Ying Hu
email: ying.hu@univ-rennes1.fr.

Shanjian Tang
email: sjtang@fudan.edu.cn.

Switching Game of Backward Stochastic Differential Equations and Associated System of Obliquely Reflected Backward Stochastic Differential Equations

Keywords: switching game, backward stochastic differential equations, oblique reflection AMS Subject Classifications. 60H10

This paper is concerned with the switching game of a one-dimensional backward stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the penalization method, the monotone convergence, and the a priori estimates.

Introduction

In this paper, we study the system of reflected BSDEs along an oblique direction arising naturally from the problem of switching game of a scalar-valued BSDE. Let us first describe precisely the switching game problem by introducing some notations and hypotheses.

Let us fix a nonnegative real number T > 0. First of all, W = {W t } t≥0 is a standard Brownian motion with values in R d defined on some complete probability space (Ω, F, P ). {F t } t≥0 is the natural filtration of the Brownian motion W augmented by the P -null sets of F. All the measurability notions will refer to this filtration. In particular, the sigma-field of predictable subsets of [0, T ] × Ω is denoted by P. Define Λ := {1, • • • , m 1 } and Π := {1, • • • , m 2 }.

We denote by S 2 (R m 1 ×m 2 ) or simply by S 2 the set of R m 1 ×m 2 -valued, adapted and càdlàg processes {Y (t)} t∈[0,T ] such that

||Y || S 2 := E sup t∈[0,T ] |Y (t)| 2 1/2 < +∞. (S 2 , || • || S 2 ) is then a Banach space.
We denote by M 2 ((R m 1 ×m 2 ) d ) or simply by M 2 the set of (equivalent classes of) predictable processes {Z(t)} t∈[0,T ] with values in (R m 1 ×m 2 ) d such that

||Z|| M 2 := E T 0 |Z(s)| 2 ds 1/2 < +∞.
M 2 is then a Banach space endowed with this norm.

We define also N 2 (R m 1 ×m 2 ) : = {K = (K ij ) ∈ S 2 : for any (i, j) ∈ Λ × Π, K ij (0) = 0, and K ij (•) is increasing }, which is abbreviated as N 2 . (N 2 , || • || S 2 ) is then a Banach space. Let ψ be a random function ψ : [0, T ] × Ω × R × R d × Λ × Π → R whose component ψ(•, i, j) is measurable with respect to P ⊗ B(R) ⊗ B(R d ) for each pair (i, j) ∈ Λ × Π, and satisfies the following Lipschitz condition.

Hypothesis 1.1. (i) The generator ψ(•, 0, 0) := (ψ(•, 0, 0, i, j)) i∈Λ,j∈Π ∈ M 2 .

(ii) There exists a constant C > 0 such that, for each (t, y, y ′ , z, z ′ , i, j)

∈ [0, T ] × R × R × R d × R d × Λ × Π, |ψ(t, y, z, i, j) -ψ(t, y ′ , z ′ , i, j)| ≤ C(|y -y ′ | + |z -z ′ |), a.s.
The functions k and l are defined on Λ × Λ and Π × Π, respectively; their values are both positive. We make the following assumption on the functions k and l, which is standard in the literature. Hypothesis 1.2. (i) For i ∈ Λ, k(i, i) = 0. For (i, i ′ ) ∈ Λ × Λ such that i = i ′ , k(i, i ′ ) > 0.

(ii) For j ∈ Π, l(j, j) = 0. For (j, j ′ ) ∈ Π × Π such that j = j ′ , l(j, j ′ ) > 0.

(iii) For any (i, i ′ , i ′′ ) ∈ Λ × Λ × Λ such that i = i ′ and i ′ = i ′′ , k(i, i ′ ) + k(i ′ , i ′′ ) > k(i, i ′′ ).

(iv) For any (j, j ′ , j ′′ ) ∈ Π × Π × Π such that j = j ′ and j ′ = j ′′ , l(j, j ′ ) + l(j ′ , j ′′ ) > l(j, j ′′ ).

Definition 1.1. An admissible switching process for Player I ( resp. II ) on [t, T ] with initial value a 0 ∈ Λ ( resp. b 0 ∈ Π ) is defined to be a pair of sequences

{a i , θ i } i≥0 (resp. {b i , τ i } i≥0 ), such that each θ i ( resp. τ i ) is an F . -stopping time with t = θ 0 ≤ θ 1 ≤ • • • ≤ T, a.s. ( resp. t = τ 0 ≤ τ 1 ≤ • • • ≤ T, a.s. ), each a i ( resp. b i ) is F θ i ( resp. F τ i ) measurable
A a t := A a [t, T ], a ∈ Λ; A t := a∈Λ A a t and B b t := B b [t, T ], b ∈ Π; B t := b∈Π B b t .
We shall identify {a

i , θ i } i≥0 ∈ A a [t, T ] with a(s) = a 0 χ {θ 0 } (s) + N i=1 a i-1 χ (θ i-1 ,θ i ] (s), s ∈ [t, T ]. (1.1)
For any a(•) ∈ A t , we define the associated (cost) process A a(•) on [t, T ] as follows:

A a(•) (s) = N -1 j=1 k(a j-1 , a j )χ [θ j ,T ] (s), s ∈ [t, T ]. (1.2)
Obviously, A a(•) (•) is a càdlàg process. In an identical way, we define

B b(•) on [t, T ] for b(•) ∈ B t .
Definition 1.2. For t ∈ [0, T ] and a ∈ Λ( resp. b ∈ Π ), an admissible strategy α a,t ( resp. β b,t ) with the initial value a ∈ Λ (resp. b ∈ Π) for player

I (resp. II) on [t, T ] is a mapping α a,t : ∪ b∈B B b [t, T ] → A a [t, T ] (resp. β b,t : ∪ a∈A A a [t, T ] → B b [t, T ]) such that b(s) = b(s) (resp. a(s) = a(s)) a.s. ∀s ∈ [t, t], implies α a,t [b(•)](s) = α a,t [ b(•)](s) (resp. β b,t [a(•)](s) = β b,t [ a(•)](s)) for s ∈ [t, t].
We denote by Γ a t (resp. ∆ b t ) all admissible strategies with the initial value a ∈ Λ (resp. b ∈ Π) for player I (resp. II) on [t, T ] . We adopt the convention that

A a T = {a}, Γ a T = {a} and B b T = {b}, ∆ b T = {b}.
Let ξ be an R m 1 ×m 2 -valued F T -measurable random variable. Now we are in position to introduce the switched BSDEs for both players. For t ∈ [0, T ], a(•) ∈ A t and b(•) ∈ B t , consider the following BSDE:

U (s) = ξ a(T )b(T ) + A a(•) (T ) -A a(•) (s) -B b(•) (T ) -B (•) (s) + T s ψ(r, U (r), V (r), a(r), b(r)) dr - T s V (r) dW (r), s ∈ [t, T ]. (1.3)
This is a (slightly) generalized BSDE: it is equivalent to the following standard BSDE:

Ū (s) = ξ a(T )b(T ) + A a(•) (T ) -B b(•) (T ) + T s ψ(r, Ū (r) -A a(•) (r) + B b(•) (r), V (r), a(r)) dr - T s V (r) dW (r), s ∈ [t, T ] (1.4)
via the simple change of variable: (•) . We note that U is only a càdlàg process.

Ū (s) = U (s) + A a(•) (s) -B b(•) (s), V (s) = V (s). Hence, for each pair (a(•), b(•)) ∈ A t × B t , BSDE (1.3) has a unique solution in S 2 × M 2 , which will be denoted by U a(•),b(•) , V a(•),b
The upper and lower switching game problems with the initial scheme (i, j) ∈ Λ×Π are defined as follows:

ess sup

β∈∆ j t ess inf a(•)∈A i t U a(•),β(a(•)) (t)
and ess inf

α∈Γ i t ess sup b(•)∈B j t U α(b(•)),b(•) (t), respectively. If Y ij (t) := ess sup β∈∆ j t ess inf a(•)∈A i t U a(•),β(a(•)) (t) = ess inf α∈Γ i t ess sup b(•)∈B j t U α(b(•)),b(•) (t)
for some (i, j) ∈ Λ × Π, we say that the switching game with the initial scheme (i, j) ∈ Λ × Π has a value Y ij (t).

The above switching game (see, e.g. [START_REF] Tang | Switching games of stochastic differential systems[END_REF]) is associated to the following Bellman-Isaacs equation, which is a new type of reflected backward stochastic differential equation (RBSDE for short) with oblique reflection: for (i, j) ∈ Λ × Π and t ∈ [0, T ],

                                         Y ij (t) = ξ ij + T t ψ(s, Y ij (s), Z ij (s), i, j) ds - T t dK ij (s) + T t dL ij (s) - T t Z ij (s) dW (s), Y ij (t) ≤ min i ′ =i {Y i ′ j (t) + k(i, i ′ )}, Y ij (t) ≥ max j ′ =j {Y ij ′ (t) -l(j, j ′ )}, T 0 Y ij (s) -min i ′ =i {Y i ′ j (s) + k(i, i ′ )} dK ij (s) = 0, T 0 Y ij (s) -max j ′ =j {Y ij ′ (s) -l(j, j ′ )} dL ij (s) = 0. (1.5)
Here, the unknowns are the processes

{Y (t)} t∈[0,T ] , {Z(t)} t∈[0,T ] , {K(t)} t∈[0,T ] ,
and {L(t)} t∈[0,T ] , which are required to be adapted with respect to the natural completed filtration of the Brownian motion W . Moreover, K and L are componentwisely increasing processes. The last two relations in (1.5) are called the upper and lower minimal boundary conditions. One-dimensional RBSDEs were first studied by El Karoui et al. [START_REF] Karoui | Reflected solutions of backward SDE's, and related obstacle problems for PDE's[END_REF] in the case of one obstacle, and then by Cvitanic and Karatzas [START_REF] Cvitanic | Backward stochastic differential equations with reflection and Dynkin games[END_REF] in the case of two obstacles. In both papers, it is recognized that one-dimensional reflected BSDEs, with one obstacle and with two obstacles, are generalizations of optimal stopping and Dynkin games, respectively. Nowadays, the literature on one-dimensional reflected BSDEs is very rich. The reader is referred to Peng and Xu [START_REF] Peng | The smallest g-supermartingale and reflected BSDE with single and double L 2 obstacles[END_REF] and Buckdahn and Li [START_REF] Buckdahn | Stochastic differential games with reflection and related obstacle problems for Isaacs equations[END_REF], among others, for the one-dimensional reflected BSDEs with two obstacles.

Multi-dimensional RBSDEs were studied by Gegout-Petit and Pardoux [START_REF] Gegout-Petit | Equations différentielles stochastiques rétrogrades réfléchies dans un convexe[END_REF], but their BSDE is reflected on the boundary of a convex domain along the inward normal direction, and their method depends heavily on the properties of this inward normal reflection (see ( 1)-(3) in [START_REF] Gegout-Petit | Equations différentielles stochastiques rétrogrades réfléchies dans un convexe[END_REF]). We note that in a very special case (e.g., ψ is independent of z), Ramasubramanian [START_REF] Ramasubramanian | Reflected backward stochastic differential equations in an orthant[END_REF] studied a BSDE in an orthant with oblique reflection. Multi-dimensional BSDEs reflected along an oblique direction rather than a normal direction, still remains to be open in general, even in a convex domain, let alone in a nonconvex domain. Note that there are some papers dealing with SDEs with oblique reflection (see, e.g. [START_REF] Lions | Stochastic differential equations with reflecting boundary conditions[END_REF][START_REF] Dupuis | SDEs with oblique reflection on nonsmooth domains[END_REF]).

In our previous work [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF], we studied the optimal switching problem for one-dimensional BSDEs, and the associated following type of obliquely reflected multi-dimensional BSDEs:

for i ∈ Λ,                          Y i (t) = ξ i + T t ψ(s, Y i (s), Z i (s), i) ds - T t dK i (s) - T t Z i (s) dW (s), Y i (t) ≤ min i ′ =i {Y i ′ (t) + k(i, i ′ )}, T 0 Y i (s) -min i ′ =i {Y i ′ (s) + k(i, i ′ )} dK i (s) = 0. (1.6)
It should be added that a less general form of RBSDE (1.6) (where the generator ψ does not depend on (y, z)) is suggested by [START_REF] Carmona | Pricing asset scheduling flexibility using optimal switching[END_REF]. But they did not discuss the existence and uniqueness of solution, which is considered to be difficult. See Remark 3 in [START_REF] Carmona | Pricing asset scheduling flexibility using optimal switching[END_REF].

Recently, Tang, Zhong and Koo [START_REF] Tang | Optimal switching of one-dimensional reflected BSDEs and associated multidimensional BSDEs with oblique reflection[END_REF] discussed the mixed switching and stopping problem for one-dimensional BSDEs, and obtained the existence and uniqueness result for the associated following type of multi-dimensional obliquely reflected BSDEs: for

i ∈ Λ and t ∈ [0, T ],                                  Y i (t) = ξ i + T t ψ(s, Y i (s), Z i (s), i) ds - T t dK i (s) + T t dL i (s) - T t Z i (s) dW (s), Y i (t) ≤ min i ′ =i {Y i ′ (t) + k(i, i ′ )}, Y i (t) ≥ S(t), T 0 Y i (s) -min i ′ =i {Y i ′ (s) + k(i, i ′ )} dK i (s) = 0, T 0 (Y i (t) -S(t)) dL i (t) = 0.
(

Here, S is a given {F t , 0 ≤ t ≤ T }-adapted process with some suitable regularity. RBSDE (1.5) is more complicated than that of RBSDE (1.6) arising from the optimal switching problem for BSDEs. For each fixed j ∈ Π, if we do not impose the following constraint:

Y ij (t) ≥ max j ′ =j {Y ij ′ (t) -l(j, j ′ )}, t ∈ [0, T ], (1.8)
and its related boundary condition:

T 0 Y ij (s) -max j ′ =j {Y ij ′ (s) -l(j, j ′ )} dL ij (s) = 0, (1.9)
then we can take L ≡ 0, and RBSDE (1.5) is reduced to RBSDE (1.6). RBSDE (1.5) evolves in the closure Q of domain Q:

Q := (y ij ) ∈ R m 1 ×m 2 : y ij < y i ′ j + k(i, i ′ )
for any i, i ′ ∈ Λ such that i ′ = i and j ∈ Π;

y ij > y ij ′ -l(j, j ′ )
for any j, j ′ ∈ Π such that j ′ = j and i ∈ Λ ,

(1.10)
which is convex and unbounded. The boundary ∂Q of domain Q consists of the boundaries ∂D - ij and ∂D + ij , (i, j) ∈ Λ × Π, with

D - ij := {(y ij ) ∈ R m 1 ×m 2 : y ij < y i ′ j + k(i, i ′ ), for any i ′ ∈ Λ such that i ′ = i} and D + ij := {(y ij ) ∈ R m 1 ×m 2 : y ij > y ij ′ -l(j, j ′ ), for any j ′ ∈ Π such that j ′ = j} for (i, j) ∈ Λ × Π. That is, ∂Q = m 1 ∪ i=1 m 2 ∪ j=1 ∂D - ij ∪ ∂D + ij .
In the interior of Q, each equation in (1.5) is independent of others. On the boundary, say ∂D - ij (resp. ∂D + ij ), the (i, j)-th equation is switched to another one (i ′ , j) (resp. (i, j ′ )), and the solution is reflected along the oblique direction -e ij (resp. e ij ), which is the negative (resp. positive) direction of the (i, j)-th coordinate axis.

The existence of solution for RBSDE (1.5) constitutes a main contribution of this paper. We prove the existence by a penalization method. Proving the existence of solution of RBSDE (1.5) presents new difficulties when one follows our previous work [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF] using the penalization method. In fact, in order to establish the a priori estimates which are essential for the proof of the existence, we have to use the representation of solutions to obliquely reflected BSDEs proved in [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF], and we have to impose the additional technical condition that the generator ψ is uniformly bounded. The question of uniqueness is still an open problem.

The rest of the paper is organized as follows: in Section 2, we prove the existence of solution for RBSDE (1.5) by a penalization method. The last section is devoted to discussions on some possible extensions.

Existence of an adapted solution to the associated RBSDE

In this section, we state and prove our existence result for RBSDE (1.5).

We need the following additional technical assumption.

Hypothesis

(R m 1 ×m 2 ) d ), t → ψ(t, Y ij (t), Z ij (t), i, j) belongs to L 1 (0, T ; R m 1 ×m 2 ) and P -a.s., RBSDE (1.5) holds for each t ∈ [0, T ].
The main result of this paper is the following existence of an adapted solution to RBSDE (1.5).

Theorem 2.1. Let Hypotheses 1.1, 1.2 and 2.1 be satisfied. Assume that

ξ ∈ L 2 (Ω, F T , P ; R m 1 ×m 2 ) takes values in Q. Then RBSDE (1.5) has an adapted solution (Y, Z, K, L) in S 2 × M 2 × (N 2 ) 2 .
We first sketch the proof. Sketch of the Proof: The proof is divided into five subsections. In Subsection 3.1, we introduce the penalized RBSDEs whose existence of solution follows from a slightly generalized result in [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF]. In Subsection 3.2, we give the (implicit) representation of these solutions. In Subsection 3.3, we state a fundamental lemma and some (uniform) a priori estimates for these solutions. In Subsection 3.4, we prove the (monotone) convergence of these solutions. And the last subsection is devoted to checking out the boundary conditions.

The penalized RBSDEs

We shall use a penalization method to construct a solution to RBSDE (1.5). We observe (as mentioned in the introduction) that RBSDE (1.5) consists of the m 2 systems of m 1 -dimensional obliquely reflected BSDEs of the form like (1.6):

                         Y ij (t) = ξ ij + T t ψ(s, Y ij (s), Z ij (s), i, j) ds - T t dK ij (s) + T t dL ij (s) - T t Z ij (s) dW (s), Y ij (t) ≤ min i ′ =i {Y i ′ j (t) + k(i, i ′ )}, T 0 Y ij (s) -min i ′ =i {Y i ′ j (s) + k(i, i ′ )} dK ij (s) = 0; i ∈ Λ, (2.1)
with the unknown processes being

(Y ij , Z ij , K ij ; i = 1, 2, . . . , m 1 )
(the process (L 1j , . . . , L m 1 j ) is taken to be previously given) for j = 1, 2, . . . , m 2 . These m 2 systems have been well studied by Hu and Tang [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF]. In RBSDE (1.5), they are coupled together by the processes (L 1j , . . . , L m 1 j ) through the constraint

Y ij (t) ≥ max j ′ =j {Y ij ′ (t) -l(j, j ′ )}, (i, j) ∈ Λ × Π (2.2)
and the minimal boundary condition:

T 0 Y ij (s) -max j ′ =j {Y ij ′ (s) -l(j, j ′ )} dL ij (s) = 0, (i, j) ∈ Λ × Π. (2.3)
Therefore, it is natural to consider the following penalized system of RBSDEs (the unknown processes are (Y ij , Z ij , K ij ; i ∈ Λ, j ∈ Π):

                                 Y ij (t) = ξ ij + T t ψ(s, Y ij (s), Z ij (s), i, j) ds +n m 2 j ′ =1 T t Y ij (s) -Y ij ′ (s) + l(j, j ′ ) -ds - T t dK ij (s) - T t Z ij (s) dW (s); Y ij (t) ≤ min i ′ =i {Y i ′ j (t) + k(i, i ′ )}; T 0 Y ij (s) -min i ′ =i {Y i ′ j (s) + k(i, i ′ )} dK ij (s) = 0; (i, j) ∈ Λ × Π.
(2.4) Note that when j ′ = j, we have, in view of Hypothesis 1.2 (ii),

(Y ij (s) -Y ij ′ (s) + l(j, j ′ )) -= 0.
(2.5)

Note also that for any integer n, the ij-th component of the generator of (2.4) depends also on y ij ′ , j ′ = j. Hence we cannot apply directly the existence result in [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF]. However, by slightly adapting the relevant arguments in [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF], we have the following assertion.

Proposition 2.1. For any integer n, RBSDE (2.4)

has an adapted solution (Y n , Z n , K n ) in the space S 2 × M 2 × N 2 .
Proof. Let the integer n be fixed. For an integer m, consider the following penalized BSDE whose solution is denoted by (Y n,m , Z n,m ):

                             Y ij (t) = ξ ij + T t ψ(s, Y ij (s), Z ij (s), i, j) ds +n m 2 j ′ =1 T t Y ij (s) -Y ij ′ (s) + l(j, j ′ ) -ds -m m 1 i ′ =1 T t Y ij (s) -Y i ′ j (s) -k(i, i ′ ) + ds - T t Z ij (s) dW (s); (i, j) ∈ Λ × Π.
(2.6)

From the comparison theorem for multi-dimensional BSDEs in [START_REF] Hu | On the comparison theorem for multi-dimensional BSDEs[END_REF], {Y n,m ij (t)} m is decreasing. Following the relevant arguments in [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF], we prove that there exists an adapted solution (Y n , Z n , K n ) in the space S 2 × M 2 × N 2 , and moreover

Y n ij (t) = lim m→∞ Y n,m ij (t).

Representation and uniqueness of solution to the penalized RBSDE

Note again that for any integer n, the ij-th component of the generator of (2.4) depends also on y ij ′ , j ′ = j. Hence we cannot apply directly the Representation Theorem 3.1 in [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF]. Nevertheless, by defining a new generator ψ(r, y, z, i, j; n) := ψ(r, y, z, i, j) + n

j ′ =j y -Y n ij ′ (r) + l(j, j ′ ) - (2.7) for (r, y, z) ∈ [t, T ] × R × R d , and (i, j) ∈ Λ × Π to include Y n ij ′ , j ′ = j in it, we have for each j ∈ Π, the triplet (Y n ij , Z n ij , K n ij ; i ∈ Λ) is an adapted solution of the following m 1 -dimensional RBSDE:                          Y ij (t) = ξ ij + T t ψ(s, Y ij (s), Z ij (s), i, j; n) ds - T t dK ij (s) - T t Z ij (s) dW (s); Y ij (t) ≤ min i ′ =i {Y i ′ j (t) + k(i, i ′ )}; T 0 Y ij (s) -min i ′ =i {Y i ′ j (s) + k(i, i ′ )} dK ij (s) = 0; i ∈ Λ.
(2.8)

Then we can apply the Representation Theorem 3.1 in [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF].

In order to state this representation theorem, first we introduce some notations. Let {θ j } ∞ j=0 be an increasing sequence of stopping times with values in [t, T ] and ∀j, α j is an F θ j -measurable random variable with values in Λ, and χ is the indicator function. We define

a(s) := α 0 χ {θ 0 } (s) + ∞ j=1 α j-1 χ (θ j-1 ,θ j ] (s), s ∈ [t, T ].
The sequence {θ j , α j } ∞ j=0 or a(•) is said to be an admissible switching strategy starting from the mode α 0 , if there exists an integer-valued random variable N such that θ N = T , P -a.s. and N ∈ L 2 (F T ).

We denote by A t the set of all these admissible switching strategies and by A i t the subset of A consisting of admissible switching strategies starting from the mode i.

For any a(•) ∈ A t , we define the associated (cost) process A a(•) as follows:

A a(•) (s) = N -1 j=1 k(α j-1 , α j )χ [θ j ,T ] (s), s ∈ [t, T ].
Obviously, A a(•) (•) is an adapted increasing càdlàg process, and A a(•) (T ) ∈ L 2 (F T ) thanks to the fact that N ∈ L 2 (F T ). Now we are in position to introduce the switched BSDE. For (t, i, j) ∈ [0, T )×Λ×Π and a(•) ∈ A i t , (U

a(•),n j , V a(•),n j
) is the unique solution to the following BSDE:

U j (s) = ξ a(T )j + [A a(•) (T ) -A a(•) (s)] + T s ψ(r, U j (r), V j (r), a(r), j; n) dr - T s V j (r) dW (r), s ∈ [t, T ].
(2.9)

On the other hand, for a(•) ∈ A i t of the following form:

a(s) = iχ t (s) + N p=1 α p-1 χ (θ p-1 ,θp] (s), s ∈ [t, T ],
(2.10) we define for s ∈ [t, T ] and j ∈ Π,

Y a(•),n j (s) := N p=1 Y n α p-1 ,j (s)χ [θ p-1 ,θp) (s) + ξ a(T ) χ {T } (s), (2.11) Z a(•),n j (s) := N p=1 Z n α p-1 ,j (s)χ [θ p-1 ,θp) (s), K a(•),n j (s) := N p=1 θp∧s θ p-1 ∧s dK n α p-1 ,j (r),
and ) is a solution to the following BSDE:

A a(•),n j (s) = N -1 p=1 [Y n αp,j (θ p ) + k(α p-1 , α p ) -Y n α p-1 ,j (θ p )]χ [θp,T ] (s), s ∈ [t, T ]. (2.
Y a(•),n j (s) (2.13) = ξ a(T )j -[( K a(•),n j (T ) + A a(•),n j (T )) -( K a(•),n j (s) + A a(•),n j (s))] + A a(•) (T ) -A a(•) (s) + T s ψ(r, Y a(•),n j (r), Z a(•),n j (r), a(r), j; n) dr - T s Z a(•),n j (r) dW (r). As Y n a(s)j ′ (s) = Y a(•),n j ′ (s), a.e. s ∈ [t, T ],
BSDE (2.9) can be rewritten as the following equation:

U j (s) = ξ a(T )j + [A a(•) (T ) -A a(•) (s)] + T s ψ(r, U j (r), V j (r), a(r), j) dr +n j ′ =j T s (U j (r) -Y a(•),n j ′ (r) + l(j, j ′ )) -dr - T s V j (r) dW (r), s ∈ [t, T ].
(2.14)

We are now ready to state the representation formula which is taken from Theorem 3.1 in [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF]. Proposition 2.2. Assume that a(•) ∈ A i t . Then we have ess sup

a(•)∈A i t Y a(•),n j (s) -U a(•),n j (s) = 0, s ∈ [t, T ], j ∈ Π, (2.15)
which, putting in particular s = t, implies

Y n ij (t) = essinf a(•)∈A i t U a(•),n j (t), j ∈ Π.
It is crucial to observe that the above representation formula is implicit since U a(•),n j still depends on Y n . However, it is sufficient for us to deduce the a priori estimates. Also, it is sufficient for us to deduce the uniqueness of the solution to the penalized RBSDE (2.4). In fact, if we have two solutions

(Y k,n ij , Z k,n ij , K k,n ij ; i ∈ Λ) with k = 1, 2, we can define (U k,a(•),n j , V k,a(•),n j ), k = 1, 2, for (t, i, j) ∈ [0, T ) × Λ × Π and a(•) ∈ A i
t , as the unique solution to BSDE (2.9) with Y n ij being replaced with Y 1,n ij and Y 2,n ij , respectively. We have the estimate for U

1,a(•),n j -U 2,a(•),n j : |U 1,a(•),n j (t) -U 2,a(•),n j (t)| 2 ≤ C n E[ T t |Y 1,n (s) -Y 2,n (s)| 2 ds|F t ]
for some constant C n . Then applying the above representation and Gronwall inequality, we obtain that Y 1,n = Y 2,n , which gives the uniqueness.

A basic lemma and a priori estimates

For each integer n, let (Y n , Z n , K n ) ∈ S 2 ×M 2 ×N 2 be the adapted solution of RBSDE (2.4). Intuitively, as n tends to +∞, we expect that the sequence of solutions

{(Y n , Z n , K n )} ∞ n=1
together with the penalty term

L n ij (t) := n m 2 j ′ =1 t 0 Y ij (s) -Y ij ′ (s) + l(j, j ′ ) -ds, (t, i, j) ∈ [0, T ] × Λ × Π
will have a limit (Y, Z, K, L), which solves RBSDE (1.5).

For this purpose, it is crucial to prove that the penalty term is bounded in some suitable sense. Then we are naturally led to compute

Y ij (t) -Y ij ′ (t) + l(j, j ′ ) -,
using Itô-Meyer's formula, as done in [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF]. However, in our present situation, the additional term K n appears in RBSDE (2.4), which gives rise to a serious difficulty to derive the bound of L n in the preceding procedure. In what follows, we shall use the representation result for Y n of Proposition 2.2 to get around the difficulty.

We have the following basic lemma.

Lemma 2.1. For j, j ′ ∈ Π and a(•) ∈ A i t , we have

n U a(•),n j (s) -Y a(•),n j ′ (s) + l(j, j ′ ) - ≤ 2 ψ ∞ , s ∈ [t, T ]. (2.16)
Here, U a(•),n j and Y a(•),n j ′ are defined by (2.9) and (2.11), respectively.

Proof.

We suppress the superscripts (a(

•), n) of U a(•),n j , U a(•),n j ′ , V a(•),n j , V a(•),n j ′ , Y a(•),n j , Y a(•),n j ′ , Y a(•),n j ′′ and Z a(•),n j ′
for simplicity. The whole proof consists of the following two steps.

Step 1. Calculation of the process U jj ′ (s) -, where

U jj ′ (s) := U j (s)-Y j ′ (s)+ l(j, j ′ ), s ∈ [t, T ] using Itô-Meyer's formula.
In view of (2.14) and (2.13), the process U jj ′ (s), s ∈ [t, T ] satisfies the following BSDE:

U jj ′ (s) = U jj ′ (T ) + T s ψ(r, U j (r), V j (r), a(r), j) -ψ(r, Y j ′ (r), Z j ′ (r), a(r), j ′ ) dr +n j ′′ =j T s (U j -Y j ′′ + l(j, j ′′ ))(r) -dr -n j ′′ =j ′ T s ( Y j ′ -Y j ′′ + l(j ′ , j ′′ ))(r) -dr + T s d K a(•),n j (r) + A a(•),n j (r) - T s (V j (r) -Z j ′ (r)) dW (r), s ∈ [t, T ].
(2.17) Applying Itô-Meyer's formula (see, e.g. Meyer [START_REF] Meyer | Un cours sur les intégrales stochastiques[END_REF]), we have

U jj ′ (s) -+ n j ′′ =j T s χ L - jj ′ (r)(U j -Y j ′′ + l(j, j ′′ ))(r) -dr -n j ′′ =j ′ T s χ L - jj ′ (r)( Y j ′ -Y j ′′ + l(j ′ , j ′′ ))(r) -dr + T s d L jj ′ (r) = - T s χ L - jj ′ (r) ψ(r, U j (r), V j (r), a(r), j) -ψ(r, Y j ′ (r), Z j ′ (r), a(r), j ′ ) dr - T s χ L - jj ′ (r-) d K a(•),n j (r) + A a(•),n j (r) + T s χ L - jj ′ (r-)(V j (r) -Z j ′ (r)) dW (r), s ∈ [t, T ] (2.18) where L - jj ′ := {(s, ω) ∈ [t, T ] × Ω : U jj ′ (s) < 0}, (2.19)
and L jj ′ is a càdlàg increasing process. The above equation can be rewritten as

U jj ′ (s) -+ n T s χ L - jj ′ (U j -Y j ′ + l(j, j ′ ))(r) -dr + T s d L jj ′ (r) = - T s χ L - jj ′
(r) ψ(r, U j (r), V j (r), a(r), j)ψ(r, Y j ′ (r), Z j ′ (r), a(r), j ′ ) dr

- T s χ L - jj ′ (r-) d( K a(•),n j (r) + A a(•),n j (r)) + T s χ L - jj ′ (r-) V j (r) -Z j ′ (r) dW (r) +n T s I 1,jj ′ (r) dr + n j ′′ =j,j ′′ =j ′ T s I 2,jj ′ j ′′ (r) dr
(2.20) where the two integrands I 1,jj ′ and I 2,jj ′ j ′′ are defined as follows

I 1,jj ′ (r) := χ L - jj ′ (r)( Y j ′ -Y j + l(j ′ , j)) -, r ∈ [t, T ]
(2.21) and

I 2,jj ′ j ′′ (r) := χ L - jj ′ (r)[( Y j ′ -Y j ′′ + l(j ′ , j ′′ ))(r) --(U j -Y j ′′ + l(j, j ′′ ))(r) -], r ∈ [t, T ].
(2.22)

Step 2. The BSDE for the process

{(U j (s) -Y j ′ (s) + l(j, j ′ )) -, s ∈ [t, T ]}.
In view of Proposition 2.2, Y j ≤ U j , we have

I 1,jj ′ ≤ χ L - jj ′
( Y j ′ -U j + l(j ′ , j)) -= 0, j, j ′ ∈ Π, (2.23) thanks to the fact that l(j, j ′ ) + l(j ′ , j) > l(j, j) = 0.

Hence, I 1,jj ′ (r) = 0. Now we can rewrite (2.20) as the following equation:

(U j (s) -Y j ′ (s) + l(j, j ′ )) - = n j ′′ =j,j ′′ =j ′ T s I 2,jj ′ j ′′ (r) dr + T s I 3,jj ′ (r) dr - T s d L jj ′ (r) - T s χ L - jj ′ (r-) d( K a(•),n j (r) + A a(•),n j (r)) -n T s (U j (r) -Y j ′ (r) + l(j, j ′ )) -dr + T s χ L - jj ′ (r-) V j (r) -Z j ′ (r) dW (r), (2.24)
where

I 3,jj ′ (r) := χ L - jj ′
(r) ψ(r, Y j ′ (r), Z j ′ (r), a(r), j ′ )ψ(r, U j (r), V j (r), a(r), j) (2.25) for r ∈ [t, T ]. We now show that I 2,jj ′ j ′′ ≤ 0.

(2.26)

In fact, for j, j ′ , j ′′ ∈ Π, taking into consideration the elementary inequality that x - 1x - 2 ≤ (x 1x 2 ) -, for any two real numbers x 1 and x 2 , we have

I 2,jj ′ j ′′ = χ L - jj ′ [( Y j ′ -Y j ′′ + l(j ′ , j ′′ ))(r) --(U j -Y j ′′ + l(j, j ′′ ))(r) -] ≤ χ L - jj ′ Y j ′ -U j + l(j ′ , j ′′ ) -l(j, j ′′ ) - = 0.
(2.27)

The last equality holds in the last relations, since {y ∈ R m : y jy j ′ + l(j, j ′ ) < 0} ∩ {y ∈ R m : y j ′y j + l(j ′ , j ′′ )l(j, j ′′ ) < 0} = ∅, thanks to Hypothesis 1.2 (iv), i.e., l(j, j ′ ) + l(j ′ , j ′′ ) > l(j, j ′′ ).

Define for r ∈ [t, T ],

I jj ′ (r) := n j ′′ =j,j ′′ =j ′ r t I 2,jj ′ j ′′ (s) ds - r t d L jj ′ (s) - r t χ L - jj ′ (s-) d( K a(•),n j (s) + A a(•),n j (s)).
Obviously, the process I jj ′ (•) is a càdlàg decreasing process for any (j, j ′ ) ∈ Π × Π. BSDE (2.24) is finally written as the following equation:

(U j (s) -Y j ′ (s) + l(j, j ′ )) - = T t dI jj ′ (r) + T s I 3,jj ′ (r) dr -n T s (U j (r) -Y j ′ (r) + l(j, j ′ )) -dr + T s χ L - jj ′ (r-) V j (r) -Z j ′ (r) dW (r), s ∈ [t, T ].
(2.28)

Then, we have the following formula

(U j (s) -Y j ′ (s) + l(j, j ′ )) -= E T s I 3,jj ′ (r) exp [-n(r -s)] dr F s +E T s exp [-n(r -s)] dI jj ′ (r) F s ≤ 2 ψ ∞ T s exp [-n(r -s)] dr. (2.29) Therefore, we have n U a(•),n j (s) -Y a(•),n j ′ (s) + l(j, j ′ ) - ≤ 2 ψ ∞ , s ∈ [t, T ].
This ends the proof.

Thanks to this basic lemma, we deduce easily the following a priori estimates.

Proof. From the RBSDE for Y n ij , in view of Hypothesis 2.1, using Itô's formula and Proposition 2.3, we have

E|Y n ij (0)| 2 + E T 0 |Z n ij (s)| 2 ds ≤ E|ξ ij | 2 + 2 m 2 j ′ =1 E T 0 |Y n ij (s)|n Y n ij (s) -Y n ij ′ (s) + l(j, j ′ ) -ds +2E T 0 |Y n ij (s)| • |ψ(s, Y n ij (s), Z n ij (s), i, j)| ds +2E T 0 |Y n ij (s)| dK n ij (s) ≤ C + CE T 0 |Y n ij (s)| ds + 2E sup t |Y n ij (t)|K n ij (T ) ≤ C ǫ + ǫE[(K n ij (T )) 2 ]
(2.33)

and

E[(K n ij (T )) 2 ] ≤ CE|ξ ij | 2 + CE|Y n ij (0)| 2 +C m 2 j ′ =1 E T 0 n Y n ij (s) -Y n ij ′ (s) + l(j, j ′ ) -2 ds +CE T 0 |ψ(s, Y n ij (s), Z n ij (s), i, j)| 2 ds + CE T 0 |Z n ij (s)| 2 ds ≤ C + CE T 0 |Z n ij (s)| 2 ds.
(2.34)

Combining the above two inequalities by taking a sufficiently small ǫ > 0, we conclude the proof. Define

β n ij (s) := n m 2 j ′ =1 Y n ij (s) -Y n ij ′ (s) + l(j, j ′ ) -. (2.35) Then L n ij (t) = t 0 β n ij (s) ds. (2.36)
Next, we prove that K n is absolutely continuous whose derivative is uniformly bounded.

Lemma 2.3. For (i, j) ∈ Λ × Π and an integer n, there is a uniformly bounded process α n ij such that K n ij has the following form:

K n ij (t) = t 0 α n ij (s) ds, t ∈ [0, T ]. ( 2

.37)

We now prove the strong convergence of Z n . Using Itô's formula, we have

|Y n ij (0) -Y m ij (0)| 2 + E T 0 |Z n ij (s) -Z m ij (s)| 2 ds = 2E T 0 Y n ij (s) -Y m ij (s) ψ(s, Y n ij (s), Z n ij (s), i, j) -α n ij (s) + β n ij (s) ds -2E T 0 Y n ij (s) -Y m ij (s) ψ(s, Y m ij (s), Z m ij (s), i, j) -α m ij (s) + β m ij (s) ds, ( 2
E T 0 |Z n (s) -Z m (s)| 2 ds = 0.
(2.44)

It is routine to show the strong convergence of {Y n } in the space S 2 . Since {α n } and {β } are uniformly bounded, the last assertion of the lemma is obvious.

Define for

(i, j, t) ∈ Λ × Π × [0, T ], K ij (t) := t 0 α ij (s) ds, L ij (t) := t 0 β ij (s) ds (2.45) and K := (K ij ), L := (L ij ). (2.46)
Finally, we shall show that (Y, Z, K, L) solves RBSDE (1.5).

In fact, it suffices to take the weak limit in L 2 (F T ) in BSDE (2.42) along a suitable subsequence to deduce that (Y, Z, K, L) solves the following BSDE: (2.47) It remains to check out the boundary conditions, which will be given in the next subsection.

Boundary conditions

Let us first prove that Y (t) ∈ Q.

On the one hand, as (Y n , Z n , K n ) satisfies (2.4), we have

Y n ij (t) ≤ min i ′ =i
{Y n i ′ j (t) + k(i, i ′ )}, from which we deduce, by taking limit when n tends to ∞, that

Y ij (t) ≤ min i ′ =i {Y i ′ j (t) + k(i, i ′ )}. ( 2
E T 0 Y ij (s) -max j ′ =j {Y ij ′ (s) -l(j, j ′ )} + β ij (s) ds = 0. (2.53)
The proof is then complete.

Concluding remarks

In this paper, we proved the existence of solution to RBSDE (1.5). But the question of uniqueness remains open. On the other hand, there exist different methods in the literature for the study of switching control and game problems. For the classical method of quasi-variational inequalities, the reader is referred to the book of Bensoussan and Lions [START_REF] Bensoussan | Impulse Control and Quasivariational Inequalities[END_REF]. See Tang and Yong [START_REF] Tang | Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach[END_REF], Pham, Ly Vath and Zhou [START_REF] Pham | Optimal switching over multiple regimes[END_REF] and Tang and Hou [START_REF] Tang | Switching games of stochastic differential systems[END_REF] and the references therein for the theory of variational inequalities and the dynamic programming for optimal stochastic switching control and switching games. But these works are restricted to the Markovian case. Recently, using the method of Snell envelope (see, e.g. El Karoui [START_REF] Karoui | Les aspects probabilistes du contrôle stochastique[END_REF]) combined with the theory of scalar valued RBSDEs, Hamadène and Jeanblanc [START_REF] Hamadène | On the starting and stopping problem: application in reversible investments[END_REF] studied the switching problem in the non-Markovian context. The obliquely reflected BSDE approach, first fully developed in Hu and Tang [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF] for optimal stochastic switching and taking the advantage of the theory and techniques of BSDEs, permits to state and solve these problems in a rather general non-Markovian framework. The link between the solution of RBSDE (1.5) and the problem of switching games constitutes another challenge.

  process due to the fact that Y n (s) satisfies the boundary condition in (2.4), ∀s ∈ [t, T ]. Then, for each j ∈ Π, the triplet ( Y

YZ

  ij (t) = ξ ij + T t ψ(s, Y ij (s), Z ij (s), i, j) dsij (s) dW (s), (i, j) ∈ Λ × Π.

  .48) On the other hand, from Proposition 2.3,Y n ij (t) -Y n ij ′ (t) + l(j, j ′ )Sending n to ∞, we deduce thatY ij (t) -Y ij ′ (t) + l(j, j ′ )

						-= 0.	(2.49)
	(2.48) and (2.49) shows that Y (t) ∈ Q.
	Now we check out the minimal boundary conditions.
	From (2.4), we have
	E	0	T	Y n ij (s) -min i ′ =i	{Y n i ′ j (s) + k(i, i ′ )}	-	α n ij (s) ds = 0.	(2.50)
	Setting n → ∞, we have
			T				-
	E			Y ij (s) -min				α ij (s) ds = 0.	(2.51)
		0
	On the other hand, from the construction, we have
	E	0	T	Y n ij (s) -max j ′ =j	{Y n ij

-≤ 2 ψ ∞ n . i ′ =i {Y i ′ j (s) + k(i, i ′ )} ′ (s)l(j, j ′ )} + β n ij (s) ds = 0. (2.52)

Setting n → ∞, we have

where C > 0 is a constant.

(2.31)

Proof. (i) According to the comparison theorem for multi-dimensional BSDEs in [START_REF] Hu | On the comparison theorem for multi-dimensional BSDEs[END_REF],

where Y n,m ij is defined via (2.6). Hence, the sequence {Y n ij (t)} ∞ n=1 is increasing by taking the limit when m tends to ∞.

We have the following two facts in view of (2.14):

(1) U

(2.32)

In view of the representation formula in Proposition 2.2, we conclude the proof.

(ii) Putting s = t in (2.16), we obtain

From Proposition 2.2, we deduce that

and the proof is complete.

Convergence of solutions

We first prove that (Z n , K n ) is bounded.

Lemma 2.2. The pair of processes

Proof. Fix the integer n. Consider the following penalized BSDEs:

with (i, j) ∈ Λ×Π and t ∈ [0, T ]. It has a unique solution, denoted by ( Ȳ n,m ij , Zn,m ij ). Proceeding similarly (in fact, much more simpler) as in Lemma 2.1, we can prove that for a constant C > 0,

From [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF], we have

(


A Brayek

B Tlilli

T Ghrib

C Nouveau

Investigation of vanadium and nitride alloys thin layers deposited by PVD

Keywords: PVD, VN, electrical resistivity, roughness, residual stress

In this work we present the technique of magnetron vapor deposition and the effect of several deposition parameters on the structural and morphological properties of prepared thin films. It was noted that the deposition time has an effect on the crystallinity, mechanical properties such as residual stress, roughness surface and the layer composition from target products. Studies were carried out on layers of vanadium (V) and the nitride vanadium (VN).

Introduction

Making materials in the thin film presents a main interest in very varied application, since the last decade we observe an increasing interest for studying surfaces for their essential role to exert a decisive influence on diverse characteristics as well as corrosion, catalyses, friction (wear, lubrication...), adherence, sintering, [1][2][3][4][5].

Alloys containing chromium and nitrogen present varied properties which makes a particularly interesting for mechanical, optical and electrically applications. CrN has been gaining a great importance due to its superior corrosion and wear resistance, friction behavior and low internal stress [4,5]. However, there are an increasing number of applications. Others authors [6] studied the efficiency of CrV and Cr coatings in reducing cutting edge wear of steel milling tools. In a first vision the objective in our present study is to analyze the effects of the deposition conditions such as the deposition duration, the rate of different alloys constituting the V and VN in physical and mechanical properties.

Sample preparation

Thin layers of vanadium and nitride vanadium are filed on silicon platelets, using a magnetron sputtering mode DC from two targets of high purity. The backings were mounted on a continuously a e-mail : taher.ghrib@yahoo.fr rotating planetary holder inside the vacuum chamber (fig. 1). The atmosphere was chosen in order to produce a V and VN layer respectively. Thin film layers are deposited onto samples of amorphous silicon, with different deposition duration. The film deposition was carried out by a dual Radio frequency (RF ) magnetron sputtering system (NORDIKO type 3500-13.56 MHz) from targets of high purity (99.999%) with 10 cm of diameters. Prior to deposition, the samples were polished and then cleaned with acetone in an ultrasonic container for 15 min. Subsequently, they were rinsed in deionized water, dried and stored in desiccators prior to coating deposition. The targets were sputter cleaned for 5 min. Then the samples were sputter-etched with argon ions (10 mbar, applied bias voltage -900 V) for approximately 10 min to remove surface oxide and pollution. All of the two targets were sputtered using RF power. During deposition, the substrate temperature was 150 °C.

The pressure in the deposition chamber was 10 -6 mbar. The distance between the target and surface backing was 10 cm. Moreover, the samples were mounted on a continuously rotating planetary holder inside the vacuum chamber. The composition of the as-deposited coatings was measured by energy dispersive analysis of X-rays (EDX), which is attached to a scanning electron microscope, at 10 keV and the INCA quantitative analysis software.

Structural properties

The observation of the morphological surface is realized by the SE microscope on the cross section gives the fig. 2 for a Vanadium layer. In the fig. 3 is reported the thickness layer the evolution versus the duration deposition. We notes that this layer increases linearly which permits to achieve an average speed deposition in the order of 10.8 nm.min -1 .

00042-p.2

EMM-FM2011 The fig. 4 gives the surface AFM photos in three dimensional of the deposited vanadium thin layer. What presents a granular structure was their grains size increase with the deposition duration and this can be traduced by the formation of the Vanadium layer in a first step with dome particle and in a second step with grains in a needles type whose diameter and maximal haughtiness respectively in the order of 38 nm and 8 nm, and however presents an increase of its surface roughness with deposition duration.

Moreover the surface arithmetic roughness (Ra) varies with the deposition duration in which their values for the different samples is given in the fig. 5 and who values vary between 0.53 and 3.7 nm.

The samples EDX spectral, Obtained with a tension of 25 kV are given in the fig. 6 where we notes the presence of two characteristics picks of the Vanadium at 4.945 and 4.952 kev whose intensities increases with the deposition duration. On top of that one can observes the picks of oxygen and silicon that can give the material atomic percentage constituting the layer which its values are reported in the table 1.

One remarks host presence of the carbon in our films, this component is an impurity probably due to the poor quality of the vacuum pump oil, we supposes that its affects in the films properties are globally homogeneous and physical properties changes are due to variation of its thickness films. In order to determine the X-ray diffraction pattern, we used BRUKER AXS D8 ADVANCE spectrometer through the beam length 1.54056 Å of the (CuK ) copper, with a bias voltage 40kV and current 20 mA, by placing the samples that are treated with different depositions durations according to Bragg Brentano ( /2 ) geometry.

The obtained XRD patterns are reported in fig. 7, when we note the presence of two Vanadium picks (200) and (110) on 2 =61.8° and 41.3°. We remark that all patterns are almost identical unless that present one pick corresponding to the (110) orientation of the V component whose intensity increase progressively. It emerge from this that the deposition duration not affect the final films structure whilst that their thickness are small and when we increases this layer it structure changes.

Mechanical properties

Like all physical properties, mechanical properties vary with the thickness such as the residual stress, the hardness [7-9,16], the wear resistance [6,16], the preferential orientation of crystalline plane [13][14][15]. The influence of the thickness to the layer properties has been studied for several materials such as the AlN [15], BN [16] and the TiN [17]. The effect of the thickness in the residual stress has been studied by Meng [15] and Mckenzie [16] has demonstrated that the residual stress hasn't a homogeneous evolution with its thickness. Besides, Quaeyhaegens and all [17] have demonstrated the thickness effect to the changes in the preferential orientation of the TiN layers.

In this studied case, as it happen the vanadium deposit, we obtain that the residual stress, determined by the Newton method; present a similar evolution with the layer thickness (fig. 9).

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In fact, we remark that the stress isn't constant where the film thickness varies, it increase until about 4.8 GPa for a thickness in order of 80nm and decrease beyond.

The maximal stress can be explained by the various effects that can appear in the course of the increasing layer such as a creation of an internal or external default, crystalline orientation changes, porosity rate.

In the deposition beginning the arrival atoms engender a formation of a smaller particle through a nucleation increases giving a growth with the (200) crystallographic direction and after an increasing of verticals columnar relatively spaced and this process is schematized by Hones [18].

Electrical resistivity

In our work we have measured the electrical resistivity of the vanadium layer with method of the four touches method.

We remark that the electrical resistivity (fig. 10) decrease greatly for petites thickness (<300nm), and stabilize for a mean value in the order of 10 µ .cm -1 . This behavior has been observed by Holland [START_REF] Holland | Vacuum deposited thin films[END_REF] for various thin metallic films.

Fig. 10. Variation of the electrical resistivity versus the layer thickness

We can explain this variation by the approach that he vanadium film can be considered as a stocking of any mono-layers infinitely thin whose conductivity increase from the surface toward the film surface. Consequently, this layer can be considered as a resistance associated with parallel disposition; that decrease when the layers number increase.

Vanadium Nitride layer

In this part we interest to study of the alloys formed by the Vanadium and a Nitride.

Structural analysis

The layers topography of the VN alloys observed by the AFM (fig. 11) is globally uniform, it constituted with heaps and minuscule craters repartee inside these lots in the surface layer and this repartition get more and more convinced if the vanadium proportion increase in the considered coating, i.e. when the columnar microstructure is favoured. The measure of the lots mean diameter is about 88 nm also the maximal depth has not pass 30 nm. The measure of the surface roughness of the prepared layer is showed in fig. 12, which gives that for a feeble thickness this roughness is almost stable and increase progressively with the deposition duration and believe toward a horizontal asymptote of a values of 1.8 nm characteristic of the massif VN alloys. We remark that if the deposition duration increases, the vanadium peak increases too, that can be explained by an increasing of the pulverization rate of the vanadium target and thus a very important quantity of the Vanadium in the layer.

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EMM-FM2011 Fig. 13. EDS analysis of the VN layer with the deposition duration: a) 30 min and b) 90 min The analysis of picks constituting the patterns gives the concentration of each chosen element and permits to obtain their atomic percentage. This values percentage are given in the table 2; we note a slight increase in the nitrogen percentage (it passes from 30.7 to 40.21%) i.e the vanadium addition favorite the film stoechiometry and the nitrogen rate brings to that of the vanadium one. In order to obtain informations on the deposition duration effect in the layer structure, we have fixed the nitrogen percentage in the plasma at 25%, a working pressure to 4 µbar we have varied the deposition duration by choosing values 10, 30 and 90mn. From the spectrogram of fig. 14 we remark that by increasing le deposition duration the film acquire a crystallized structure. The (111) pick amplitude increases and its (FWHM) have tendency to increase with the deposition duration and after the crystallite grain size increase too. Others study realized to the TiN and CrN alloys shows a crystallographic structure with a (111) ; translating a strongly residual stresses within the layer [START_REF] Oden | [END_REF]. 00042-p.9

Residual stress

The thickness is one from various parameters that can influence the level of the residual stress of the coatings. The study of the profile of the residual stress gives that this latter is very higher in the beginning of the growth and decrease greatly with the deposition duration and highlights an existence of a maximum of a thin layer of about 88 nm thick.

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Globally this stress is feeble and this can be explained by the amelioration of the cristallinity and textures of these layers how believe toward an equilibrium state and this same conclusion has been observed beforehand during the works of Nouveau and all [21].

Conclusion

This preliminary study had been an objective to verify the effect of the deposition conditions on the VN film properties. For this reason we have beginning by study the vanadium layer and after we have realized VN coatings by varying the duration deposition. It demonstrates that the increasing in V or VN layers give a crystalline structure of a very important and regular electrical and mechanical properties. This study is in agreement with others works studying another films like the TiN and CrN alloys.

Fig. 1 .

 1 Fig.1. Experimental set-up used to deposited V or VN thin layers

Fig. 2 .Fig. 3 .

 23 Fig. 2. Cross section by SE microscope of the vanadium layer

Fig. 4 .

 4 Fig.4. The surface AFM photos in three dimensional of the deposited vanadium thin layer carry out to a tapping mod for various deposition duration: (a) 5min, (b) 7min, (c) 12min

Fig. 5 .

 5 Fig. 5. Variation of the surface roughness with the deposition duration

Fig. 6 .Fig. 7 .

 67 Fig.6. EDS analysis of the deposited vanadium layer with the deposition duration: 5min (a), 7min (b), 30 min (c) et 90 min (d)The mean crystal size has been (fig.8) calculated using the Debye Scherrer formula. The half pick width is determined from the spectrogram recording a -2 mod and the calculation of the mean size of the formed particle is realized by considering the (110) pick.

Fig. 8 .

 8 Fig. 8. Grains size evolution versus the deposition duration

Fig. 9 .

 9 Fig. 9. Variation of the residual stress with the layer thickness

Fig. 11 .

 11 Fig. 11. Three dimensional AFM realized in tapping mod of the deposited VN alloys for different deposition duration: a) 5min, b) 10min, c) 15min, d) 20min, e) 30min, f) 90min

Fig. 12 .

 12 Fig. 12. Variation of the arithmetic roughness of the VN layers with the deposition duration Patterns obtained by EDS analysis for the VN alloys whatever theirs vanadium content are summarized in fig. 13, which give elements constituting the coatings. In fact, we observe one nitrogen pick in 0,392 keV and two vanadium picks K 1 and K 2 respectively in 4.945 keV and in 4.952 keV.We remark that if the deposition duration increases, the vanadium peak increases too, that can be explained by an increasing of the pulverization rate of the vanadium target and thus a very important quantity of the Vanadium in the layer.

Fig. 14 .

 14 Fig. 14. XRD patterns of the VN alloys for deposition duration equal to: a) 10min, b) 30min and c) 90min

Fig. 15 .

 15 Fig. 15. Variation of the residual stress of the VN alloy versus the layer thickness

Table 1 .

 1 Chemical composition of the deposited vanadium layer

		atomic %	atomic %	atomic %	atomic %
	Component
		(5min)	(10min)	(30min)	(90min)
	C	3.25	4	12.55	6.43
	O	1.7	2.77	12.79	28.29
	Si	92.85	88.71	68.75	28.22
	V	2.2	4.52	10.91	37.06
	Total	100	100	100	100

Table 2 .

 2 Chemical composition of the deposited vanadium nitride alloys layer

	Component	atomic % (30min)	atomic % (90min)
	N	18.79	26.17
	O	7.85	12.55
	Si	68.72	49.52
	V	4.63	11.75
	Total	100	100