σ(t, x) ∈ IR k×d ; f ij : (t, x, y, z) ∈ [0, T ] × IR k+Λ+d → f ij (t, x, y, z) ∈ IR ; g ik : (t, x) ∈ [0, T ] × IR k → g ik (t, x) ∈ IR (k ∈ (Γ 1 ) -i ); ḡjl : (t, x) ∈ [0, T ] × IR k → ḡjl (t, x) ∈ IR (l ∈ (Γ 2 ) -j ); h ij : x ∈ IR k → h ij (x) ∈ IR. A function Φ : (t, x) ∈ [0, T ] × IR k → Φ(t, x)
∈ IR is called of polynomial growth if there exist two non-negative real constants C and γ such that

|Φ(t, x)| ≤ C(1 + |x| γ ), (t, x) ∈ [0, T ] × IR k .
Hereafter, this class of functions is denoted by Π

g . Let C 1,2 ([0, T ] × IR k ) (or simply C 1,2
) denote the set of real-valued functions defined on [0, T ] × IR k , which are once (resp. twice) differentiable w.r.t. t (resp.

x) and with continuous derivatives.

The following assumptions on the data of the systems (4.1.1) and (4.1.2) are in force throughout the paper.

(H0) The functions b and σ are jointly continuous in (t, x) and Lipschitz continuous w.r.t. x uniformly in t, meaning that there exists a non-negative constant C such that for any (t, x, x

′ ) ∈ [0, T ] × R k+k we have |σ(t, x) -σ(t, x ′ )| + |b(t, x) -b(t, x ′ )| ≤ C|x -x ′ |.
Therefore, they are also of linear growth w.r.t. x, i.e., there exists a constant C such that for any

(t, x) ∈ [0, T ] × IR k , |b(t, x)| + |σ(t, x)| ≤ C(1 + |x|).
(H1) Each function f ij (i) is continuous in (t, x) uniformly w.r.t. the other variables ( y, z) and, for any (t, x), the mapping (t, x) → f ij (t, x, 0, 0) is of polynomial growth.

(ii) is Lipschitz continuous with respect to the variables ( y := (y ij ) (i,j)∈Γ1×Γ2 , z) uniformly in (t, x), i.e.

∀ (t, x) ∈ [0, T ] × R k , ∀ ( y 1 , y 2 ) ∈ R Λ × R Λ , (z 1 , z 2 ) ∈ R d × R d , |f ij (t, x, y 1 , z 1 ) -f ij (t, x, y 2 , z 2 )| ≤ C (| y 1 -y 2 | + |z 1 -z 2 |) ,
where, | y| stands for the standard Euclidean norm of y in IR Λ .

(H2) Monotonicity: Let y = (y kl ) (k,l)∈Γ 1 ×Γ 2 . For any (i, j) ∈ Γ 1 × Γ 2 and any (k, l) = (i, j) the mapping

y kl → f ij (s, y, z) is non-decreasing.
(H3) The functions h ij , which are the terminal conditions in the systems (4.1.1) and (4.1.2), are continuous with respect to x, belong to class Π g and satisfy

∀ (i, j) ∈ Γ 1 × Γ 2 and x ∈ IR k , max k∈(Γ 1 ) -i h kj (x) -g ik (T, x) ≤ h ij (x) ≤ min l∈(Γ 2 ) -j h il (x) + ḡjl (T, x) .
(H4) The no free loop property: The switching costs g ik and ḡjl are non-negative, jointly continuous in (t, x), belong to Π g and satisfy the following condition:

For any loop in Γ 1 × Γ 2 , i.e., any sequence of pairs (i 1 , j 1 ), . . . , (i N , j N ) of Γ 1 × Γ 2 such that (i N , j N ) = (i 1 , j 1 ), card{(i 1 , j 1 ), . . . , (i N , j N )} = N -1 and any q = 1, . . . , N -1, either i q+1 = i q or j q+1 = j q , we have

∀(t, x) ∈ [0, T ] × IR k , q=1,N -1 ϕ iqiq+1 (t, x) = 0, (4.1.3)
where, ∀ q = 1, . . . , N -1, ϕ iqiq+1 (t, x) = -g iqiq+1 (t, x)1

1 iq =iq+1 + ḡjqiq+1 (t, x)1 1 jq =jq+1 .
This assumption implies in particular that

∀ (i 1 , . . . , i N ) ∈ (Γ 1 ) N such that i N = i 1 and card{i 1 , . . . , i N } = N -1, N -1 p=1 g i k ,i k+1 > 0 (4.1.4)
and

∀ (j 1 , . . . , j N ) ∈ (Γ 2 ) N such that j N = j 1 and card{j 1 , . . . , j N } = N -1, N -1 p=1 ḡj k ,j k+1 > 0. ( 4.1.5)
By convention we set ḡj,j = g i,i = 0.

Conditions (4.1.4) and (4.1.5) are classical in the literature of switching problems and usually referred to as the no free loop property.

We now introduce the probabilistic tools we need later. Let (Ω, F, P ) be a fixed probability space on which is defined a standard d-dimensional Brownian motion B = (B t ) 0≤t≤T whose natural filtration is (F 0 t := σ{B s , s ≤ t}) 0≤t≤T . Let F = (F t ) 0≤t≤T be the completed filtration of (F 0 t ) 0≤t≤T with the P-null sets of F, hence (F t ) 0≤t≤T satisfies the usual conditions, i.e., it is right continuous and complete. On the other hand let P be the σ-algebra on [0, T ] × Ω of F-progressively measurable sets.

Next, let (i) H 2,ℓ (ℓ ≥ 1) be the set of P-measurable and IR ℓ -valued processes w = (w t ) t≤T such that

E[ T 0 |w s | 2 ds] < ∞; (ii) S 2 (resp. S 2 d ) be the set of P-measurable continuous (resp. RCLL) processes such that E[sup t≤T |w t | 2 ] < ∞.
(iii) A 2 i be the subset of S 2 of non-decreasing processes K = (K t ) t≤T such that K 0 = 0. For (t, x) ∈ [0, T ] × IR k , let X t,x be the diffusion process solution of the following standard SDE:

∀s ∈ [t, T ], X t,x s = x + s t b(r, X t,x r )dr + s t σ(r, X t,x r )dB r ; X t,x s = x, s ∈ [0, t]. (4.1.6)
Under Assumption (H0) on b and σ, the process X t,x exists and is unique. Moreover, it satisfies the following estimates: For all p ≥ 1,

E[sup s≤T |X t,x s | p ] ≤ C(1 + |x| p ). (4.1.7)
Its infinitesimal generator L X is given, for every (t, x) ∈ [0, T ] × IR k and φ ∈ C 1,2 , by

L X φ(t, x) := 1 2 k i,j=1 (σσ * (t, x)) i,j ∂ 2 xixj φ(t, x) + i=1,k b i (t, x)∂ xi φ(t, x) = 1 2 T r[σσ ⊤ (t, x)D 2 xx φ(t, x)] + b(t, x) ⊤ D x φ(t, x).
(v ij ) (i,j)∈Γ 1 ×Γ 2 (resp. (v ij ) (i,j)∈Γ 1 ×Γ 2 ) of the following system: ∀(i, j) ∈ Γ 1 × Γ 2 ,        min (v ij -L ij ( v))(t, x), max (v ij -U ij ( v))(t, x), -∂ t vij (t, x) -L X (v ij )(t, x) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x)D x vij (t, x)) = 0, vij (T, x) = h ij (x) (4.1.9) (resp.        max (v ij -U ij ( v))(t, x); min (v ij -L ij ( v))(t, x); -∂ t v ij (t, x) -L X (v ij )(t, x) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x)D x v ij (t, x)) = 0, v ij (T, x) = h ij (x).

)

In order to obtain the solutions of the systems (4.1.9) and (4.1.10) Djehiche et al. ( [START_REF] Djehiche | Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles[END_REF]) introduce the following sequences of backward reflected BSDEs with inter-connected obstacles: ∀m, n ≥ 0, ∀(i, j) ∈

Γ 1 × Γ 2 ,            Ŷ ij,m ∈ S 2 , Ẑij,m ∈ H 2,d and Kij,m ∈ A 2 i ; Ŷ ij,m s = h ij (X t,x T ) + T s f ij,m (r, X t,x r , ( Ŷ kl,m r ) (k,l)∈Γ 1 ×Γ 2 , Ẑij,m r )dr + T s d Kij,m s - T s Ẑij,m r dB r , s ≤ T ; Ŷ ij,m s ≥ max k∈(Γ 1 ) -i { Ŷ kj,m s -g ik (s, X t,x s )}, s ≤ T ; T 0 ( Ŷ ij,m s -max k∈(Γ 1 ) -i { Ŷ kj,m s -g ik (s, X t,x s )})d Kij,m s = 0 (4.1.11) and          Y ij,n ∈ S 2 , Z ij,n ∈ H 2,d and K ij,n ∈ A 2 i ; Y ij,n s = h ij (X t,x T ) + T s f ij,n (r, X t,x r , (Y kl,n r ) (k,l)∈Γ 1 ×Γ 2 , Z ij,n r )dr - T s Z ij,n r dB r - T s dK ij,n r , s ≤ T ; Y ij,n s ≤ min l∈(Γ 2 ) -j Y il,n s + ḡjl (s, X t,x s ) , s ≤ T ; T 0 (Y ij,n s -min l∈(Γ 2 ) -j Y il,n s + ḡjl (s, X t,x s ) )dK ij,n s = 0 (4.1.
12) where, for any (i, j) ∈ Γ 1 × Γ 2 , n, m ≥ 0 and (s, x, y, z ij ),

f ij,m (s, x, y, z ij ) := f ij (s, x, (y kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ) -m y ij -min l∈(Γ 2 ) -j (y il + ḡjl (s, x)) + (4.1.13)
and

f ij,n (s, x, y, z ij ) := f ij (s, x, (y kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ) + n y ij -max k∈(Γ 1 ) -i (y kj -g ik (s, x)) -. (4.1.14)
Under Assumptions (H0)-(H4) it is shown in [START_REF] Hamadène | Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems[END_REF] (see also [START_REF] Chassagneux | A note on existence and uniqueness for solutions of multidimensional reflected BSDEs[END_REF] or [START_REF] Hamadène | Switching problem and related system of reflected backward SDEs[END_REF]) that each one of the systems (4.1.11) and (4.1.12) has a unique solution ( Ŷ ij,m , Ẑij,m , Kij,m ) and (Y ij,m , Z ij,m , K ij,m ) respectively. In addition, they enjoy the following properties:

(i) For any m, n ≥ 0 and (i,

j) ∈ Γ 1 × Γ 2 Ŷ ij,m ≥ Ŷ ij,m+1 ≥ Y ij,n+1 ≥ Y ij,n . (4.1.15)
(ii) For any n, m ≥ 0 and (i, j) ∈ Γ 1 × Γ 2 there exist deterministic continuous functions vij,m and v ij,n such that for any (t, x) ∈ [0, T ] × IR k and s ∈ [t, T ], we have Ŷ ij,m s = vij,m (s, X t,x s ) and Y ij,n s = v ij,n (s, X t,x s ). Moreover, from (4.1.15) we easily deduce that, for any n, m ≥ 0 and (i,

j) ∈ Γ 1 × Γ 2 , vij,m ≥ vij,m+1 ≥ v ij,n+1 ≥ v ij,n . (4.1.16)
Finally, for any m ≥ 0 (resp. n ≥ 0), vm := (v ij,m ) (i,j)∈Γ 1 ×Γ 2 (resp. v n := (v ij,n ) (i,j)∈Γ 1 ×Γ 2 ) is the unique continuous viscosity solution, in the class Π g , of the following system of PDEs with inter-connected obstacles:

∀(i, j) ∈ Γ 1 × Γ 2 , ∀(t, x) ∈ [0, T ] × IR k ,        min (v ij,m -L ij ( vm )(t, x); -∂ t vij,m -L X (v ij,m )(t, x) -f ij,m (t, x, (v kl,m (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ(t, x) ⊤ D x vij,m (t, x)) = 0, vij,m (T, x) = h ij (x) (resp.        max (v ij,n -U ij ( v n ))(t, x); -∂ t v ij,n (t, x) -L X (v ij,n )(t, x) -f ij,n (t, x, (v kl,n (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x)D x v ij,n (t, x)) = 0, v ij,n (T, x) = h ij (x)). (iii) For (i, j) ∈ Γ 1 × Γ 2 and (t, x) ∈ [0, T ] × IR k , let us set vij (t, x) := lim m→∞ ց vij,m (t, x) and v ij (t, x) := lim n→∞ ր v ij,n (t, x).
Then, using Perron's method, it is shown that (v ij ) (i,j)∈Γ 1 ×Γ 2 (resp. (v ij ) (i,j)∈Γ 1 ×Γ 2 ) is continuous, belongs to Π g and is the unique viscosity solution, in class Π g , of system (4.1.9) (resp. (4.1.10)). Finally, by construction and in view of (4. 1.16), it holds that, for any (i, j)

∈ Γ 1 × Γ 2 , v ij ≤ vij .
(4.1.17)

Equality of min-max and max-min solutions

In [START_REF] Djehiche | Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles[END_REF], the question whether or not for any (i, j) ∈ Γ 1 × Γ 2 , v ij ≡ vij was left open. This was mainly due to the fact we have not been able to compare the inter-connected obstacles neither in (4.1.9) nor in (4.1.10).

Actually, had we known that

(i) ∀(i, j) ∈ Γ 1 × Γ 2 , L ij ( v) ≤ U ij ( v) or (ii) ∀(i, j) ∈ Γ 1 × Γ 2 , L ij ( v) ≤ U ij ( v) (4.2.1)
then we would have deduced from Theorem 5.5.1 in Appendix and the uniqueness of the solution of (4.1.9) or (4.1.10) that for any (i, j) ∈ Γ 1 × Γ 2 , vij = v ij . In this section we are going to investigate under which additional regularity assumptions on the data of the problem, one of the inequalities in (4.2.1) is satisfied to be able to conclude that vij = v ij , for any (i, j) ∈ Γ 1 × Γ 2 , i.e., the solutions of (4.1.9) and (4.1.10) are the same.

For that let us introduce the following assumption. (H5):

(i) For any (i, j) ∈ Γ 1 × Γ 2 , the functions ḡij are C 1,2 . Moreover, D x ḡij and D 2 xx ḡij belong to Π g . Furthermore, for any j 1 , j 2 , j 3 ∈ Γ 2 such that |{j 1 , j 2 , j 3 }| = 3, ḡj1j3 (t, x) < ḡj1j2 (t, x) + ḡj2j3 (t, x), ∀(t, x) ∈ [0, T ] × IR k .

(ii) For any (i, j) ∈ Γ 1 × Γ 2 , the function f ij verifies the following estimate:

|f ij (t, x, y, z ij )| ≤ C(1 + |x| p )
for some real constants C and p.

(i, j) ∈ Γ 1 × Γ 2 , ḡij (s, X t,x s ) = ḡij (t, x) + S t L X (ḡ ij )(s, X t,x s )ds + s t D x ḡij (s, X t,x s )σ(s, X t,x s )dB s , s ∈ [t, T ] ḡij (s, X t,x s ) = ḡij (s, x), s ≤ t.
Hereafter, we denote by

a ij (s) := L X (ḡ ij )(s, X t,x s ), b ij (s) := D x ḡij (s, X t,x s )σ(s, X t,x s ), s ≤ T.
Proposition 4.2.1. Under Assumptions (H0)-(H5) we have, for every (i, j)

∈ Γ 1 × Γ 2 , L ij ( v) ≤ vij ≤ U ij ( v).
Proof. We derive this inequality through the following three steps.

Step 1: For any m ≥ 0, (i, j) ∈ Γ 1 × Γ 2 and (t, x) ∈ [0, T ] × IR k , let us consider the system of reflected BSDEs with one inter-connected obstacles:

           Y ij,m ∈ S 2 , Žij,m ∈ H 2,d and Ǩij,m ∈ A 2 i ; Y ij,m s = h ij (X t,x T ) + T s f ij,m (r, X t,x r , ( Y kl,m r ) (k,l)∈Γ 1 ×Γ 2 , Žij,m r )dr + T s d Ǩij,m s - T s Žij,m r dB r , s ≤ T ; Y ij,m s ≥ max k∈(Γ 1 ) -i { Y kj,m s -g ik (s, X t,x s )}, s ≤ T ; T 0 ( Y ij,m s -max k∈(Γ 1 ) -i { Ŷ kj,m s -g ik (s, X t,x s )})d Ǩij,m s = 0, (4.2.2) where, f ij,m (s, x, y, z ij ) := f ij (s, x, y, z ij ) -m l∈(Γ 2 ) -j (y ij -y il -ḡjl (s, x)) + . (4.2.3)
By Corollary 2, in [START_REF] Hamadène | Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems[END_REF], the solution of this system exists and is unique and there exist deterministic continuous functions (v ij,m ) (i,j)∈Γ 1 ×Γ 2 , which belong also to Π g such that, for any i, j and m ≥ 0, it holds that ∀s ∈ [t, T ], Y ij,m s = vij,m (s, X t,x s ).

Moreover, the family of functions vm := (v ij,m ) (i,j)∈Γ 1 ×Γ 2 is the unique continuous solution in viscosity sense in Π g of the following system of PDEs with obstacles:

       min (v i,j,m -L i,j ( vm ))(t, x); -∂ t vi,j,m (t, x) -L X (v i,j,m )(t, x) -f ij,m (t, x, (v kl,m (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ(t, x) ⊤ D x vi,j,m (t, x)) = 0, vij,m (T, x) = h ij (x).
Finally, by the Comparison Theorem (see [START_REF] Hamadène | Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems[END_REF], Remark 1), since

f ij,m+1 ≤ f ij,m and f ij,|Γ2|m ≤ f ij,m ≤ f ij,m , we have, ∀(i, j) ∈ Γ 1 × Γ 2 and m ≥ 0, Y ij,m+1 ≤ Y ij,m and Ŷ ij,|Γ2|m ≤ Y ij,m ≤ Ŷ ij,m , which implies that, for any (i, j) ∈ Γ 1 × Γ 2 and m ≥ 0, vij,m+1 ≤ vij,m and vij,|Γ2|m ≤ vij,m ≤ vij,m .
Then, for any (i, j) ∈ Γ 1 ×Γ 2 , the sequence (v ij,m ) m≥0 is decreasing and converges, uniformly on compact subsets of [0, T ] × IR k , to vij since lim m→∞ vij,m (t, x) = vij (t, x), for any (t, x) ∈ [0, T ] × IR k .

Step 2: The following estimate holds: For every (i, j) ∈ Γ 1 × Γ 2 and m ≥ 0,

E m T 0 l∈(Γ 2 ) -j { Y ij,m s -Y il,m s -ḡjl (s, X t,x s )} + ds ≤ C(1 + |x| p ), (4.2.4)
where, the constant C is independent of m and x.

We first give a representation of Y ij,m as the optimal payoff of a switching problem. Indeed, let δ := (τ n , ζ n ) n≥0 be an admissible strategy of switching, i.e., Then, E[(A a T ) 2 ] < ∞. The quantity A δ T stands for the switching cost at terminal time T when the strategy δ is implemented. Next, with an admissible strategy δ := (τ n , ζ n ) n≥0 we associate a piecewise constant process a = (a s ) s∈[0,T ] defined by

a s := ζ 0 {τ0} (s) + ∞ j=1 ζ j-1 ]τj-1,τj ] (s), s ≤ T. (4.2.5)
For any s ≥ τ 0 , a s is the mode indicator at time s. Note that there is a bijection between the processes a and the admissible strategies δ, therefore hereafter A a is nothing else but A δ . Finally, for any fixed i ∈ Γ 1 and a real constant θ ∈ [0, T ], we denote by A i θ the following set: where, for any s ≥ τ 0 and (ȳ, z) ∈ IR 1+d , f aj,m (s, X t,x s , ȳ, z) (resp. f aj (s, X t,x s , ȳ, z)) is equal to f ℓj,m (s, X t,x s , [(v kl,m (s,

A i θ := δ = (τ n , ζ n ) n≥0
X t,x s )) (k,l)∈Γ 1 ×Γ 2 -{(ℓ,j)} , ȳ], z) ) -f al (r, X t,x r , U al,m r ) -a jl r )dr - T s 1 B j,l (r) V aj,m r -V al,m r -b jl r dB r
where, L a,jl,m is the local time at 0 of the semimartingale W a,jl,m . Splitting the difference ∆ a,j,l,m (r) := m

j ′′ =j 1 B j,l (r)(W a,jj ′′ ,m r ) + -m j ′′ =l 1 B j,l (r)(W a,lj ′′ ,m r ) + as ∆ j,l,m (r) = m1 B j,l (r)(W a,jl,m r ) + -1 B j,l (r)(W a,lj,m r ) + + m j ′′ =j,j ′′ =l 1 B j,l (r) (W a,jj ′′ ,m r ) + -(W a,lj ′′ ,m r ) + ,
the previous formula can be rewritten as follows: ∀s ∈ [θ, T ],

(W a,jl,m s

) + + 1 2 T s dL a,jl,m r + m T s 1 B j,l (r)(W a,jl,m r ) + dr = T s 1 B j,l (r)( f aj (r, X t,x r , Ǔ aj,m r , V aj,m r ) -f al (r, X t,x r , Ǔ al,m r , V al,m r ) -a jl r )dr +m T s 1 B j,l (r)(W a,lj,m r ) + dr - T s 1 B j,l (r) V aj,m r -V al,m r -b jl r dB r -m T s dr{ j ′′ =j,l 1 B j,l (r) [(W a,j ′′ ,m r ) + -(W a,lj",m r ) + ]} (4.2.9)
But, ḡjl (t, x) + ḡlj (t, x) > ḡjj (t, x) = 0. Thus, we obtain that, for every (t, x) ∈ [0, T ] × IR k , {y ∈ IR m , y jy lḡj,l (t, x) ≥ 0} ∩ {y ∈ IR m , y ly jḡl,j (t, x) ≥ 0} = ∅, from which we deduce that 1 B j,l (r)(W a,lj,m r Using here that the family of penalty costs satisfies ḡj,j " < ḡjl + ḡlj " we deduce that {y ∈ R m , y jy lḡjl (t, x) ≥ 0} ∩ {y ∈ R m , y ly jḡlj ′′ (t, x) + ḡjj ′′ (t, x) ≥ 0} = ∅ which therefore yields

) + = 0, ∀r ∈ [θ, T ]. ( 4
∀r ∈ [θ, T ], 1 B j,l (r) Ǔ al,m r -Ǔ aj,m r
ḡlj ′′ (r, X t,x r ) + ḡjj ′′ (r, X t,x r ) + = 0. (4.2.12)

Going back now to (4.2.9), applying Itô's formula to e -ms (W a,jl,m s ) + and taking into account of (4.2.10), (4.2.11) and (4.2.12) to obtain: ∀s ∈ [θ, T ], (W a,jl,m s

) + ≤ T s 1 B j,l (r)e -m(r-s) ( f aj (r, X t,x r , Ǔ aj,m r , V aj,m r ) -f al (r, X t,x r , Ǔ al,m r , V al,m r ) -u jl r )dr
As θ is arbitrary in [0, T ] then by integration with respect to dθ in the previous inequality we obtain (4.2.4).

Step 3 : For any (t 0 , x 0 ) ∈ [0, T ] × IR k and (i,

j) ∈ Γ 1 × Γ 2 , L ij ( v)(t 0 , x 0 ) ≤ vij (t 0 , x 0 ) ≤ U ij ( v)(t 0 , x 0 ).
We first claim that vij (t 0 , x 0 ) ≥ L ij ( v)(t 0 , x 0 ) holds. Indeed, by construction of vm := (v ij,m ) (i,j)∈Γ 1 ×Γ 2 one has vij,m (t 0 , x 0 ) ≥ L ij ( vm )(t 0 , x 0 ). Therefore, taking the limit w.r.t. m, we obtain vij (t 0 , x 0 ) ≥ L ij ( v)(t 0 , x 0 ).

We now show that vij (t 0 , x 0 ) ≤ U ij ( v)(t 0 , x 0 ). First, assume that vij (t 0 , x 0 ) > L ij ( v)(t 0 , x 0 ). Then, relying on the viscosity subsolution property of vij yields

min (v ij -L ij ( v))(t 0 , x 0 ); max (v ij -U ij ( v))(t 0 , x 0 ); -∂ t vij (t 0 , x 0 ) -L X (v ij )(t 0 , x 0 ) -f ij (t 0 , x 0 , (v kl (t 0 , x 0 )) (k,l)∈Γ 1 ×Γ 2 , σ(t 0 , x 0 ) ⊤ D x vij (t 0 , x 0 )) ≤ 0, which implies that max (v ij -U ij )(t 0 , x 0 ); -∂ t vij (t 0 , x 0 ) -L X (v ij ) -f ij (t 0 , x 0 , (v kl (t 0 , x 0 )) (k,l)∈Γ 1 ×Γ 2 )) ≤ 0.
Hence, (v ij -U ij ( v))(t 0 , x 0 ) ≤ 0.

Suppose now that at (t 0 , x 0 ) we have vij (t 0 , x 0 ) = L ij ( v)(t 0 , x 0 ). Proceeding by contradiction we suppose in addition that

∃ ǫ > 0, (v ij -U ij ( v))(t 0 , x 0 ) > ǫ. (4.2.14)
Using both the continuity of (t, x) → vi,j (t, x) and (t, x) → U ij ( v)(t, x) and the uniform convergence on compact subsets of (v ij,m ) m to vij we claim that for some strictly positive ρ and for m 0 large enough it holds that ∀m ≥ m 0 , ∀(t, x) ∈ B((t 0 , x 0 ), ρ),

(v ij,m -U ij ( vm ))(t, x) ≥ ǫ 2 , with B((t 0 , x 0 ), ρ) = {(t, x) ∈ [0, T ] × IR k s.t. |t -t 0 | ≤ ρ, |x -x 0 | ≤ ρ}.
Without loss of generality we can now assume [t 0 , t 0 + ρ] ⊂ [t 0 , T ]. By the definition of U ij ( vm ), there exists one index l 0 = j such that the inequalities vij,m -(v il0,m + ḡjl0 ) ≥ ǫ 2 and

l∈(Γ 2 ) -j vij,m -vil,m -ḡjl + ≥ ǫ 2 , (4.2.15)
hold on the ball B((t 0 , x 0 ), ρ).

Let us now introduce the following stopping time τ X :

τ X = inf{s ≥ t 0 , X t0,x0 s ∈ B((t 0 , x 0 ), ρ)} ∧ t 0 + ρ .
We then have, for all m ≥ m 0 ,

mE   τ X t0 l =j (v ij,m (s, X t0,x0 s ) -vil,m (s, X t0,x0 s ) + ḡjl (s, X t0,x0 s ) + ds   ≥ m ǫ 2 E(τ X -t 0 ) → ∞, (4.2.16
) as m → ∞. But, this is contradictory to (4.2.4). Then vij (t 0 , x 0 ) ≤ U ij ( v))(t 0 , x 0 ) and the proof is complete.

As a by product of Proposition 4.2.1 and Theorem 5.5.2 (displayed in the appendix), we have: (ii) From (4.2.13) and Doob's inequality we have, for every

(t, x) ∈ [0, T ] × IR k , m 2 E l =j [( Y ij,m s -Y il,m s -ḡjl (s, X t,x s )) + ] 2 ≤ C(1 + |x| 2p ), s ≤ T, (4.2.17)
where, C is a constant.

4.3

The min-max solution as a the value of the zero-sum switching game

Let us consider now the following assumption which is used later:

(H6):

(i) For any (i, j) ∈ Γ 1 × Γ 2 , the function f ij does not depend on z ij .

(ii) For any (i, j) ∈ Γ 1 × Γ 2 , the function f ij does not depend on ( y, z ij ).

Once for all, in this section we suppose that Assumptions (H0)-(H5) hold.

Set Y ij s = v ij (s, X t,x s ), s ∈ [t, T ] and (i, j) ∈ Γ 1 × Γ 2 .
We then have the following representation of Y ij as the value function of a Dynkin game. This is a by-product of Theorems 4.1.1, 5.5.2 and Propositions 4.2.1 and 5.5.10 (displayed in the appendix) since the barriers

L ij ( v) = max k =i v kj -g ik and U ij ( v)= min l =j v il + ḡjl are comparable, i.e., L ij ( v) ≤ U ij ( v)
for any i, j.

Proposition 4.3.1. Assume that Assumptions (H0)-(H5) and (H6)-(i) are fulfilled. For any (i, j) ∈ Γ 1 × Γ 2 and s ∈ [t, T ] we have,

v ij (s, X t,x s ) = Y ij s = ess sup σ≥s ess inf τ ≥s E σ∧τ s f ij (r, X t,x r , (v kl (r, X t,x r )) (k,l)∈Γ 1 ×Γ 2 )dr +{max k∈(Γ 1 ) -i {v kj (σ, X t,x σ ) -g ik (σ, X t,x σ )}}1 1 [σ<τ ] + min l∈(Γ 2 ) -j {v il (τ, X t,x τ ) + ḡjl (τ, X t,x τ )}1 1 [τ ≤σ<T ] +h ij (X t,x T )1 1 [τ =σ=T ] |F s = ess inf τ ≥s ess sup σ≥s E σ∧τ s f ij (r, X t,x r , (v kl (r, X t,x r )) (k,l)∈Γ 1 ×Γ 2 )dr +{max k∈(Γ 1 ) -i {v kj (σ, X t,x σ ) -g ik (σ, X t,x σ )}}1 1 [σ<τ ] + min l∈(Γ 2 ) -j {v il (τ, X t,x τ ) + ḡjl (τ, X t,x τ )}1 1 [τ ≤σ<T ] +h ij (X t,x T )1 1 [τ =σ=T ] |F s . (4.3.1)
On the other hand, it is shown in ( [START_REF] Hamadène | BSDEs with general discontinuous reflecting barriers without Mokobodski's condition[END_REF], Theorem 3.1) , that Y ij is the unique local solution of the two barriers reflected BSDEs associated with (f ij (s, X t,x s , y), h ij (X t,x T ), L ij ( v)(s, X t,x s ), U ij ( v)(s, X t,x s )). Precisely we have:

Proposition 4.3.2. Let (i, j) ∈ Γ 1 × Γ 2 be fixed. For any stopping time τ ≥ t, there exists another stopping time δ τ ≥ τ , Pa.s. (δ τ depends also on i, j but we omit it as far as there is no confusion) and three processes Z ij,τ , K ij,±,τ such that:

(i) Y ij T = h ij (X t,x T ) ; (ii)        Z ij,τ ∈ H 2,d , K ij,±,τ ∈ A 2 i and non-decreasing ; ∀s ∈ [τ, δ τ ], Y ij s = Y ij δτ + δτ s f ij (r, X t,x r , (Y kl r ) (k,l)∈Γ 1 ×Γ 2 )dr - δτ s Z ij r dB r + δτ s dK ij,+,τ r - δτ s dK ij,-,τ r L ij ( v)(s, X t,x s ) ≤ Y ij s ≤ U ij ( v)(s, X t,x s ), ∀s ∈ [t, T ] ; δτ τ Y ij r -L i,j ( v)(r, X t,x r ) dK ij,+,τ r = 0 and δτ τ Y ij r -U ij ( v)(r, X t,x r ) dK ij,-,τ r = 0; (4.3.
2) (iii) Let γ τ and θ τ be the following two stopping times:

γ τ := inf{s ≥ τ, Y ij s = L ij ( v)(s, X t,x s )} ∧ T and θ τ := inf{s ≥ τ, Y ij s = U ij ( v)(s, X t,x s )} ∧ T.
Then, Pa.s., γ τ ∨ θ τ ≤ δ τ .

Description of the zero-sum switching game

We now address the issue of the relationship between the value function of a zero-sum switching game and the functions (v ij ) (i,j)∈Γ 1 ×Γ 2 solution of system (4.1.9). We first suppose that Assumption (H6)-(ii) is satisfied, i.e., f ij does not depend on ( y, z ij ), for any (i, j) ∈ Γ 1 × Γ 2 .

To begin with let us describe briefly the zero-sum switching game. Assume we have two players π 1 and π 2 who intervene on a system with the help of switching strategies. An admissible switching strategy for π 1 (resp. π 2 ) is a sequence δ := (σ n , ξ n ) n≥0 (resp. ν := (τ n , ζ n ) n≥0 ) where for any n ≥ 0, (i) σ n (resp. τ n ) is an F-stopping times such that P -a.s., σ n ≤ σ n+1 ≤ T (resp. τ n ≤ τ n+1 ≤ T ) ; (ii) ξ n (resp. ζ n ) is a random variable with values in Γ 1 (resp. Γ 2 ) which is F σn (resp. F τn )-measurable ;

(iii) P [σ n < T, ∀n ≥ 0] = P [τ n < T, ∀n ≥ 0] = 0 ; (iv) If (A δ s ) s≤T and (B ν s ) s≤T are the F-adapted RCLL processes defined by: Then,

∀ s ∈ [t, T ), A δ s = n≥1 g ξn-1ξn (σ n , X
E[(A δ T ) 2 + (B ν T ) 2 ] < ∞.
For any s ≤ T , A δ s (resp. B ν s ) is the cumulative switching cost at time s for π 1 (resp. π 2 ) when she implements the strategy δ (resp. ν).

Next, for t ∈ IR, i ∈ Γ 1 (resp. j ∈ Γ 2 ), we say that the admissible strategy δ := (σ n , ξ n ) n≥0 (resp.

ν := (τ n , ζ n ) n≥0 ) belongs A i π1 (t) (resp. A i π2 (t)) if σ 0 = t, ξ 0 = i, E[(A δ T ) 2 ] < ∞ (resp. τ 0 = t, ζ 0 = j, E[(B ν T ) 2 ] < ∞).
Given an admissible strategy δ (resp. ν) of π 1 (resp. π 2 ) one associates a stochastic process (u s ) s≤T (resp. (v s ) s≤T ) which indicates along with time the current mode of π 1 (resp. π 2 ) and which is defined by:

∀s ≤ T, u s = ξ 0 1 {σ0} (s) + n≥1 ξ n-1 1 ]σn-1,σn] (s) (resp. v s = ζ 0 1 {τ0} (s) + n≥1 ζ n-1 1 ]τn-1,τn] (s)). (4.3.3) Let now δ = (σ n , ξ n ) n≥0 (resp. ν = (τ n , ζ n ) n≥0
) be a strategy for π 1 (resp. π 2 ) which belongs to A i π1 (t) (resp. A j π2 (t)). The interventions of the players are not free and generate a payoff which is a reward (resp. cost) for π 1 (resp. π 2 ) and whose expression is given by

J t (δ, ν) := E[h u T v T (X T ) + T t f (r, X t,x r , u r , v r )dr -A δ T + B ν T |F t ], (4.3.4)
where, for any (k, l) ∈ Γ 1 × Γ 2 , we set f (s, x, k, l) = f kl (s, x), since f kl is assumed to not depend on ( y, z ij ).

As usual in the literature of zero-sum games, we are interested in the following issue: Does this zero-sum switching game have a value, that is, does the following equality hold? ess inf ν∈A j π 2 (t) ess sup δ∈A i π 1

(t) J t (δ, ν) = ess sup δ∈A i π 1

(t) ess inf ν∈A j π 2 (t) J t (δ, ν) In the remaining part of this section, we focus on this issue.

For later use, let us introduce two new families of auxiliary processes ( Û δ,j ) j∈Γ 2 (resp. ( Û i,ν ) i∈Γ 1 ) associated with a given admissible strategy δ (resp. ν) of π 1 (resp π 2 ). They are defined by: ∀j ∈ Γ 2 ,

             Û δ,j ∈ S 2 d , Ẑδ,j ∈ H 2,d , K -,δ,j ∈ A 2 i ; Û δ,j s = h u(T )j (X t,x T ) + T s f (r, X t,x r , u r , j)dr - T s Z δ,j r dB r -(A δ T -A δ s ) -(K -,δ,j T -K -,δ,j s ), s ∈ [t, T ];
∀s ∈ [t, T ], Û δ,j s ≤ min l =j Û δ,l s + ḡjl (s, X t,x s ) and T t { Û δ,j rmin l =j { Û δ,l r + ḡjl (r, X t,x r )}}dK -,δ,j r = 0. (4.3.5) and for any

i ∈ Γ 1              U i,ν ∈ S 2 d , Z i,ν ∈ H 2,d , K +,i,ν ∈ A 2 i ; U i,ν s = h iv(T ) + T s f (r, X t,x r , i, v r )dr - T s Z i,ν r dB r + (B ν T -B ν s ) + (K +,i,ν T -K +,i,ν s ), s ∈ [t, T ] ; ∀s ∈ [t, T ], U i,ν s ≥ max k =i {U k,ν s -g ik (s, X t,x s )} and T s Û δ,j r -max k =i {U k,ν r -g ik (r, X t,x r )} dK +,i,ν r = 0. (4.3.6)
These equations are actually not of standard form, but by an obvious change of variables one can easily show that they have unique solutions. On the other hand, let us point out that thanks to the connection between the standard switching problem and multidimensional RBSDE with a lower interconnected obstacle (see e.g. [START_REF] Djehiche | A finite horizon optimal multiple switching problem[END_REF] or [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF]) the family ( Û δ,j -A δ ) j∈Γ 2 (resp. (U i,ν + B ν ) i∈Γ 4.3.2 the relationship between the zero-sum switching game and the minmax solution

We now give the main result of this section. It relates (Y ij s ) s≤T = (v ij (s, X t,x s )) s≤T , (i, j) ∈ Γ 1 × Γ 2 , with the value of the zero-sum switching game described above.

(i 0 , j 0 ) ∈ Γ 1 × Γ 2 , v i0j0 (t, x) = Y i0j0 t = ess sup δ∈A i 0 π 1 (t) ess inf ν∈A j 0 π 2 (t) J t (δ, ν) = ess inf ν∈A j 0 π 2 (t) ess sup δ∈A i 0 π 1 (t) J t (δ, ν). ( 4
Y ij,m , Z ij,m , K ij,m ) (i,j)∈Γ 1 ×Γ 2 , m ≥ 0: ∀(i, j) ∈ Γ 1 × Γ 2 ,            Y ij,m ∈ S 2 , Z i,j,m ∈ H 2,d and K ij,m ∈ A 2 i ; Y ij,m s = h ij (X t,x T ) + T s f ij,m (r, X t,x r , (Y kl,m r ) (k,l)∈Γ 1 ×Γ 2 )dr + (K ij,m T -K ij,m s ) - T s Z ij,m r dB r , ∀s ∈ [t, T ]; Y ij,m s ≥ max k∈(Γ 2 ) -i {Y kj,m s -g ik (s, X t,x s )}, ∀s ∈ [t, T ]; T t (Y ij,m s -max k∈(Γ 1 ) -i {Y kj,m s -g ik (s, X t,x s )})dK ij,m s = 0 (4.3.9
) where, we recall that, for any s ∈

[t, T ], m ≥ 0 and (i, j) ∈ Γ 1 × Γ 2 , f ij,m (s, X t,x s , y) = f ij (s, X t,x s ) -m l∈(Γ 2 ) -j y ij -(y il + ḡjl (s, X t,x s )) + .
As already mentioned above, we know that, for any (i, j)

∈ Γ 1 × Γ 2 , Y ij,m → m Y ij in S 2 .
For sake of clarity, we divide the proof into two steps

Step 1: For any (i 0 , j

0 ) ∈ Γ 1 × Γ 2 , Y i0j0 t = ess sup δ∈A i 0 π 1 (t) { Û δ,j0 t -A δ t }. (4.3.10)
Let δ = (σ l , ξ l ) l≥0 be a strategy of A i0 π1 (t). We are going first to show that Y i0j0 t ≥ Û δ,j0 t -A δ t . So let us define the processes (Y δj,m ) j∈Γ 2 and ( Û δ,j,m ) j∈Γ 2 as follows:

(i) ∀j ∈ Γ 2 , ∀s ∈ [t, T ), Y δ,j,m s = l≥0 Y ξ l j,m s 1 1 [σ l ≤s<σ l+1 ] and Y δ,j,m T = h u(T )j (X t,x T ), where, ∀s ∈ [t, T ], Y ξ l j,m s = q∈Γ 1 Y qj,m s 1 1 [ξ l =q] . (4.3.11)
The process Y δ,j,m is well defined since the sum contains only finite many terms since the strategy δ is admissible and then P [σ l < T, ∀l ≥ 0] = 0. On the other hand, at time 0 < σ l < T , Y δ,j,m has a jump which is equal to

Y ξ l j,m σ l -Y ξ l-1 j,m σ l .
(ii) The processes ( Û δ,j,m ) j∈Γ 2 are defined as the solution of the following non standard multi-dimensional BSDE:

∀j ∈ Γ 2 , Û δ,j,m s = h u(T )j (X t,x T ) + T s f (r, X t,x r , u r , j) -m l =j ( Û δ,j,m r -Û δ,l,m r -ḡjl ) + dr -(A δ T -A δ s ) - T s V δ,j,m u dB u , s ∈ [t, T ].
(4.3.12)

Note that ( Û δ,j,m + A δ ) j∈Γ 2 is a solution of a standard multidimensional BSDE whose coefficient is Lipschitz. As those latter processes exist, then so are ( Û δ,j,m ) j∈Γ 2 . On the other hand, as for the system given in (4.1.12), the sequence of processes (( Û δ,j,m ) j∈Γ 2 ) m≥0 converges in S 2 d toward ( Û δ,j ) j∈Γ 2 . We now prove the following: for any m ≥ 0, j ∈ Γ 2 , Y δ,j,m 0 ≥ Û δ,j,m 0 .

(4.3.13)

For any j ∈ Γ 2 let us define K δ,j,m and Z δ,j,m as follows: ∀s ∈ [t, T ],

Z δ,j,m s := l≥0 Z ξ l j,m s 1 1 [σ l ≤s<σ l+1 [ and K δ,j,m s = l≥0 s∧σ l+1 s∧σ l dK ξ l j,m s ,
where, Z ξ l j,m s and K ξ l j,m s are defined in the same way as in (4.3.11). Once more there is no definition problem of those processes since δ is admissible. Therefore the triple of processes (Y δ,j,m , Z δ,j,m , K δ,j,m ) j∈Γ 2 verifies: ∀s ∈ [t, T ),

Y δ,j,m s = Y δ,j,m t - s t f urj (r, X t,x r )dr + m l =j Y δ,j,m r -Y δ,l,m r -ḡjl (r, X t,x r ) + dr + Z δ,j,m r dB r -dK δ,j,m r + l≥1 (Y ξ l j,m σ l -Y ξ l-1 j,m σ l )1 1 [σ l ≤s] = Y δ,j,m t - s t f urj (r, X t,x r )dr + m l =j Y δ,j,m r -Y δ,l,m r -ḡjl (r, X t,x r ) + dr + Z δ,j,m r dB r -dK δ,j,m r -l≥1 (Y ξ l-1 j,m σ l -Y ξ l j,m σ l + g ξ l-1 ξ l (σ l , X t,x σ l ))1 1 [σ l ≤s] + A δ s .
Next, let us define Ãδ,j,m by: Ãδ,j,m

s := l≥1 (Y ξ l-1 j,m σ l -Y ξ l j,m σ l + g ξ l-1 ξ l (σ l , X t,x σ l ))1 1 [σ l ≤s] for s ∈ [t, T ) and Ãδ,j T = lim s→T Ãδ,j s
which is an F-adapted non-decreasing process. As the strategy δ is admissible, then writing backwardly between s ∧ σ k and σ k ∨ s the equation for the process Y δ,j,m and taking the limit k → ∞, we obtain: As the processes Ãδ,j,m and K δ,j,m are non-decreasing then

∀j ∈ Γ 2 , Y δ,j,m s = h u(T )j (X t,x T ) + T s f urj (r, X t,x r )dr -m l =j Y δ,j,m r -Y δ,l,m r -ḡjl (r, X t,x r ) + dr -Z δ,
Y δ,j,m s ≥ Û δ,j,m s , ∀s ∈ [t, T ].
Taking now the limit w.r.t. m, we obtain that

Y i0j t = lim m→∞ Y i0j,m t ≥ lim m→∞ {Y δ,j,m t -A δ t } ≥ lim m→∞ { Û δ,j,m t -A δ t } = Û δ,j t -A δ t , ∀j ∈ Γ 2 .
Step 2: In order to complete the proof of the claim we construct a strategy δ of

A i0 π1 (t) such that Y i0,j0 t = Û δ,j0 t .
Let us first define the strategy δ= (ξ * l , σ * l ) l≥0 as follows: (i) ξ * 0 = i, σ * 0 = t. (ii) Next, for any l ≥ 1, we define σ * l and ξ * l by:

         σ * l = inf s ≥ σ * l-1 , Y ξ * l-1 j0 s = max k =ξ * l-1 Y kj0 s -g ξ * l-1 k (s, X t,x s ) ∧ T, ξ * l ∈ argmax k, k =ξ * l-1 Y kj0 σ * l -g ξ * l-1 k (σ * l , X t,x σ * l ) . (4.3.16)
We first prove that δ verifies P ({ω, ∀ l ≥ 0, σ * l (ω) < T }) = 0. (4.3.17)

We proceed by contradiction. Assume that the last property does not hold. As the set Γ 1 is finite then one can find a loop (i 1 , i 2 , • • • , i l = i 1 ) of exactly l -1 (l ≥ 2) indices and a subsequence (l p ) p≥0 (which may depend on ω) satisfying l p+1l p ≥ l and such that:

P Y i1j0 σ * lp = Y i2j0 σ * lp -g i1i2 (σ * lp , X t,x σ * lp ), • • • , Y i l-1 j0 σ * lp+l-1 = Y i l j0 σ * lp+l-1 -g i l-1 i l (σ * lp+l-1 , X t,x σ * lp+l-1
), ∀p ≥ 0 > 0.

Next, let us set τ * by τ * (ω) := lim p σ * lp (ω), then by taking the limit in the previous equalities we obtain

P Y i1j τ * = Y i2j0 τ * -g i1i2 (τ * , X t,x τ * ), • • • , Y i l-1 j0 τ * = Y i l j0 τ * -g i l-1 i l (τ * , X t,x τ * ) > 0 Since i 1 = i l , we obtain P l-1 k=1 g i k i k+1 (τ * , X t,x τ * ) = 0 > 0
which contradicts to the so called non free loop property and then δ satisfies (4.3.17).

Let us show that E[(A δ T ) 2 ] < ∞. First note that due to the non-free loop property E[(A δ t ) 2 ] < ∞. Next let us introduce the process Y δ,j0 by setting

∀ s ∈ [t, T ), Y δ,j0 s = l≥0 Y ξ * l j0 s 1 {σ * l ≤s<σ * l+1 } and Y δ,j0 T = h u δ (T )j0 , (4.3.18)
where, (u δ (s)) s∈[t,T ] , as in (4.3.12), is the RCLL process associated with δ which indicates the mode of π 1 at time s when the strategy δ is implemented. Next, by the local solution property of Proposition 4.3.2, for any l ≥ 0, we have

       Y ξ * l j0 s = Y ξ * l j0 σ * l+1 + σ * l+1 s f ξ * l j0 (r, X t,x r )dr -(K ξ * l j0,- σ * l+1 -K ξ * l j0,- s ) - σ * l+1 s Z ξ * l j0 r dB r , ∀s ∈ [σ * l , σ * l+1 ]; Y ξ * l j0 s ≤ min p∈(Γ 2 ) -j 0 {Y ξ * l j0 s + ḡj0p (s, X t,x s )}, ∀s ∈ [σ * l , σ * l+1 ]; σ * l+1 σ * l (Y ξ * l j0 s -min p∈(Γ 2 ) -j 0 {Y ξ * l p s + ḡj0p (s, X t,x s )})dK ξ * l j0,- u = 0, (4.3.19
) where, Z ξ * l j0 and K ξ * l j0,are fixed processes which depend actually on σ * l , for all l ≥ 0. Let us now define Z δ,j0 and K δ,j0,-by:

Z δ,j0 s := l≥0 Z ξ * l j0 s 1 1 [σ * l ≤s<σ * l+1 [ and K δ,j0,- s := l≥0 s∧σ * l+1 s∧σ * l dK ξ * l j0,- s , s ∈ [t, T ].
We note that, by definition, we have, for any l ≥ 0,

{Y ξ * l j σ * l+1 -Y ξ * l+1 j σ l+1 + g ξ * l ξ * l+1 (σ * l+1 , X t,x σ * l+1 )}1 {σ * l+1 <T } = 0.
Then, taking into account the jump of Y δ,j0 at σ * l+1 (when smaller than T ) which is equal to

Y ξ * l+1 j0 σ * l+1 - Y ξ * l j0 σ * l+1
and by (4.3.17), we have, for every s ∈

[t, T ], Y δ,j0 s = h u δ (T )j0 (X t,x T ) -(A δ T -A δ s ) + T s f (r, X t,x r , u δ r , j 0 )dr -(K δ,j0,- T -K δ,j0,- s ) - T s Z δ,j0 r dB r , (4.3.20) which implies that Y δ,j0 s = Y δ,j0 t + (A δ s -A δ t ) + K δ,j0,- s - s t f (r, X t,x r , u δ r , j 0 )dr + s t Z δ,j0 r dB r , ∀s ∈ [t, T ].
As Y δ,j0 belongs to S 2 d then a localization procedure and Fatou's Lemma permit to deduce that

E[A δ T + K δ,j0,- T ] < ∞.
Thus, for any s ∈

[t, T ], E[(A δ T -A δ s ) +(K δ,j0,- T -K δ,j0,- s )|F s ] = E[h u δ (T ),j0 (X t,x T ) + T s f (r, X t,x r , u δ r , j 0 )dr|F s ] -Y δ,j0 s = E[h u δ (T ),j0 (X t,x T ) + T t f (r, X t,x r , u δ r , j 0 )dr|F s ] -{Y δ,j0 s + s t f (r, X t,x r , u δ r , j 0 )dr}. ( 4
Y δ,j0,m s -E[( Ãδ ,j0,m T - Ãδ ,j0,m s ) + (K δ,j0,m T -K δ,j0,m s )|F s ] = Û δ,j0,m s , ∀s ∈ [t, T ]. (4.3.22) Therefore, the process (Y δ,j0,m s - Û δ,j0,m s ) s≤T is a supermartingale which satisfies sup m≥0 E[sup s≤T |Y δ,j0,m s - Û δ,j0,m s | 2 ] < ∞, since, for any (i, j) ∈ Γ 1 × Γ 2 , sup m≥0 E[sup s≤T {|Y ij,m s | + | Û ij,m s |} 2 ] < ∞.
Once more, by a Dellacherie-Meyer's result ( [START_REF] Dellacherie | Probabilités et Potentiel[END_REF], pp. 220-221), we obtain

sup m≥0 E[{ Ãδ ,j0,m T + K δ,j0,m T } 2 ] < ∞. ( 4
{Y ξ * l j0 σ * l+1 -Y ξ * l+1 j0 σ l+1 + g ξ * l ξ * l+1 (σ * l+1 , X σ * l+1 )}1 σ * l+1 <T = 0.
As the strategy δ is admissible (i.e. for ω fixed there is only a finite many σ * l such that σ * l < T ) and Y ij0,m ց Y ij0 as m → ∞ in S

→ 0 in L 1 (dP ).
Next, we shall show that there exists a subsequence of {m} which we still denote by {m} such that for any l ≥ 0, the random variable

p=l p=0 σ * p+1 σ * p dK ξ * p j0,m s → m 0 weakly in L 2 (dP ).
To begin with, by using (4.2.17), let {m} be a subsequence such that for any (i,

j) ∈ Γ 1 × Γ 2 (m l =j (Y ij,m r -Y il,m r -ḡjl (r, X t,x r )) + ) r∈[t,T ] converges weakly in H 2,d to a process (α ij r ) r∈[t,T ] .
We only consider the sequence (

σ * 1 σ * 0 dK ξ * 0 j0,m s ) m≥0 = ( σ * 1 t dK ij0,m s
) m≥0 since for the other cases a similar procedure applies (keep in mind that we should have P [σ * 1 > σ * 0 ] > 0, otherwise this case is irrelevant and then one should begin with the next case, i.e, taking p = 1). For s ∈ [t, σ * 1 ] and from (4.3.9) we have

Y ij0,m s = Y ij0,m σ * 1 + σ * 1 s f ij0 (r, X t,x r )dr -m σ * 1 s l =j0 (Y ij0,m r -Y il,m r -ḡj0l (r, X t,x r )) + dr - σ * 1 s Z ij0,m r dB r + K ij0,m σ * 1 -K ij0,m s .
As Y ij0,m converges to Y ij0 in S 2 , by Itô's formula, we have:

(i) sup m≥0 E[(K ij0,m σ * 1 ) 2 ] < ∞; (ii) the sequence (Z ij0,m s 1 1 [s≤σ * 1 ]
) converges in H 2,d to some process Zij0 . Now, for s ≤ σ * 1 , define

K ij0 s = Y ij0 t -Y ij0 s + s t Zij0 r dB r + s t α ij0 r dr - s t f ij0 (r, X t,x r )dr.
Then, the process K ij0 is continuous on [t, σ * 1 ]. Moreover, using the weak convergence pointed out previously, for any any stopping time τ

∈ [t, σ * 1 ], K ij0,m τ → m K ij0 τ weakly in L 2 (dP ).
Next, let τ be a stopping time such that t ≤ τ < σ * 1 . The properties of K ij0,m (especially the Skorokhod condition) combined with the uniform convergence of (Y ij,m ) m to Y ij and the definition of σ * 1 , i.e.,

∀s < σ * 1 , Y ij0 s > max k =i {Y kj0 s -g ik (s, X t,x s )}
imply the existence of some m 0 (ω) such that if m ≥ m 0 then K ij0,m τ = 0. Therefore, the sequence (K ij0,m τ ) m converges Pa.s. to 0 and by (i) above it converges also in L 2-ǫ (dP ) to 0 and then K ij0 τ = 0. Finally by continuity we have K ij0 s = 0 for any s ∈ [t, σ * 1 ] and then the sequence (

σ * 1 t dK i,j0,m s
) m≥0 converges weakly in L 2 (dP ) to 0. As we can do the same for the other sequences, the claim holds.

Let l be fixed. By using (4.3.22) between t and σ * l one obtains:

Y δ,j0,m t -E[Y δ,j0,m σ * l + ( Ãδ ,j0,m σ * l - Ãδ ,j0,m t ) + K δ,j0,m σ * l |F t ] = Û δ,j0,m t -E[U δ,j0,m σ * l |F t ].
Taking now the weak limit w.r.t. m (at least through the subsequence constructed above) we obtain that

Y δ,j0 t -E[Y δ,j0 σ * l |F t ] = Û δ,j0 t -E[U δ,j0 σ * l |F t ].
Finally, taking the limit as l → ∞, noting that Y δ,j0

T = U δ,j0 T = h u δ (T )j0 , we obtain Y δ,j0 t = Û δ,j0 t .
Thus, in view of the definition of δ, we have

Y i0,j0 t = Y δ,j0 t -A δ t = Û δ,j0 t -A δ t .
Now, taking into account of (4.3.7), the first equality holds.

Finally, in order to obtain the second equality of (4.3.8), it is enough to consider the approximating increasing scheme (which is the opposite of (4.3.9)) and which can be transformed into a decreasing scheme by taking its opposite sign. Then, from the result of Step 1, we have

-v i0j0 (t, x) = -Y i0j0 t = ess sup ν∈A j 0 π 2 ess inf δ∈A i 0 π 1 -J t (δ, ν)
and the proof is finished. As a by product of Theorem (4.3.1) and the uniqueness of the solution of system (4.1.9) we have the following result in the case when the functions f ij depend also on y.

Corollary 4.3.1. Suppose Assumptions (H0)-(H5) and (H6)-(i) are satisfied and let (v ij ) (i,j)∈Γ 1 ×Γ 2 be the unique solution of system (4.1.9) and (4.1.10). Then for any (t, x) ∈ [0, T ]×IR k and (i 0 , j

0 ) ∈ Γ 1 ×Γ 2 , v i0j0 (t, x) = ess sup δ∈A i 0 π 1 ess inf ν∈A j 0 π 2 Jt (δ, ν) = ess inf ν∈A j 0 π 2 ess sup δ∈A i 0 π 1 Jt (δ, ν). (4.3.24)
where,

Jt (δ, ν) := E[h u T v T (X T ) + T t f urvr (r, X t,x r , (v kl (r, X t,x r )) (k,l)∈Γ 1 ×Γ 2 )dr -A δ T + B ν T |F t ]. ( 4

.3.25)

Proof: Let (w ij ) (i,j)∈Γ 1 ×Γ 2 be the unique solution in viscosity sense of the following system of PDEs with inter-connected obstacles:

∀(i, j) ∈ Γ 1 × Γ 2 ,        min (w ij -L ij ( w))(t, x); max (w ij -U ij ( w))(t, x); -∂ t w ij (t, x) -L X (w ij )(t, x) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 ) = 0; w ij (T, x) = h ij (x). ( 4
∀(i 0 , j 0 ) ∈ Γ 1 × Γ 2 , v i0j0 (t, x) = sup δ∈A i 0 π 1 (t) inf ν∈A j 0 π 2 (t) E[ Jt (δ, ν)] = inf ν∈A j 0 π 2 (t) sup δ∈A i 0 π 1 (t) E[ Jt (δ, ν)]. (4.3.28)

Conclusion

In this paper, we have given appropriate conditions on the data of both the min-max and max-min systems so that their respective unique viscosity solutions coincide. These unique continuous viscosity solution have been constructed by means of a penalization procedure in the recent paper [START_REF] Djehiche | Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles[END_REF]. The main difficulty faced in that paper is that the two obstacles are interconnected and therefore not comparable. For this reason and without the separation of the two barriers, we cannot apply the classical relationship between doubly reflected BSDEs, system of PDEs with lower and upper obstacles and the underlying game obtained e.g. in [START_REF] Hamadene | BSDEs with two reflecting barriers driven by a Brownian and a Poisson noise and related Dynkin game[END_REF]). By providing appropriate regularity conditions so that comparison holds, we establish in the present paper that the solutions of the Min-Max and Max-Min systems coincide.

Finally, under further conditions on the drivers, this solution can be interpreted as the value function of a switching game.

We note that to obtain the required condition of comparison, we rely on the regularity of penalty costs. We also need to get precise estimates of penalized terms which can be obtained by controlling the growth of the driver. Our analysis deeply relies on the Markovian setting, therefore it seems quite natural to ask whether one can study the switching game in the general non-Markovian case. We leave this question for future research.

Therefore

Y Γ,j t0 = Y Γ,j θ1 + θ1 t0 f j (r, X t,x r , -→ Γ r , U Γ,j r )dr - ∞ i=1 θ1 t0 U j,i r dH (i) r + (K Γ,j θ1 -K Γ,j t0 ) ≥ (Y Γ,α1 θ1 -g i,α1 (θ 1 , X t,x θ1 ))1 [θ1<T ] + 1 [θ1=T ] h α0 (X t,x T ) + θ1 t0 f a(r) (r, X t,x r , -→ Γ r , U a r )dr - ∞ i=1 θ1 t0 U a,i r dH (i) r + (K Γ,j θ1 -K Γ,j t0 ) = Y Γ,α1 θ2
1 [θ1<T ] + θ2 t0 f a(r) (r, X t,x r , -→ Γ r , U a r )dr - ∞ i=1 θ2 t0 U a,i r dH (i) r + (K Γ,i θ1 -K Γ,i t0 ) + (K Γ,α1 θ2 -K Γ,α1 θ1 ) -g i,α1 (θ 1 , X t,x θ1 )1 [θ1<T ]
. Repeat now this procedue as many times as necessary and since a is an admissible startegy (i.e. P[θ n < T, ∀n ≥ 0] = 0) we obtain:

Y Γ,j t0 ≥ h a(T ) (X t,x T ) + T t0 f a(r) (r, X t,x r , -→ Γ r , U a r )dr - ∞ i=1 T t0 U a,i r dH (i) r -A a T + Ka T . (5.1.4)
As Ka T ≥ 0 and by (5.1.2) we have

Y Γ,j t0 -V a t0 + A a t0 ≥ T t0 (f a(r) (r, X t,x r , -→ Γ r , U a r ) -f a(r) (r, X t,x r , -→ Γ r , N a r ))dr - ∞ i=1 T t0 (U a,i r -N a,i r )dH (i) r ≥ T t0 V a,U a ,N a , U a -N a p s ds - ∞ i=1 T t0 (U a,i r -N a,i r )dH (i) r
Next by Girsanov's Theorem ( [START_REF] Protter | Stochastic integration and differential equations[END_REF], pp.136), under the probability measure d

P := ε( ∞ j=1 • t0 V a,U a ,N a ,i r dH (i) r ) T dP, (M t := t t0 V a,U a ,N a , U a -N a p s ds - ∞ i=1 t 0 (U a,i r -N a,i r )dH (i)
r ) t∈[t0,T ] is a martingale, and by taking conditional expectation of Y Γ,j t0 -V a t0 + A a t0 , we obtain

E P [Y Γ,j t0 -V a t0 + A a t0 |F s ] ≥ E P [ T t0 V a,U a ,N a , U a -N a p s ds - ∞ i=1 T t0 (U a,i r -N a,i r )dH (i) r |F t0 ] = 0.
Thus Y Γ,j t0 ≥ V a t0 , Pa.s. and then, since P and P are equivalent, for any a ∈ A j t0 , Y Γ,j t0 ≥ V a t0 -A a t0 , Pa.s..

Next let us consider a * the strategy defined by a *

(r) = α * 0 {t0} (r) + ∞ j=1 α * j-1 ]θ * j-1 θ * j ] (r), r ≤ T , where θ * 0 = t 0 , α * 0 = j and for n ≥ 0, θ * n+1 = inf{r ≥ θ * n , Y Γ,α * n r = max k∈A α * n (Y Γ,k r -g α * n ,k (r, X t,x r ))} ∧ T, and
α * n+1 = arg max k∈A α * n {Y Γ,k θ * n+1 -g α * n ,k (θ * n+1 , X t,x θ * n+1 )}.
Let us show that a * ∈ A j s . We first prove that P[θ * n < T, ∀n ≥ 0] = 0. We proceed by contradiction assuming that P[θ * n < T, ∀n ≥ 0] > 0. By definition of θ * n , we then have

P[Y Γ,α * n θ * n+1 = Y Γ,α * n+1 θ * n+1 -g α * n ,α * n+1 (θ * n+1 , X t,x θ * n+1 ), α * n+1 ∈ A α * n , ∀n ≥ 0] > 0.
But A is finite, then there is a loop i 0 , i 1 , • • • , i k , i 0 (i 1 = i 0 ) of elements of A and a subsequence (n q (ω)) q≥0 such that:

P[Y Γ,i l α * n q+l = Y Γ,i l+1 α * n q+l -g i l ,i l+1 (α * n q+l , X t,x α * n q+l ), l = 0, • • • , k, (i k+1 = i 0 ), ∀q ≥ 0] > 0. (5.1.6) t0 f a * (r) (r, X t,x r , -→ Γ r , U a * r )dr - ∞ k=1 θ * 2 t0 U a * ,k r dH (k) r -A a * θ * 2 .
(5.1.9)

Repeating now this procedure as many times as necessary and since P[θ * j < T, ∀j ≥ 0] = 0 to get

Y Γ,j s = h a * (T ) (X t,x T ) + T t0 f a * (r) (r, X t,x r , -→ Γ r , U a * r )dr - ∞ k=1 T t0 U a * ,k r dH (k) r -A a * T . (5.1.10) Now since Γ ∈ [H 2 ] m , U a * ∈ H 2 (l 2 ) and Y Γ,j ∈ S 2 , we deduce from (5.1.10) that E[(A a * T ) 2 ] < ∞. Next by (5.1.2), V a * t0 -A a * t0 -Y Γ,j t0 = T t0 f a * (r) (r, X t,x r , -→ Γ r , N a * r )dr - T s f a * (r) (r, X t,x r , -→ Γ r , U a * r )dr - ∞ k=1 T t0 (N a * r -U a * ,k r )dH (k) r ≥ T t0 V a * ,N a * ,U a * , N a * -U a * p r dr - ∞ k=1 T t0 (N a * ,k r -U a * ,k r )dH (k) r .
Once more using Girsanov's theorem as previously, to obtain

E P [V a * t0 -A a * t0 -Y Γ,i t0 |F t0 ] ≥ 0 and then V a * t0 -A a * t0 -Y Γ,i t0
≥ 0, Pa.s. Taking now into account (5.1.5) leads to the desired result. Remark 5.1.1. As a by product of (5.1.3) we have also:

∀j ∈ A, E[Y Γ,j t0 ] = sup a∈A j t 0 E[V a t0 -A a t0 ].
the equation (5.3.1) can be expressed as a BSDE with two reflecting barrier in the following manner. For all t ≤ T ,

         (Y n , Z n , U n , K n ) ∈ S 2 × H 2 × H 2 ( N ) × A 2 ; Y n s = ξ - T s g r dr + T s Z n r dB r + T s E U n r (e) N (drde) + K n T -K n s -(K n,- T -K n,- s ) L s ≤ Y n s ≤ Y n s ∨ H s , s ≤ T, T 0 (Y n s -L s )dK n s = T 0 (Y n s -Y n s ∨ H s )dK n,- s = 0, (5.3.2)
Thus, a result by Hamadène and Hassani [START_REF] Hamadene | BSDEs with two reflecting barriers driven by a Brownian and a Poisson noise and related Dynkin game[END_REF] allows to represent Y n as a value function of a Dynkin game, i.e., it holds true that for any t ≤ T ,

Y n t = ess sup σ≥t ess inf τ ≥t E[ σ∧τ t g s ds + L σ [σ<τ ] + (Y n τ ∨ H τ ) [τ ≤σ<T ] + ξ [τ =σ=T ] |F t ] = ess inf τ ≥t ess sup σ≥t E[ σ∧τ t g s ds + L σ [σ<τ ] + (Y n τ ∨ H τ ) [τ ≤σ<T ] + ξ [τ =σ=T ] |F t ],
where τ and σ are F -stopping times. Now let (Y s , Z s , U s , K + s , K - s ) s≤T be the solution of the BSDE with two reflecting barriers associated with (g(s, X t,x s , y, z, u), h(X t,x T ), L(s, X t,x s ), H(s, X t,x s ), which exists and is unique by Theorem 2.2 [START_REF] Hamadene | BSDEs with two reflecting barriers driven by a Brownian and a Poisson noise and related Dynkin game[END_REF].(Their function g does not depend on u, the reason is to use comparison theorem, but in our case, if g satisfies the assumption (A1), we have also the comparison theorem by [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF].) For n ≥ 0, let ( n Y s ) s≤T (resp.( n Ȳs ) s≤T ) be the first component of the unique solution of the BSDE with on reflecting lower (resp. upper) barrier associated with (g(s, X t,x s , y, z, u)n(H(s, X t,x sy) -, h(X t,x T ), L(s, X t,x s )) (resp. (g(s, X t,x s , y, z, u)n(L(s, X t,x s y) -, h(X t,x T ), H(s, X t,x s )))). It has been shown in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF], that for any n ≥ 0 there exist functions n u(t, x) and n ū, (t, x) ∈ [0, T ] × R, such that ∀s ∈ [t, T ], n Y s = n u(s, X t,x s ) and n Ȳs = n ū(s, X t,x s ),

where n u (resp. n ū) is continuous with polynomial growth and be a viscosity solution for the following obstacle prblem: where, δ t,x := inf {s ≥ t, Y s = H(s, X t,x s )} ∧ T and θ t,x := inf {s ≥ t, Y s = L(s, X t,x s )} ∧ T . Proof. First we show that u is a viscosity subsolution of (5.5.4). Note that u(T, x) = h(x) and L(t, x) ≤ u(t, x) ≤ H(t, x), for any φ ∈ C 1,2 ([0, T ] × R) Π g and for any local maximum point (t, x) ∈ [0, T ] × R of uφ such that u(t, x) > L(t, x), let (t n , x n ) be a sequence of local maximum points of n uφ such that (t n , x n ) converges to (t, x). For n large enough we have n u(t n , x n ) > L(t n , x n ), then we have ≤ 0.

   min{v(t,
-∂ t φ(t n , x n ) -b(t n , x n )D x φ(t n , x n ) + 1 2 σ 2 (t n , x n )D
In a similar way we can show that u is also a viscosity supersolution.

Viscosity solution of system of variational inequalities with interconnected bilateral obstacles and connections to multiple modes switching game of jump-diffusion processes

First let A := {1, • • • , m}, and let us introduce the following functions f i , h i and g ij , i, j ∈ A:

g i : [0, T ] × IR k × IR m × IR -→ IR (t,
x, (y i ) i=1,m , u) -→ g i (t, x, (y i ) i=1,m , u) h i (resp. g ij ) : [0, T ] × IR k -→ IR (t, x) -→ h i (t, x) (resp. g ij (t, x))

which satisfy:

Assumption (H1) (I) For any i ∈ A, g i (t, x, -→ y , z, q) : R × R × R m × R × R → R, (i) is continuous in (t, x) uniformly w.r.t. the other variables ( -→ y , z, q) and for any (t, x) the mapping (t, x) → g i (t, x, 0, 0, 0) is of polynomial growth.

(ii) satisfies the standard hypothesis of Lipschitz continuity w.r.t. the variables ( -→ y , z, q), i.e. ∀(t, x) ∈ [0, T ] × R, ∀( -→ y 1 , -→ y 2 ) ∈ R m × R m , (z 1 , z 2 ) ∈ R × R, (q 1 , q 2 ) ∈ R × R, |g i (t, x, -→ y 1 , z 1 , q 1 )g i (t, x, -→ y 2 , z 2 , q 2 )| ≤ C(| -→ y 1 --→

y 2 | + |z 1 -z 2 | + |q 1 -q 2 |),
where, | -→ y | stands for the standard Euclidean norm of -→ y in R m .

(iii) q → g i (t, x, y, z, q) is non-decreasing, for all (t, x, y, z) (iv) Monotonicity: For any i ∈ A and any k = i the mapping y k → g i (t, x, y 1 , • • • , y k , • • • , y m , z, u) is non-decreasing.

∈ [0, T ] × R × R m × R.
(II) ∀i ∈ A, g ii ≡ 0 and for i = j, g ij (t, x) is non-negative, continuous with polynomial growth and satisfy the following non-free loop property: ∀(t, x) ∈ [0, T ] × R and for any sequence of indices i 1 , • • • , i k such that i 1 = i k and card{i 1 , • • • , i k } = k -1 we have:

g i1i2 (t, x) + g i2i3 (t, x) + • • • + g i k i1 (t, x) > 0, ∀(t, x) ∈ [0, T ] × IR k .
(III) ∀i ∈ A, h i is continuous with polynomial growth and satisfies the following coherance conditions: h i (x) ≥ max j∈A -i (h j (x)g ij (T, x)), ∀x ∈ IR.

Now let (t, x) ∈ [0, T ] × IR k and let us consider the following system of reflected BSDEs with oblique reflection: ∀j = 1, ..., m Proof. It will be given in two steps.

           Y j ∈ S 2 , Z j ∈ H 2 , U j ∈ H 2 ( N ), K j ∈ A 2 Y j s = h j (X
Step 1: Let us consider the following BSDEs : First note that the above BSDEs have unique solutions thanks to Theorem 2.1 in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]. For j = 1, • • • , m and n ≥ 1, let us define (Y j,n , U j,n , K j,n ) by:

Ȳs = max
              
Y j,n ∈ S 2 , Z j,n ∈ H 2 , U j,n ∈ H 2 ( N ), K j,n ∈ A 2 Y j,0 = Y s Y j,n s = h j (X t,x T ) + g jk (r, X t,x r ))]dK j,n r = 0.

(5.4.2) By induction on n we can show that system (5.4.2) has a unique soluton for any fixed n ≥ 1. On the other hand it is easy to show that ( Ȳ , Ū , 0) is also a solution of : | Ȳs | 2 ] < ∞. Then sequence (Y j,n ) n≥0 , has a limit which we denote by Y j , for any j ∈ A. By the monotonic limit theorem in [START_REF] Essaky | Reflected backward stochastic differential equation with jumps and RCLL obstacle[END_REF](Theorem 3.1, pp697), Y j ∈ S 2 and there exist Z j ∈ H 2 , U j ∈ H 2 ( N ), K j ∈ A 2 such that      Y j s = h j (X t,x T ) + where for any j ∈ A, U j is the weak limit of (U j,n ) n≥1 in H 2 ( N ), Z j is the weak limit of Z j,n in H 2 and for any stopping time τ , K j τ is the weak limit of K j,n τ in L 2 (Ω, F τ , P). Finally note that K j is predictable since the processes K n,j are so, ∀n ≥ 1. Let us now consider the following RBSE: {Y k sg jk (s, X t,x s )}dK j s = 0.

   Ȳs = max
               Ŷ j ∈ S 2 , Ẑj ∈ H 2 , Û j ∈ H 2 ( N ), Kj ∈ A 2 Ŷ j s = h j (X
(5.4.4)

According to Theorem 1.2 in [START_REF] Hamadène | Reflected backward stochastic differential equation with jumps and random obstacle[END_REF], this equation has a unique solution. Apply Tanaka-Meyer's formula (see e.g. [START_REF] Protter | Stochastic integration and differential equations[END_REF] on ( Ŷ j t -Y j t ) + in [s, T ], we can prove that P-a.s., Ŷ j ≤ Y j for any j ∈ A. On the other hand, since ∀n ≥ 1, ∀j ∈ A, Y j,n-1 ≤ Y j , we have max k∈Aj (Y k,n-1 s g jk (s, X t,x s )) ≤ max k∈Aj (Y k sg jk (s, X t,x s )), ∀s ≤ T.

Then by Comparison Theorem, we obtain Y j,n ≤ Ŷ j , thus Y j ≤ Ŷ j which implies Y j = Ŷ j , ∀j ∈ A.

Next using Itô's formula with Y j -Ŷ j and taking expectation in both-hand sides, we obtain It implies that Z j = Ẑj , U j = Û j , dt ⊗ dP and finally K j = Kj for any j ∈ A, i.e. (Y j , Z j , U j , K j ) j∈A verify (5.4.4).

Next we will show that the predictable process K j does not have jumps. So assume there exists j 1 ∈ A and a predictable stopping time τ such that △Y j1 τ = -△ K j1 τ = -△ Kj1 τ < 0. Then by the second equality in (5.4.4) we have

Y j1 τ -= max k∈A -j 1
(Y k τ -g j1k (τ, X t,x τ -)).

(5.4.5)

Now let j 2 ∈ A -j1 be the optimal index in (5.4.5), i.e., Y j2 τ -g j1,j2 (τ, X t,x τ ) = Y j1 τ -> Y j1 τ ≥ Y j2 τg j1,j2 (τ, X t,x τ ).

Note that g j1,j2 (τ, X t,x τ -) = g j1,j2 (τ, X t,x τ ) since the stopping time τ is predictable, and the process (X t,x s ) t≤s≤T does not have predictable jump. Thus △Y j2 τ < 0 and once more we have,

Y j2 τ -= max k∈A -j 2
(Y k τ -g j2k (τ, X t,x τ -)).

(5.4.6)

We can now repeat the same argument as many times as necessary, and then we deduce the existence of a loop ℓ 1 , ..., ℓ p-1 , ℓ p = ℓ 1 and l 2 = l i such that

Y ℓ1 τ -= Y ℓ2 τ --g ℓ1,ℓ2 (τ, X t,x τ -), • • • , Y ℓp-1 τ - = Y ℓp τ --g ℓp-1,ℓp (τ, X t,x τ -).
Therefore g ℓ1,ℓ2 (τ, X t,x τ -) + • • • + g ℓp-1,ℓp (τ, X t,x τ -) = 0 which is contradictory with Assumption (A4)(II). It implies that ∆K j1 τ = 0 and then K j1 is continuous since it is predictable. As j is arbitrary in A, then the processes K j is continuous and taking into account (5.4.4), we deduce that the triples (Y j , Z j , U j , K j ), j ∈ A, is a solution for system (5.4.1).

Next for s ≤ T let us define the process (a s ) s≤T for:

a s := α 0 {θ0} (s) + ∞ j=1 α j-1 ]θj-1θj ] (s), (5.4.7)
where {θ j } j≥0 is an increasing sequence of stopping times with values in [0,T] (θ 0 = 0) and for j ≥ 0, α j is a random variable F θj -measurable with values in A = {1, ..., m}. If P[lim n θ n < T ] = 0, then the pair {θ j , α j } j≥0 is called an admissible strategy of switching . Next we denote by (A a s ) s≤T the switching cost process associated with the process a, which is defined as following: N a r (e) N (dedr) -A a (T, X t,x T ) + A a (s, X t,x s ).

As the same way in [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], we can prove that Y j s = V a * s = ess sup which is a norm of H 2 . Let Γ 1 and Γ 2 be two processes in the Banach space (([H 2 ]) m , . 2,β ), and for k = 1, 2, let (Y k,j , U k,j , K k,j ) j∈A satisfy:

       Y k,j s = h j (X t,x
T ) + T s f j (r, X t,x r , Γ k r , Z j,k r , U j,k r )dr - Let (a s ) t≤s≤T be a admissible strategy defined in (5.4.7) and define Ŷ i s , V a s , V j,a s as follows:

      
Ŷ j s = h j (X t,x T ) + T s f j (r, X t,x r , Γ 1 r , Ẑj r , Û j r ) ∨ f j (r, X t,x r , Γ 2 r , Ẑj r , Û j r )dr - T s e βr △ V a * r-[f a * (r, X t,x r , Γ 1 r , Ẑa * r , N a * r ) ∨ f a * (r, X t,x r , Γ 2 r , Ẑa * r , N a * r )

f a * (r, X t,x r , Γ 1 r , Ẑ1,a * Under Assumptions (H1), there exist deterministic lower semi-continuous functions (u j (t, x)) j∈A of polynomial growth such that ∀(t, x) ∈ [0, T ] × IR k , ∀s ∈ [t, T ], Y j s = u j (s, X t,x s ), ∀j ∈ A.

The proof is same as Corollary 3.1 in [START_REF] Hamadène | Systems of integro-PDEs with interconnected obstacles and multimodes switching problem driven by Lévy process[END_REF]. Now let us consider now the following system of IPDEs: ∀i ∈ A,

         min{u i (t, x) -max j∈Ai (u j (t, x) -g ij (t, x));
-∂ t u i (t, x) -Lu i (t, x)g i (t, x, (u j ) m j=1 (t, x), σ(t, x)D x u i (t, x), I B i (t, x, u i (t, x, )))} = 0, (t, x) ∈ [0, T ] × [0, T ] × IR k ; u i (T, x) = h i (x) (5.4.13) where Lu(t, x) = L 1 u(t, x) + I(t, x, u) Proof. (i) The existence follows from Theorem 5.5.1 and uniqueness follows from Proposition 5.5.1.

(ii) The construction of the function u implies that w = -u is the unique viscosity solution in the class Π g of the following system:

      
min w(t, x) + H(t, x), min w(t, x) + L(t, x), -∂ t w(t, x) -Lw(t, x) + f (t, x, -w(t, x), -σ(t, x) ⊤ D x w(t, x)) = 0; w(T, x) = -g(x).

(5.5.8)

Thus -w = u is the unique solution in the class Π g of system (5.5.8) (see e.g. [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF], pp.18).

In terms of BSDEs the process Y t,x defined in (5.5.3) is a local solution for the two barriers reflected BSDE associated with (f (s, X t,x s , y, z), g(X t,x T ), L(s, X t,x s ), H(s, X t,x s )). Namely we have the following result: Proposition 5.5.2. ([28], Theorem 3.1) For any stopping time τ , there exists another stopping time δ τ ≥ τ , Pa.s. and three processes Z τ , K ±,τ such that: (i) Y t,x T = g(X t,x T ) ; (ii) (5.5.9) (iii) Let γ τ and θ τ be the following two stopping times: γ τ := inf{s ≥ τ, Y t,x s = L i,j (s, X t,x s )} ∧ T and θ τ := inf{s ≥ τ, Y t,x s = U (s, X t,x s )} ∧ T.

       Z τ ∈ H
Then Pa.s., γ τ ∨ θ τ ≤ δ τ . The process Y t,x is unique to satisfy (i)-(iii).

Finally in the case when f does not depend on z we have the following charaterization of Y t,x as the value function of a zero-sum Dynkin game. f (r, X t,x r , Y t,x r )dr +L(σ, X t,x σ )1 1 [σ<τ ] + H(τ, X t,x τ )1 1 [τ ≤σ<T ] + g(X t,x T )1 1 [τ =σ=T ] |F θ = ess inf τ ≥θ ess sup σ≥θ E σ∧τ θ f (r, X t,x r , Y t,x r )dr +L(σ, X t,x σ )1 1 [σ<τ ] + H(τ, X t,x τ )1 1 [τ ≤σ<T ] + g(X t,x T )1 1 [τ =σ=T ] |F θ .

(5.5.10)

Problèmes de Switching Optimal, Equation Différentielles Stochastiques Rétrogrades et Equations Différentielles Partielles

Xuzhe ZHAO

Abstract

1 2 < ∞}; Definition 1 . 1 . 1 .

 1111 Let ξ ∈ L 2 m (F T ) be an R m -valued terminal condition and g(t, ω, y, z):[0, T ] × Ω × R m ×R m×d → R m , P m ⊗B(R m ×R m×d )-measurable.A solution for the m-dimensional BSDE associated with parameters (g, ξ) is a pair of progressively measurable processes (Y,Z) := (Y t , Z t ) t≤T with values in R m × R m×d such that Y ∈ S 2 m , Z ∈ H 2 m×d ; Y t = ξ + T t g(s, Y s , Z s )ds -T t Z s dB s , ∀0 ≤ t ≤ T.(1.1.1)

  (H2): (i) b and σ are uniformly Lipschitz continuous with respect to x; (ii) there exists a constant C s.t. for any (s, x),|σ(s, x)| + |b(s, x)| ≤ c(1 + |x|); (iii) The function g : [0, T ] × R d × R m × R m×d → R m is uniformly Lipschitz in (y, z) with Lipschitz constant C, i.e., |g(s, x, y 1 , z 1 )g(s, x, y 2 , z 2 )| ≤ C(|y 1y 2 | + |z 1z 2 |);(iv) There exists two constants c and p ≥ 0 such that, |g(s, x, y, z)| + |Ψ(x)| ≤ c(1 + |x| p );

  |h(x)| + |Ψ(t, x)| + |f (t, x, 0, 0)| ≤ C(1 + |x| p ).

i≥1 sup t≤T |θ i t | 2 <

 2 ∞ Pa.s.,

  , ∀(s, j) ∈ [0, T ] × {1, • • • , m} and the uniqueness follows (see Appendix Theorem 5.1.1).

Theorem 2 . 3 . 3 .

 233 Assume that Assumptions [A1]-[A3], [A5] are fulfillef and (-f j ) j∈A verify [A4]. Then the system of IPDEs (2.2.20) has a unique continuous and of polynomial growth solution which is moreover unique.

0

  and (3.3.6), it implies that the sequence lim ǫ→0 (t 0 , x 0 , y 0 ) = (t * , x * , x * ) (3.3.7) and once more from (3.3.6) we deduce lim ǫ→0 |x 0y 0 | 2 ǫ = 0. (3.3.8)

  .3.10) and w i0,j0 (t 0 , y 0 ) < min l∈A 2 j 0

( 4 . 1 . 8 )

 418 Under Assumptions (H0)-(H4), we have Theorem 4.1.1. ([17], Theorems 5.4 and 5.5) There exists a unique continuous viscosity solution in the class Π g

( 4 .

 4 1.10)

Remark 4 . 2 . 1 .

 421 By Itô's formula, for ant

  (τ n ) n≥0 is an increasing sequence of stopping times such that P [τ n < T, ∀n ≥ 0] = 0; (b) ∀n ≥ 0, ζ n is a random variable with values in Γ 1 and F τn -measurable; (c) Let (A δ s ) s≤T be the RCLL F t -adapted process defined by ∀s ∈ [0, T ), A δ s = n≥1 g ζn-1ζn (τ n , X t,x τn )1 1 {τn≤s} and A a T = lim s→T A a s .

  admissible strategy such that τ 0 = θ and ζ 0 = i . Now, for an admissible strategy δ = (τ n , α n ) n≥0 , or equivalently a, let us define the pair of processes ( Ǔ aj,m , V aj,m ) which belongs to S 2 d × H 2,d solution of the following BSDE (which is of non standard form): For every s ≤ T , Ǔ aj,m s = h a(T )j (X T ) + T s 1 {r≥τ0} f aj,m (r, X t,x r , Ǔ aj,m r , V aj,m r )dr -T s V aj,m r dB r -A a T -A a s , (4.2.6)

  .2.10)Relying next on the elementary inequality a+b + ≤ (ab) + , it holds 1 B j,l (r) [(W a,jj",m r ) + -(W a,lj",m r ) + ] ≤ 1 B j,l (r) Ǔ al,m r -Ǔ aj,m r ḡlj ′′ (r, X t,xr ) + ḡjj ′′ (r, X t,x r ) + . (4.2.11)

Theorem 4 . 2 . 1 .

 421 Under Assumptions (H0)-(H5), for any (i, j) ∈ Γ 1 × Γ 2 , it holds that vij = v ij . Remark 4.2.2. (i) The result of Theorem 5.5.2 (see the appendix) is still valid if (H0)-(H4) are in force and the functions (g ij ) (i,j)∈Γ 1 ×Γ 2 verify (H5).

Theorem 4 . 3 . 1 .

 431 Suppose Assumptions (H0)-(H5) and (H6)-(ii) are satisfied. Then, for any

T-T) 2 ]

 2 j,m r dB r + dK δ,j,m r -(A δ T -A δ s ) + ( Ãδ,j,m Ãδ,j,m s ), ∀s ∈ [t, T ]. (4.3.14) This equation implies also that E[( Ãδ,j,m < ∞. Comparing now equation (4.3.14) for (Y δ,j,m s , Z δ,j,m s ) s∈[t,T ] and the one satisfied by ( Û δ,j,m s , V δ,j,m s ) s∈[t,T ] we have, by uniqueness of the solution of the multidimensional BSDE (4.3.12), that Y δ,j,m s -E[( Ãδ,j,m T -Ãδ,j,m s ) + (K δ,j,m T -K δ,j,m s )|F s ] = Û δ,j,m s , ∀s ∈ [t, T ] and j ∈ Γ 2 . (4.3.15)

1

 1 [σ * l ≤s] for s ∈ [t, definition of the strategy δ, for any l ≥ 0, it holds

Futhermore, let γ

  i : R × B E → R such that there exists C > 0, 0 ≤ γ i (x, e) ≤ C(1 ∧ |e|), x ∈ R, e ∈ B E |γ i (x, e)γ i (x ′ , e ′ )| < C|xx ′ |(1 ∧ |e|), x, x ′ ∈ R, e ∈ E.

  We set f i (t, x, y, z, u) = g i (t, x, y, z, E u(e)γ i (x, e)n(de),for (t, x, y, z, u) ∈ [0, t] × R × R m × R × L 2 (R, B E , n).

Theorem 5 . 4 . 1 .

 541 Assume that Assumption (H1) is fulfilled. Then system of reflected BSDE with oblique reflection (5.4.1) has a unique solution.

  f j (r, X t,x r , Ȳr , • • • , Ȳr , Zr , Ūr )drf j (r, X t,x r , Y r , • • • , Y r , Z r , U r )dr -

Ts

  f j (r, X t,x r , Y 1,n-1 r , • • • , Y j-1,n-1 r , Y j,n r , Y j+1,n-1 r , • • • , Y m,n-1 , Z j,n r , U j,n r )dr -T s Z j,n r dB r -T s E U j,n r (e) N (drde) + K j,n T -K j,n s , ∀s ≤ T ; Y j,n s ≥ max k∈Aj (Y k,n-1 s g jk (s, X t,xs )), ∀s ≤ T ;

s≤

  f j (r, X t,x r , Ȳr , • • • , Ȳr , Zr , Ūr )dr -T Zr dB r -T s E Ūr (e) N (drde) + KT -Ks Ȳs ≥ max k∈Aj ( Ȳsg jk (s, X t,x s )), ∀s ≤ T ; T 0 [ Ȳrmax k∈Aj ( Ȳsg jk (s, X t,x s ))]d Kr = 0.Next since ∀i ∈ A, f i verifies the Assumption H1(I), by induction we have: ∀n, j, ∀s ≤ T, Y j,nȲs , Pa.s., and E[ sup s∈[0,T ]

Ts

  f j (r, X t,x r , Y 1 r , Y 2 r , • • • , Y m r , Z j r , U j r )dr -T s Z j r dB r -T s E U j r (e) N (drde) + K j T -K j s Y j s ≥ max k∈Aj (Y k sg jk (s, X t,x s ));(5.4.3)

  r (e) -Û j r (e)) 2 N (dedr)] = 0.

∀s < T, A a s = j≥1 g

 j≥1 αj-1,αj (θ j , X t,x θj ) [θj <s] , A a T = lim s→T A a s and E[(A a T ) 2 ] < ∞,where X t,x is the process given in(3.1.4). Now, for t ≤ T we denote byA i t := {a s := α 0 {θ0} (s) + ∞ j=1 α j-1 ]θj-1θj ] (s) is admissible strategy, α 0 = i, θ 0 = 0, θ 1 ≥ t and E[(A a T ) 2 ] < ∞}. For Γ := ((Γ i s ) s∈[0,T ] ) i∈A , such that ∀i ∈ A, Γ i ∈ H 2 ,we introduce the unique solution of the switched BSDE:∀s ≤ T ,V a(.) s = h a(T ) (X t,x T ) + T sf a(r) (r, X t,x r , -→ Γ r , M a r , N a r )dr -

Step 2 :

 2 , ∀(s, j) ∈ [0, T ] × {1, • • • , m} and the uniqueness follows. Now we deal with the general case. Let Θ :[H 2 ] m be H 2 × • • • × H 2 (m times) and we introduce the operator Θ : [H 2 ] m → [H 2 ] m , Γ → Y : = h j (X t,x T ) + T s f j (r, X t,x r , Γ r , Z j r , U j r )dr -T s Z j r dB r -T s E U j r (e) N (drde) + K j T -K j s , ∀s ≤ T. Y j s ≥ max k∈Aj {Y k sg jk (s, Y j s )}, ∀s ≤ T ; T 0 [Y j smax k∈Aj {Y k sg jk (s, Y j s )}]dKj s = 0 (5.4.8) By Step 1, Θ is well defined. Next for Y ∈ H 2 let us define || • || 2,β by: Y 2,β := (E[ T 0 e βs |Y s | 2 ds])

Ts

  Z j,k r dB r -T s E U j,k r (e) N (drde) +K k,j T -K k,j s Y k,j s ≥ max q∈Aj {Y k,q s g jq (s, Y k,j s )}; T 0 [Y k,j s max q∈Aj {Y k,q s g jq (s, Y k,j s )}]dK k,j s = 0.

e

  sg jq (s, Ŷ j s )}]d Kj s = 0, V a s = h a(T ) (X t,x T ) + T s f a(r) (r, X t,x r , Γ 1 s , M a r , N a r ) ∨ f a(r) (r, X t,x r , Γ 2 s , M a r , N a r )dr e βs | △ V a * s | 2 + βr | △ N a * r (e)| 2 n(de)dr = -T s βe βr | △ V a * r-| 2 ds -2 T s e βr △ V a * r-M a * r dB r -2 T s Ee βr △ V a * r-△ N a * r (e) N (drde) + 2

, N 1 12 ) 1 )

 1121 property of f and the fact that |x ∨ y -y| ≤ |x -y|,|f a * (r, X t,x r , Γ 1 r , Ẑa * r , N a * r ) ∨ f a * (r, X t,x r , Γ 2 r , Ẑa * r , N a * r )f a * (r, X t,x r , Γ 1 r , Ẑ1,a * r 1 r -Γ 2 r | + | Ẑa * r -Ẑ1,a * r | + || N a * r -N 1,a * r || L 2 ( N ) )Combine these two estimates with the inequality 2xy≤ 1 β x 2 + βy 2 (β > 0) e βs | △ V a * s | 2 ≤ L β T s e βr |Γ 1 r -Γ 2 r | 2 dr -2 T s e βr △ V a * r-M a * r dB r -2 T s E e βr △ V a * r-△ N a * r (e) N (drde)for β ≥ L.We deduceE[e βs | △ V a * s | 2 ] ≤ L β E[ T s e βr |Γ 1 r -Γ 2 r |dr]. Similarly, we get also ∀s ≤ T , E[e βs | V a * s -V 2,a * s | 2 ] ≤ L β E[ T s e βr |Γ 1 r -Γ 2 r |dr].Since s is arbitary, we going back to (5.4.10), squaring and taking the expectation, we haveE[sup η≤T e βη |Y 1,i η -Y 2,i η | 2 ]For β large enough ,Θ is contraction on the Banach space (([H 2 ]) m , . 2,β ), then the fixed point theorem ensures the existence of a unique Y such that Θ(Y ) = Y , which is the unique solution of system of RBSDE (5.4.Corollary 5.4.1.

with L 1

 1 u(t, x) := b(t, x)∂ x u(t, x) + 1 2 σ(t, x) 2 D 2xx u(t, x) andI(t, x, φ) = E (φ(t, x + β(x, e))φ(t, x) -D x φ(t, x)β(x, e))n(de);I B i (t, x, φ) = E (φ(t, x + β(x, e))φ(t, x))γ i (x, e)n(de).

r

  -L(r, X t,x r ) dK +,τ r = 0 and δτ τ Y τ r -U (r, X t,x r ) dK -,τ r = 0;

Proposition 5 . 5 . 3 .

 553 ([28], pp.894) The process Y t,x verifies: for any stopping time θ ≥ t, Y t,x θ = ess sup σ≥θ ess inf τ ≥θ E σ∧τ θ

  There are three main results in this thesis. The first is existence and uniqueness of the solution in viscosity sense for a system of nonlinear m variational integral-partial differential equations with interconnected obstacles. From the probabilistic point of view, this system is related to optimal stochastic switching problem when the noise is driven by a Lévy process. As a by-product we obtain that the value function of the switching problem is continuous and unique solution of its associated Hamilton-Jacobi-Bellman system of equations. Next, we study a general class of minmax and max-min nonlinear second-order integral-partial variational inequalities with interconnected bilateral obstacles, related to a multiple modes zero-sum switching game with jumps. Using Perron's method and by the help of systems of penalized unilateral reflected backward SDEs with jumps, we construct a continuous with polynomial growth viscosity solution, and a comparison result yields the uniqueness of the solution. At last, we deal with the solutions of systems of PDEs with bilateral inter-connected obstacles of min-max and max-min types in the Brownian framework. These systems arise naturally in stochastic switching zero-sum game problems. We show that when the switching costs of one side are smooth, the solutions of the min-max and max-min systems coincide. Furthermore, this solution is identified as the value function of the zero-sum switching game. Key Words : IPDE with interconnected obstacles ; Lévy process ; Multi-modes switching problem ; Backward stochastic differential equations; Switching zero-sum game; Hamilton-Jacobi-Bellman-Isaacs equation; Perron's method. Résumé Cette thèse est composée de trois parties. Dans la première nous montrons l'existence et l'unicité de la solution continue et à croissance polynomiale, au sens viscosité, du système non linéaire de m équations variationnelles de type intégro-différentiel à obstacles unilatéraux interconnectés. Ce système est lié au problème du switching optimal stochastique lorsque le bruit est dirigé par un processus de Lévy. Un cas particulier du système correspond en effet à l'équation d'Hamilton-Jacobi-Bellman associé au problème du switching et la solution de ce système n'est rien d'autre que la fonction valeur du problème. Ensuite, nous étudions un système d'équations intégro-différentielles à obstacles bilatéraux interconnectés. Nous montrons l'existence et l'unicité des solutions continus à croissance polynomiale, au sens viscosité, des systèmes min-max et max-min. La démarche conjugue les systèmes d'EDSR réfléchies ainsi que la méthode de Perron. Dans la dernière partie nous montrons l'égalité des solutions des systèmes max-min et minmax d'EDP lorsque le bruit est uniquement de type diffusion. Nous montrons que si les coûts de switching sont assez réguliers alors ces solutions coincident. De plus elles sont caractérisées comme fonction valeur du jeu de switching de somme nulle. Mots clés : Equations intégro-différentielles à obstacles interconnectés; Processus de Lévy; Switching optimal ; Equations différentielles stochastiques rétrogrades; Jeu à somme nulle; Équation de Hamilton-Jacobi-Bellman-Isaacs; Méthode de Perron.

  General Results on Backward Stochastic Differential Equations . . . . . . . . . . . . . . . 1.2 Systems of Integro-PDEs with Interconnected Obstacles and Multi-Modes Switching Problem Driven by Lévy Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Recalling some results for RBSDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Viscosity solution of system of variational inequalities with interconnected bilateral obstacles and connections to multiple modes switching game of jump-diffusion processes . . Second existence and uniqueness result . . . . . . . . . . . . . . . . . . . . . . . . . 3 Viscosity solution of system of variational inequalities with interconnected bilateral obstacles and connections to multiple modes switching game of jump-diffusion processes 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Approximation schemes of the solution of systems of reflected BSDEs . . . . . . . . . . . 3.3 Uniqueness and Existence of viscosity solution for system of IPDEs . . . . . . . . . . . . B t ) t≤T the natural filtration of B and (F t ) t≤T its completion with the P -null sets of F . Define the following spaces: • P n the set of F t -progressively measurable, R n -valued processes on Ω × [0, T ]; • L 2

	Contents 1 Introduction 1.1 Chapter 1 Introduction
	1.1 General Results on Backward Stochastic Differential Equa-
	tions

1.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Two approximating schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 On the identity of min-max and max-min solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Assumptions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Systems of Integro-PDEs with Interconnected Obstacles and Multi-Modes Switching Problem Driven by Lévy Process. 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Systems of Reflected BSDEs with Oblique Reflection driven by a Lévy process . . . . . . 2.2.1 Reflected BSDE driven by a Lévy process and their relationship with IPDEs . . . 2.2.2 Systems of reflected BSDEs with inter-connected obstacles driven by a Lévy process and multi-modes switching problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Existence and uniqueness of the solution for the system of IPDEs with inter-connected obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Existence of the viscosity solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Uniqueness of the viscosity solution . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Let (Ω, F, P ) be a probability space on which is defined a d-dimensional Brownian motion B := (B t ) t≤T . Let us denote by (F n (F t )={η : F tmeasurable R nvalued random variable s.t. E[|η| 2 ] < ∞}; • S 2 n (0, T )={ϕ : P nmeasurable with continuous paths, s.t. E[sup s≤T [|ϕ| 2 ] < ∞}; • H 2 n (0, T ) = {Z : P nmeasurable s.t. E[ T 0 |Z s | 2 ds] < ∞}; • H 1 n (0, T ) = {Z : P nmeasurable s.t. E[ T 0 |Z s | 2 ds]

  2 1 (F T ). Consider the following linear BSDE: dY t = (φ t + Y t β t + Z t µ t )dt -Z t dB t ; Y T = ξ. (1.1.4) (i) Equation (1.1.4) has a unique solution (Y, Z) ∈ S 2 1 (0, T ) × H 2 d (0, T ), and Y is given explicitly by Y t = E[ξΓ t,T +

T t Γ t,s φ s |F t ],

  2,ℓ (ℓ ≥ 1) be the set of P-measurable and IR ℓ -valued processes w = (w t ) t≤T such that E[

	∞.	T 0 |w s | 2 ds] < ∞; (ii) S 2 (resp. S 2 d ) be the set of P-measurable continuous (resp. RCLL) processes such that E[sup t≤T |w t | 2 ] <

  Remark 2.2.2. Conditions (2.2.8) and (2.2.9) can be replaced respectively by

1 t ≥ Y 2 t since Y 1 and Y 2 are RCLL. The proof of the claim is now complete.

  Definition 2.2.1. A continuous function u : [0, T ] × IR → IR is said to be a viscosity subsolution (resp.

	x)y)p k (y)Π(dy)	k≥1
	where c 1,1 is defined in (2.1.3).
	We are going to consider solutions of (2.2.20) in viscosity sense whose definition is as follows:

supersolution) of (2.2.20) if for any (t, x) ∈ [0, T ) × IR and for any ϕ ∈ C 1,2 ([0, T ] × IR) ∩ Π g such that ϕ(t, x) = u(t, x) and ϕu attains its global minimum (resp. maximum) at (t,x),

  2.35) By the result proved in Step 1, Θ is well defined. Next for η ∈ H 2 let us define • 2,β by which is a norm of H 2 , equivalent to . and (H 2 , • 2,β ) is a Banach space. Let now Γ 1 and Γ 2 be two processes of [H 2

	η 2,β := (E[	0	T	e βs |η s | 2 ds])	1 2 ,

  2.1. Theorem 2.3.1. Assume Assumptions [A4] and [A5] and (2.2.14), (2.2.15) as well, then (u j (t, x)) j∈A is a viscosity solution of (2.3.1).

  2 and of compact support, then the two non-local terms are bounded and∂ t v n , ∂ x v n , D 2 xx v n are so. Since ∂ t ( √ Tt)→ -∞, when t → T , then we can choose t k large enough in front of δ k and the derivatives of v n to ensure that

  progressively measurable process s.t. ||ϕ|| 2 H ∞} ; A 2 is the subspace of S 2 of continuous non-decreasing processes null at t = 0 ; 57 (v) H 2 ( Ñ ) := {U : Ω×[0, T ]×E → R, P⊗B E -measurable and s.t. U 2 H 2 ( Ñ ) := E(

	∞}.	T 0	E |U t (e)| 2 n(de)dt) <

2 := E( T 0 |ϕ t | 2 dt) < ∞}; (iv) S 2 := {{ϕ t , 0 ≤ t ≤ T } is an IR-valued, F t -adapted RCLL process s.t. ||ϕ|| 2 S 2 := E[sup 0≤t≤T |ϕ t | 2 ] <

  They are the bounds of the first and second derivatives of y → φ(t, y) in B(x, C β ) where C β is a bound of the function β.

	Now consider the following SDE:

2 , and |φ(t, x + β(x, e))φ(t, x)| ≤ C 2 t,x |β(x, e)| ≤ C 2 t,x (1 ∧ |e|) where C 1 t,x and C 2 t,x are bounded constants.

  1 ) of processes verifies:

	Û δ,j t	= ess inf ν∈A j π 2 (t) {J t (δ, ν) + A δ t } and U i,ν t	= ess sup δ∈A i π 1	(t) {J t (δ, ν) -B ν t }.	(4.3.7)

  .3.8) Proof: Recall the definition of ( Y ij,m , Žij,m , Ǩij,m ) (i,j)∈Γ 1 ×Γ 2 , m ≥ 0, given in (4.2.2). In order to alleviate notations, we denote it simply by (

  2 then Pa.s.,

	with (4.3.23) we deduce that	Ãδ ,j0,m T	Ãδ ,j0,m s	→ 0 as m → ∞, for any s ∈ [t, T ]. Therefore,

  .3.26) Then, by Theorem 4.3.1 we have, for any (t, x) ∈ [0, T ] × IR k and (i 0 , j 0 ) ∈ Γ 1 × Γ 2 , ij ) (i,j)∈Γ 1 ×Γ 2 is also solution of the system (4.3.26), then by uniqueness for any (i, j) ∈ Γ 1 × Γ 2 , v ij = w ij . Plug now this equality in (4.3.27) to obtain the desired result.Remark 4.3.1. We have also the following relation:

	w i0j0 (t, x) = ess sup δ∈A i 0 π 1	ess inf ν∈A j 0 π 2 Jt (δ, ν) = ess inf ν∈A j 0 π 2	ess sup δ∈A i 0 π 1 Jt (δ, ν).	(4.3.27)
	But (v

2

  Relation with double obstacle variational inequality Let g(t, x, y, z, u) satisfies the Assumption (A1), L : (t, x) → L(t, x) and H : (t, x) → H(t, x) are jointly continuous and of polynomial growth, i.

e., there exist positve constant C and p such that:

∀(t, x) ∈ [0, T ] × R, |L(t, x)| + H(t, x)| ≤ C(1 + |x| p ).

The functions h(x) : R → R are continuous w.r.t. x, belong to class Π g . Furthermore, for any (s, x) ∈ [0, T ] × R: L(s, x) ≤ H(s, x), and L(T, x) ≤ h(x) ≤ H(T, x).

  -H(t, x), -∂ t v(t, x) -Lv(t, x) -g(t, x, v(t, x), σ(t, x)D x v(t, x), Bv(t, x))+ n(L(t, x)v(t, x)) -} = 0 v(T, x) = h(x),and n ū are so. It implies that the convergence of n u and n ū to u are uniform on compact subsets of [0, T ] × R. Now consider the following obstacle problem:-L(t, x), max[v(t, x) -H(t, x), -∂ t v(t, x) -Lv(t, x)g(t, x, v(t, x), σ(t, x)D x v(t, x), Bv(t, x))]} = 0 v(T, x) = h(x)(5.3.3) Theorem 5.3.1. The function u defined above is a viscosity solution of (5.5.4) and for any (t, x) ∈ [0, T ] × R and any stopping times ν, σ ∈ T t where T t is the set of F t stopping time take values in [t, T ], (ν, σ) = J t,x (δ t,x , θ t,x ), , Y s , Z s , E U (e) s λ(X t,x s , e)n(de))ds + L(σ, X t,x σ ) [σ≤ν<T ] +H(ν, X t,x ν ) [ν<σ] + g(X t,x T ) [ν=σ=T ] ] if E[ ν∧σ t|g(s, X t,x s , Y s , Z s , E U (e) s λ(X t,x s , e)n(de))|ds] < ∞ +∞ else;

	   J t,x where min{v(t, x) u(t, x) = inf ν∈Tt sup σ∈Tt J t,x (ν, σ) = sup σ∈Tt inf ν∈Tt
	  	E[	ν∧σ t	g(s, X t,x s
	
	J t,x (ν, σ) :=
	
	
	
	(resp.
	   max{v(t, x) where,
	Lφ(t, x) := b(t, x)D x φ(t, x) +	1 2	σ 2 (t, x)D 2 xx φ(t, x) +	E	(φ(t, x + β(x, e)) -φ(t, x) -D x φ(t, x)β(x, e))n(de),
			Bφ(t, x) =

x) -L(t, x), -∂ t v(t, x) -Lv(t, x) -g(t, x, v(t, x), σ(t, x)D x v(t, x), Bv(t, x)) + n(H(t, x)v(t, x)) -} = 0 v(T, x) = h(x) E (φ(t, x + β(x, e))φ(t, x))γ(x, e)n(de).

The comparison result allows us to infer that ( n Y ) n≥0 (resp. ( n Ȳ ) n≥0 ) is a decreasing (resp. increasing) sequence, moreover they converge in S 2 to Y . Therefore for any (t, x) ∈ [0, T ] × R, the sequence ( n Y (t, x)) n≥0 (resp. ( n ū(t, x)) n≥0 ) converges decreasingly (resp. increasingly) to the same limit u(t, x) := Y t which satisfies Y s = u(s, X t,x s ) for any s ∈ [s, T ]. u is continuous with polynomial growth since n u

  (φ(t n , x n + β(x n , e))φ(t n , x n ))γ(x n , e)n(de) =Now from the continuity of the functions and the uniform convergence, we have

				2 1 (1 + |e|) 2 +	1 4	K 2 2 (1 + |e|) 2 .
	These combine with Dominated convergence theorem,
	lim n→∞ E (φ(t and
	lim n→∞ E
	-∂ t φ(t, x) -b(t, x)D x φ(t, x) +	1 2	σ 2 (t, x)D 2 xx φ(t, x)
	+	E	(φ(t, x + β(x, e)) -φ(t, x) -D x φ(t, x)β(x, e))n(de)

2 xx φ(t n , x n ) + E (φ(t n , x n + β(x n , e))φ(t n , x n ) -D x φ(t n , x n )β(x n , e))n(de) g(t n , x n , n u (t n , x n ), σ(t n , x n )D x φ(t n , x n ), E (φ(t n , x n + β(x n , e))φ(t n , x n ))γ(x n , e)n(de)) + n(H(t n , x n ) -n u(t n , x n )) -≤ 0 Note that since lim n→∞ x n = x, and φ is a C 1,2 function |φ(t n , x n + β(x n , e))φ(t n , x n ) -D x φ(t n , x n )β(x n , e)| ≤ C|β(x n , e)| 2 ≤ K 2 (1 + |e|) 2 ,

and

|(φ(t n , x n + β(x n , e))φ(t n , x n ))γ(x n , e)| ≤ C|β(x n , e)|γ(x n , e) ≤ K n , x n + β(x n , e))φ(t n , x n ) -D x φ(t n , x n )β(x n , e))n(de) = E (φ(t, x + β(x, e))φ(t, x) -D x φ(t, x)β(x, e))n(de), E (φ(t, x + β(x, e))φ(t, x))γ(x, e)n(de).

g(t, x, u(t, x), σ(t, x)D x φ(t, x), E (φ(t, x + β(x, e))φ(t, x))γ(x, e)n(de))

  2,d , K ±,τ ∈ S 2 and non-decreasing; ∀s ∈ [τ, δ τ ], Y t,x

	s s ) ≤ Y t,x = Y t,x δτ + L(s, X t,x	δτ s f (r, X t,x r , Y t,x r , Z τ r )dr -	δτ s Z τ r dB r +	δτ s dK +,τ r	-	δτ s dK -,τ r

s ≤ H(s, X t,x s ), ∀s ∈ [t, T ] ;

Acknowledgements

I would like to express my gratitude to all those who helped me during the writing of this thesis.

(iii)There exsits a unique A 1 × A 2 -uplet of deterministic continuous functions (u k,l,n ) (k,l)∈A 1 ×A 2 in Π g such that, for every t ≤ T , Y i,j,n s = u i,j,n (s, X t,x s ), s ∈ [t, T ]. (3.2.15) Moreover, ∀(i, j) ∈ A 1 × A 2 and (t, x) ∈ [0, T ] × R k , u i,j,n (t, x) ≤ u i,j,n+1 (t, x).

Finally, (u i,j,n ) (i,j)∈A 1 ×A 2 is a viscosity solution in the class Π g of the following system of variational inequalities with inter-connected obstacles. ∀(i, j)

max{u i,j,n (t, x)min l∈A 2 j (u i,l,n (t, x) + g jl (t, x)); -∂ t u i,j,n (t, x) -Lu i,j,n (t, x) -g ij,-,n (t, x, (u k,l,n (t, x)) (k,l)∈A 1 ×A 2 , σ(t, x)D x u i,j,n (t, x), B ij u i,j,n (t, x))} = 0 u i,j,n (T, x) = h i,j (x).

(3.2.16)

We define ūij (t, x) := lim m→∞ ūi,j,m (t, x), u ij (t, x) := lim n→∞ u i,j,n (t, x).

Then, we can prove a by-product of Proposition 3.2 and 3.3:

Corollary 3.2.1. ∀(i, j) ∈ A 1 × A 2 , the function ūij (resp. u ij is usc (resp. lsc). Moreover, ūij and u ij belong to Π g , for any (t, x) ∈ [0, T ] × R, u ij (t, x) ≤ ūij (t, x).

Proof. For any (i, j) ∈ A 1 × A 2 , the function ūij (resp. u ij ) is obtained as a decreasing (resp. increasing) limit of continuous functions. Therefore, it is usc (resp. lsc). Next, for any (i, j) and n, m, u ij,n,m (t, x) ≤ u ij,n,0 (t, x), (t, x) ∈ [0, T ] × R, as the sequence (u ij,n,m ) m≥0 is decreasing. Thus, taking the limit as m → ∞ we obtain, u ij,n ≤ u ij,n,0 . Now using (3.2.3) and (3.2.6), it follows that, for any t ≤ T and s ∈ [t, T ], Y ij,n,0 s = u ij,n,0 (s, X t,x s ) and the processes Y ij,n,0 converges in S 2 , as n → ∞, to Ȳ ij,0 which is solution of (3.2.5) with m = 0. Furthermore, by (3.2.15), there exists a deterministic continuous function ūij,0 with polynomial growth such that for any t ≤ T and s ∈ [t, T ], Y ij,0 s = ūij,0 (s, X t,x s ). Then taking s = t and the limit as n → ∞ to obtain u ij (t, x) := lim n→∞ u ij,n (t, x) ≤ lim n→∞ u ij,n,0 (t, x) = ūij,0 (t, x), ∀(t, x) ∈ [0, T ] × R.

But ūij,0 and u ij,n belong to Π g and u ij,n ≤ u ij,n+1 . Thus , u ij ∈ Π g , for any (i, j) ∈ A 1 × A 2 . In the same way one can show that ūij ∈ Π g , for any (i, j) ∈ A 1 × A 2 . The last inequality follows from (3.2.4) and the definitions of ūij and u ij .

3.3

Uniqueness and Existence of viscosity solution for system of IPDEs

In this section we will show the uniqueness of viscosity solution of (3.1.2) as the corollary of a comparison result. In the same way with [START_REF] Djehiche | Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles[END_REF], we can prove the following lemma.

) is an usc subsolution (resp. lsc supersolution) of (3.1.2) which belongs to Π g . For all (t, x) ∈ [0, T ] × R and let Γ(t, x) be the following set: Γ(t, x) := {(i, j) ∈ A 1 × A 2 , u ij (t, x)w ij (t, x) = max (k,l)∈A 1 ×A 2 (u kl (t, x)w kl (t, x))}. Then there exists (i 0 , j 0 ) ∈ Γ(t, x) such that u i0j0 (t, x) > L i0j0 [ -→ u ](t, x), w i0j0 (t, x) < U i0j0 [ -→ w ](t, x).

(3.3.1)

Now we give the comparison theorem.

since u ij,m0 is continuous and u ij is lsc. Therefore, for any x ∈ R, u ij * (T, x) = h ij (x) (3.3.29) since u ij (T, x) = h ij (x) = u ij,m0 (T, x). Assume that (u ij ) (i,j)∈A 1 ×A 2 is not a supersolution for (3.1.2), then taking into account of (3.3.29) and Remark 2.1, there exists at least on pair (i, j) such that u ij does not satisfy the viscosity supersolution property: means that for some point (t 0 , x 0 ) ∈ [0, T ] × R there exists a triple (p, q, M ) in

) Now for any positive δ, γ and r, set u δ,γ and B r as follows:

and

Using (3.3.30) and continuity of all the data, choosing δ, γ small enough we obtain

where,

First we note that from (3.3.31), Υ(t 0 , x 0 ) < 0, since u δ,γ (t 0 , x 0 ) = u ij * (t 0 , x 0 ) + δ. On the other hand by the continuity of u δ,γ , Assumptions (A1), (A2) on f ij and finally the lower semi-continuity of (u kl * ) (k,l)∈A 1 ×A 2 , we can check that Υ is usc. Thus for any ε > 0, there exists η > 0 s.t. for any (t, x) ∈ B η we have Υ(t 0 , x 0 ) ≥ Υ(t, x)ε. Since Υ(t 0 , x 0 ) < 0, choosing ε small enough we deduce that Υ(t, x) < 0 for any (t, x) ∈ B η with an appropriate η. It follows that the function u δ,γ is a viscosity subsolution on B η of the following system.

and f ij satisfies the monotonicity condition (A2), u δ,γ is also a viscosity solution on B η of the system

as soon as r 2 < |xx 0 | ≤ r and r small enough. Take r ≤ η and consider the function ũij :

Then taking into account of (3.3.32), and using Lemma 4.2 in Crandall et al. (1992)( [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]), it follows that ũij is a subsolution of (3.3.28). Next, as ũij ≥ u ij and using once more the monotonicity assumption (A2) on f kl , we get that [(u kl ) (k,l) =(i,j) , ũij ] a also a subsolution of (3.1.2) which belongs to Π g . Thus, thanks to the comparison theorem 4.1, [(u kl ) (k,l) =(i,j) , ũij ] belongs also to U m0 . Finally in view of the definition of u ij * , there exists a sequence (t n , x n , u ij (t n , x n )) n≥1 that converges to (t 0 , x 0 , u ij * (t 0 , x 0 )) n≥1 . This implies that lim

It means that there are points (

, ũij ] belongs also to U m0 . Therefore, (u ij ) (i,j)∈A Chapter 4

On the identity of min-max and max-min solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles.

This chapter is a joint work with Boualem Djehiche, Said Hamadène and Marie-Amelie Morlais.

Notations and first results

Let us consider the following two systems of partial differential equations (PDEs) with bilateral interconnected obstacles (i.e., the obstacles depend on the solution) of min-max and max-min types: for any (i, j)

2) where (i) Γ 1 and Γ 2 are finite sets (possibly different);

(iii) L X is a second order generator associated with a diffusion process described below.

The systems (4.1.1) and (4.1.2) are of min-max and max-min types respectively. The barriers L ij ( v), U ij ( v) and L ij ( v), U ij ( v) depend on the solution v and v of (4.1.1) and (4.1.2) respectively. They are related to zero-sum switching game problems since actually, specific cases of these systems, stand for the Hamilton-Jacobi-Bellman-Isaacs equations associated with those games.

Let T (resp. k, d) be a fixed positive constant (resp. two integers) and Γ 1 (resp. Γ 2 ) denote the set of switching modes for player 1 (resp. 2). For later use, we shall denote by Λ the cardinal of the product set Γ 1 ×Γ 2 and for (i, j) ∈ Γ 1 ×Γ 2 , (Γ 1 )

-i := Γ 1 -{i} and (Γ 2 ) -j := Γ 2 -{j}. For y = (y kl ) (k,l)∈Γ 1 ×Γ 2 ∈ IR Λ , 77

Chapter 5 APPENDIX

Representation of the value function of the stochastic optimal switching problem

Let Υ := (θ n , α n ) n≥0 be an admissible strategy of switching and let a = (a s ) s∈[0,T ] be the process defined by

(5.1.1)

as the solution of the following BSDE:

k∈A and A a is the cumulative switching cost associated with the strategy a or Υ (see (2.2.22) for its definition). This BSDE is not a standard one, but in assuming that E[(A a T ) 2 ] < ∞ and by setting up V a = V a -A a , it becomes a standard one and therefore we deduce the existence and uniqueness of the process (V a , N a ) since the RCLL process A a is adapted and square integrable. Obviously if for some j ∈ A, a belongs to A j t0 then (V a , N a ) exists and is unique.

where (Y Γ,j ) j∈A is the first component of the solution of the BSDE (2.2.35). Thus the solution of (2.2.35) is unique. Moreover there exists a

Proof. By Theorem 3.4, let (Y Γ,j , U Γ,j , K Γ,j ) j∈A be the solution of the system (2.2.35). Let a ∈ A j t0 and let us define

, is said to be a viscosity subsolution of (4.1) (resp. supersolution

are respectively viscosity supersolution and viscosity subsolution of (2.3.1). Proposition 5.2.1. Definitions (5.2.1) and (2.3.1) are equivalent.

Proof. First let us show that Def. 5.2.1 implies Def. 2.3.1. We prove it only for the subsolution case since the supersolution one is similar. Let (u i ) i∈A be a subsolution of system (2.2.20) according to Definition 5.2.1. Wlog assume that u i , i ∈ A, is usc. Thus for any and verifies φ(t 0 , x 0 ) := u i (t 0 , x 0 ). Applying now Def. 5.2.1 with φ yields:

for any δ > 0. Next since (t 0 , x 0 ) ∈ (0, T ) × IR is global maximum point of u iϕ, we then have u i (t 0 , x 0 + σ(t 0 , x 0 )y)u i (t 0 , x 0 ) ≤ ϕ(t 0 , x 0 + σ(t 0 , x 0 )y)ϕ(t 0 , x 0 ) which implies that I 2,δ (t 0 , x 0 , D x ϕ(t 0 , x 0 ), u i ) ≤ I 2,δ (t 0 , x 0 , D x ϕ(t 0 , x 0 ), ϕ) and then We are going now to show that Def.2.3.1 implies Def. 5.2.1. Once more let us consider (u i ) i∈A a subsolution of system (2.2.20) according to Definition 2.3.1 which, wlog, we assume that they are usc functions. Then for any

) and u i (t 0 , x 0 ) = ϕ(t 0 , x 0 ). But there exists a function φ which belongs to Π g C 1,2 ([0, T ]×IR) such that u-φ attains a global maximum in (t 0 , x 0 ) on [0, T ] × IR and satisfying φ(s, y) = ϕ(s, y), for any (s, y) such that |(s, y) -(t 0 , x 0 )| < Cσδ 2 . Consequently we have also

(5.2.1) Next for any ǫ > 0, there exists ϕ ǫ element of C 1,2 ([0, T ] × IR) such that u i ≤ ϕ ǫ ≤ φ and ϕ ǫ → u i as ǫ → 0, a.e. (see e.g. Lemma 4.7 in [START_REF] Jakobsen | A "maximum principle for semicontinuous functions" applicable to integro-partial differential equations[END_REF] or [START_REF] Awatif | Equqtions D'Hamilton-Jacobi Du Premier Ordre Avec Termes Intégro-Différentiels: Partie 1: Unicité Des Solutions De Viscosité[END_REF]). It implies that u iϕ ǫ and ϕ ǫφ have a global maximum at (t 0 , x 0 ) on [0, T ] × IR k . Therefore, on the one hand, we have

and, on the other hand, by Def. 2.3.1 it holds

(5.2.3)

and by (5.2.1) and (5.2.2)

On the other hand

(5.2.5)

Plug now (5.2.4) and (5.2.5) in (5.2.3) to get

(5.2.6) Take now the limit as ǫ → 0 in (5.2.1), using the Lebesgue dominated convergence theorem and by the following inequality (since

we obtain

which is the desired result.

BSDE with two reflected barriers

1 Representation of a penalization scheme of two barriers reflected BSDE For n ≥ 0 let (Y n , Z n , u n , K n ) be the solution of the following one barrier reflected BSDE.

And define:

and of cause, for any φ ∈ Π g , (t, x, p) ∈ R 3 , I 2 δ (t, x, p, φ) and I 2,B ij δ (t, x, φ) are well defined.

Theorem 5.4.2. Assume Assumptions (H1), then (u j (t, x)) j∈A is a viscosity solution of (5.4.13).

Proof. We just show the main different steps with the proof of Theorem 4.1 in [START_REF] Hamadène | Systems of integro-PDEs with interconnected obstacles and multimodes switching problem driven by Lévy process[END_REF].

First we show that (u j ) m j=1 is a supersolution of (5.4.13). Noting that for all j ∈ A, as u j is lsc, we have

, where (Y j,n,t,x ; U j,n,t,x , K j,n,t,x ) j∈A is the unique solution of (5.4.2). As pointed out in Corollary 5.4.1, for any n ≥ 0, (t,

Y j,n,t,x s = u n j (s, X t,x s ) and u n j (t, x) ր u j (t, x).

Additionally by induction, (u n j ) j∈A , n ≥ 0, are continuous, belong to Π g and by Barles et al.'s result ( [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]) verify in viscosity sense the following system (n ≥ 1): ∀j ∈ A,

σ(t, x)D x u j,n (t, x), I B i (t, x, u j,n (t, x, ))) = 0; u j,n (T, x) = h j (x).

(5.4. [START_REF] Dellacherie | Capacités et processus stochastiques[END_REF] First note that for any j ∈ A, u j verifies

Next let (t, x) ∈ (0, T )×IR and let j ∈ A be fixed. Let φ be a function which belongs to C 1,2 ([0, T ]×IR)∩Π g such that u jφ has a strict global minimum in (t, x) on [0, T ] × IR k and wlog we assume that u j (t, x) = φ(t, x). Now let δ > 0 be fixed. Then (t, x) is a global strict minimum of u jφ in [0, T ] × B(x, Kδ). Next let (t n , x n ) be the global minimum of u n jφ on [0, T ] × B(x, Kδ). Therefore lim

Actually let us consider a convergent subsequence of (t n , x n ), which we still denote by (t n , x n ), and let set (t * , x * ) its limit. Then

(5.4.15)

Taking the limit wrt n and since u j * = u j is lsc to obtain

As the minimum (t, x) of u jφ on [0, T ] × IR k is strict then (t * , x * ) = (t, x). It follows that the sequence ((t n , x n )) n converges to (t, x). Going back now to (5.4.15) and in sending n to infintite we obtain

which implies that u n j (t n , x n ) → u j (t, x) as n → ∞. Now for n large enough (t n , x n ) ∈ (0, T ) × B(x, Kδ) and it is the global minimum of u n jφ in [0, T ] × B(x, Kδ). As u n j is a supersolution of (5.4.14), then by Definition 2.2, we have

(5.4.16) But there exists a subsequence of {n} such that:

(i) for any k ∈ A j , (u n-1 k

Sending now n to infinite (through the previous subsequence) in (5.4.16), using the fact that g j is continuous and verifies (A4)(I)(v) and finally by Fatou's Lemma to obtain:

But u j (t, x) = φ(t, x) and u j ≥ φ, then I 2,δ (t, x, D x φ(t, x), u j ) ≥ I 2,δ (t, x, D x φ(t, x), φ) and I 2,B j (t, x, u j ) ≥ I 2,B j (t, x, φ). Plugging now this inequality in the previous one to obtain

As j is arbitrary then (u j ) j∈A is a viscosity supersolution of (5.4.13). Now we will now show that (u * j ) j∈A is a subsolution of (5.4.13). First we have: ∀j ∈ A, min{u * j (T, x)h j (x); u * j (T, x)max k∈Aj (u * k (T, x)g jk (T, x))} = 0, and u * j (T, x) = h j (x), ∀j ∈ A. The proof is same as Theorem 4.1 in [START_REF] Hamadène | Systems of integro-PDEs with interconnected obstacles and multimodes switching problem driven by Lévy process[END_REF]. Now let us show (u * j ) j∈A is a subsolution of (5.4.13). First note that since u n j ր u j and u n j is continuous, we have

Besides ∀j ∈ A and n ≥ 0 we deduce from the construction of u n j that

and by taking the limit in n we obtain:

(5.4.17)

Let φ be a C 1,2 ([0, T ] × IR) ∩ Π g function such that u * jφ has a global maximum at (t, x) in [0, T ] × IR k which wlog we suppose strict and u j (t, x) = φ(t, x). Therefore (t, x) is a global strict maximum of u jφ in [0, T ] × B(x, Kδ). On the other hand there exist subsequences {n k } and ((t

Actually let us consider a convergent subsequent of (t n k , x n k ), which we still denote by (t n k , x n k ), and let ( t, x) be its limit. Then for some k 0 and for k ≥ k 0 we have

(5.4.18)

Taking the limit wrt k to obtain u * j ( t, x)φ( t, x) ≥ u * j (t, x)φ(t, x). As the maximum (t, x) of u jφ on [0, T ] × IR k is strict then ( t, x) = (t, x). It follows that the sequence ((t n k , x n k )) k converges to (t, x). Going back now to (5.4.18) and in sending k to infinite we obtain

As u n k j is a subsolution of (5.4.14), then by Definition 2.2, we have

(5.4.19) But there exists a subsequence of {n k } (which we still denote by {n k }) such that:

(i) for any l ∈ A j , (u

Sending now k to infinite (through the previous subsequence) in (5.4.19), using the fact that g j is continuous and verifies (H1) and finally by Lebesgue's Theorem to obtain

But u j (t, x) = φ(t, x) and u * j ≤ φ, then I 2,δ (t, x, D x φ(t, x), u * j ) ≤ I 2,δ (t, x, D x φ(t, x), φ) and I 1,B j δ (t, x, u * j ) ≤ I 1,B j δ (t, x, φ). Plugging now this inequality in the previous one to obtain

As j is arbitrary then (u j ) j∈A is a viscosity subsolution of (5.4.13).

Viscosity solution of PDE with two obstacle of min-max type

Let (t, x) ∈ [0, T ] × IR k and (X t,x s ) s≤T be the solution of the standard SDE given in (4.1.6) where the functions b and σ satisfy Assumption (H0). Let us now consider the following functions:

We assume that all those functions are continuous and satisfy the following assumptions (A1)-(A2).

where C and p are some positive constants.

(A2): For each R > 0, there is a continuous function ϕ R such that ϕ R (0) = 0 and

Next for n ≥ 0, let ( n Y t,x s ) s≤T (resp. ( n Ȳ t,x s ) s≤T ) be the first component of the unique solution of the BSDE with one reflecting lower (resp. upper) barrier associated with the triple (f (s, X t,x s , y)n(H(s, X t,x s )y) -, g(X t,x T ), L(s, X t,x s )) (resp. (f (s, X t,x s , y, z) + n(L(s, X t,x s )y) + , g(X t,x T ), H(s, X t,x s ))), which exists and is unique (see e.g. [START_REF] El Karoui | Les aspects probabilistes du contrôle stochastique[END_REF]). It has been shown in [START_REF] El Karoui | Les aspects probabilistes du contrôle stochastique[END_REF] that, under Assumptions (H0) and (A1)-(A2), for any n ≥ 0 there exist deterministic functions n u(t, x) and n ū, (t,

s ) and n Ȳ t,x s = n ū(s, X t,x s ), where n u (resp. n ū) is continuous with uniform polynomial growth i.e. there exist two non negative real constants C and p such that

Moreover it is a unique viscosity solution, in the classe Π g , of the following PDE with obstacle:

(5.5.2) By comparison (see e.g. [START_REF] Hamadene | BSDEs with two reflecting barriers driven by a Brownian and a Poisson noise and related Dynkin game[END_REF]) we easily deduce that the sequence of processes ( n Y t,x ) n≥0 (resp. ( n Ȳ t,x ) n≥0 ) is decreasing (resp. increasing), moreover they converge in S 2 to a same processes (Y t,x s ) s≤T which satisfies L(s, X t,x s ) ≤ Y t,x s ≤ H(s, X t,x s ), ∀s ≤ T. Therefore for any (t, x) ∈ [0, T ] × IR k , the sequence ( n u(t, x)) n≥0 (resp. ( n ū(t, x)) n≥0 ) converges decreasingly (resp. increasingly) to the same limit

Next as n u and n ū are continuous and belong to Π g , then the function u belongs also to Π g and is also continuous since it is both lsc and usc. By Dini's Theorem we deduce that the convergence of the sequence ( n u) n≥0 (resp. ( n ū) n≥0 ) is uniform on compact subsets of [0, T ] × IR k .

Next let us consider the following PDE with two obstacles of min-max type:

(5.5.4)

To begin with, we are going to give the notion of viscosity solution of (5.5.4). Definition 5.5.1. Let v be a function which belongs to C([0, T ] × IR k ). It is called a viscosity: (i) subsolution of (5.5.4) if v(T, x) ≤ g(x) and for any φ ∈ C 1,2 ([0, T ] × IR k ) and any local maximum point (t, x) ∈ (0, T ) × IR k of vφ, we have min v(t, x) -L(t, x); max v(t, x) -H(t, x); -∂ t φ(t, x) -L X φ(t, x)f (t, x, v(t, x), σ(t, x) ⊤ D x φ(t, x)) ≤ 0.

(ii) supersolution of (5.5.4) if v(T, x) ≥ g(x) and for any φ ∈ C 1,2 ([0, T ] × IR k ) and any local minimum point (t, x) ∈ (0, T ) × IR k of vφ, we have min v(t, x)-L(t, x); max v(t, x)-H(t, x); -∂ t φ(t, x)-L X φ(t, x)-f (t, x, v(t, x), σ(t, x) ⊤ D x φ(t, x)) ≥ 0.

(iii) solution of (5.5.4) if it is both a viscosity subsolution and supersolution.

Theorem 5.5.1. The function u defined in (5.5.3) is a viscosity solution of (5.5.4).

Proof. First we show that u is a viscosity subsolution of (5.5.4). Note that u(T, x) = g(x) and L(t, x) ≤ u(t, x) ≤ H(t, x). Let now φ ∈ C 1,2 and (t, x) ∈ (0, T ) × IR k be a local maximum of uφ in [0, T ] × IR k such that u(t, x) > L(t, x). Let (t n , x n ) be a sequence of local maximum points of n uφ such that (t n , x n ) converges to (t, x) (such a sequence exists because of the uniform convergence of n u to u on compact subsets (see e.g. [START_REF] Kamizono | On a variational inequality associated with a stopping game combined with a control[END_REF], pp.117). For n large enough we have n u(t n , x n ) > L(t n , x n ) and since n u is a viscosity solution of (5.5.1) then

Now by the continuity of the functions and the uniform convergence, we have

Thus u is a viscosity subsolution of (5.5.4). In a similar way we can show that u is also a viscosity supersolution.

The following result is of comparison type between sub. and supersolutions of (5.5.4). Namely we have: Proposition 5.5.1. Assume that Assumptions (H0), (A1)-(A2) are in force. Then if v (resp. u) is a viscosity supersolution (resp. subsolution) of (5.5.4) with polynomial growth, then for all (t, x) ∈ [0, T ] × IR k we have u(t, x) ≤ v(t, x).

Proof.

Step (i): First we proof that v ≥ L and u ≤ H.

By the definition of supersolution, it is clear that v ≥ L. Let us now show that u(t, x) ≤ H(t, x). Suppose that for t ≤ T , u(t, x) > H(t, x). Therefore we have t < T and u(t, x) -L(t, x) > 0 since

Step (ii): Let us define v ′ := v ∧H and u ′ = u∨L, then v ′ is a viscosity supersolution and u ′ is a viscosity subsolution of (5.5.4).

In fact, since

) and by continuity, (t, x) is also a local minimum point of v -φ. Since v is a supersolution of (5.5.4), then v ′ verifies min

Thus v ′ is a viscosity supersolution of (5.5.4). In the same way we can prove u ′ is a subsolution of (5.5.4).

Step (iii): Modification of the problem.

Let λ ∈ IR and ξ, η and κ be the functions defined on IR k as

where p is chosen in such a way that ū and v below are bounded and converge uniformly to 0 as x → ∞.

It exists since u and v are both in Π g . Next let us consider the followings

x, e -λt ξ(x)y, e -λt ξ(x)z + e -λt D x ξ(x)σ(t, x)y).

Therefore one can easily check that ū (resp. v) is a viscosity subsolution (resp. supersolution) of

(5.5.5) Let now F be the function from [0, T ] × IR k+1+d × S k (S k is the space of symmetric real matrices of dimension k) which with (t, x, y, z, M ) associates F (t, x, y, z, M ) ∈ IR and verifying

We choose λ great enough in such a way that the mapping y ∈ IR -→ F (t, x, y, z, M ) ∈ IR is strictly decreasing for all (t, x, z, M ) ∈ [0, T ] × IR k+1 × S k . Finally note that for all ǫ > 0 the function v + ǫ t is also a supersolution solution of (5.5.5). Therefore in order to obtain the comparison result it is enough to show that ū ≤ v + ǫ t and then to take the limit as ǫ → 0. Step (iv): Last part of the proof.

We are going to show by contradiction that: ∀R > 0 Next as v ′ is continuous, combined with point (i), we deduce the existence of N > 0 such that for any n ≥ N , |v ′ (t n , x n )v ′ (t n , y n )| < δ 2 . Therefore for n ≥ N 0 ,

which implies that for any n ≥ N 0 , u ′ (t n , x n ) > v ′ (t n , x n ). On the other hand, by the results obtained in Steps (i)-(ii),

and then v ′ (t n , x n ) < H(t n , x n ) and u ′ (t n , x n ) > L(t n , x n ). As u ′ (resp. v ′ + ǫ t ) is a sub (resp. super) solution of ((5.5.4) modified), we then have -p n -1 2 T r(σσ ⊤ (t n , x n )X n )-< b, n(x ny n ) > -f (t n , x n , u ′ (t n , x n ), n(x ny n )) ≤ 0, and With the same argument as in ( [START_REF] El Karoui | Les aspects probabilistes du contrôle stochastique[END_REF], pp. 734), under (H0),(A1)-(A2), we obtain that lim inf n→∞ Λ n ≤ 0 and then ε ≤ 0 which is contradictory. Finally taking the limits in (5.5.6), first when R → ∞ then ε → ∞, we obtain u ′ ≤ v ′ and then u ≤ v.

As a by-product we have:

Theorem 5.5.2. Under (H0),(A1) and (A2) we have:

(i) There is a unique continuous viscosity solution of (5.5.4) with polynomial growth ;

(ii) The function u is also a unique viscosity solution, in the class Π g , for the following max-min problem:

max v(t, x) -H(t, x) ; min v(t, x) -L(t, x); -∂ t v(t, x) -L X v(t, x)f (t, x, v(t, x), σ(t, x) ⊤ D x v(t, x)) = 0; v(T, x) = g(x).

(5.5.7)