Infinite Horizon Multi-Dimensional BSDE with Oblique Reflection and Switching Problem
aa r X i v : . [ m a t h . P R ] F e b Infinite Horizon Multi-Dimensional BSDE with ObliqueReflection and Switching Problem
Brahim EL ASRI ∗ and Nacer OURKIYA † Abstract
This paper studies a system of multi-dimensional reflected backward stochastic differ-ential equations with oblique reflections (RBSDEs for short) in infinite horizon associatedto switching problems. The existence and uniqueness of the adapted solution is obtainedby using a method based on a combination of penalization, verification method and con-traction property.
Keywords : Reflected backward stochastic differential equations, Switching problem, Backwardstochastic differential equations, Infinite horizon, Oblique reflection.
In this paper we study a system of multi-dimensional RBSDE with oblique reflection in infinitehorizon.For i ∈ I := { , ..., m } and t ≥ , we define the multi-dimensional RBSDE with obliquereflections by, ∀ r ∈ R + e − rt Y it = R + ∞ t e − rs f i ( s, X s , Y s , ..., Y ms , Z is ) ds + K i ∞ − K it − R + ∞ t e − rs Z is dB s ;lim t → + ∞ e − rt Y it = 0 , ∀ t ≥ , e − rt Y it ≥ e − rt max j ∈I − i ( Y jt − g ij ( X t )) , R + ∞ e − rs { Y is − max j ∈I − i ( Y js − g ij ( X s )) } dK is = 0 , (1.1)where I − i := I − i .Here, B is a standard Brownian motion on a complete probability space (Ω , F , P ) , f i thegenerator which is Lipschitz continuous w.r.t. y and z , X := ( X t ) t ≥ be a P -measurable, R k -valued continuous stochastic process and g ij represent the switching costs from mode i tomode j. We aim at finding a m -triples of ( F t ) t ≥ -adapted processes ( Y i , Z i , K i ) i ∈I , which solves P -a.s. the system of multi-dimensional RBSDE (1.1).One-dimensional RBSDEs were first studied by El Karoui et al. [ ] in finite horizon case.Later, Hamadène et al. [ ] proved existence and uniqueness results of the solution for infinitehorizon RBSDEs. The authors applied these results to get the existence of optimal controlstrategy for the mixed control problem. The literature on this specific form of equation has ∗ Université Ibn Zohr, Equipe. Aide á la decision, ENSA, B.P. 1136, Agadir, Maroc. e-mail:[email protected] † Université Ibn Zohr, Equipe. Aide á la decision, ENSA, B.P. 1136, Agadir, Maroc. e-mail:[email protected]. [ ] , but their BSDEis reflected on the boundary of a convex domain along the inward normal direction, and theirmethod depends heavily on the properties of this inward normal reflection (see (1) − (3) in [ ] ).And then, by Ramasubramanian [ ] in an orthant with some restriction on the direction ofoblique reflection and the driver f .Another type of multi-dimensional RBSDEs occurs in the context of the optimal switchingproblem. This kind of BSDEs, which are reflected along an oblique reflection rather a normalone in a convex domain, were first introduced by Hamadène and Jeanblanc [ ] , where theyused its solution to characterize the value of an optimal switching problem, in particular in thesetting of power plant management. The related equation was solved by Hu and Tang [ ] usingthe penalization method and by Hamadène and Zhang [ ] using the Picard iteration method,they generalized the preceding work. See also Chassagneux et al. [ ] and the references therein.In the case when the horizon is infinite, there is still much to be done and this is the noveltyof this paper. So our problem is how to generalize the multi-dimensional RBSDEs to an infinitehorizon. In that case, El Asri [ ] studied RBSDEs with the generator does not depend on ( y, z ) and provided an application to optimal switching problem. The latter is a problem in whicha decision controller controls a system which may operate in different modes (e.g., a powerplant). The aim of the controller is to maximise some performance criterion by optimallychoosing controls of the form a := ( τ n , ζ n ) n ≥ . Here ( τ n ) n ≥ denotes an increasing sequence of(stopping) times at which the controller switches the system across different operating modes.Moreover, ( ζ n ) n ≥ is a sequence of random variables taking their values in I . Each ζ n representsthe system’s new operating mode after a switch has occurred at time τ n . In this setting it iswell known that Y it is the value of an optimal switching strategy, i.e., e − rt Y it = ess sup ( τ n ,ζ n ) ∈D it E (cid:20)Z + ∞ t e − rs f a s ( s, X s ) ds − A at (cid:21) ; (1.2)where D it = { ( τ n , ζ n ) n ≥ such that τ = t and ζ = i } , the process ( a t ) t ≥ is indicating themode of the system at time s and A at stands for the total switching cost when the strategy a isimplemented.The main contribution of this paper is to establish the existence and uniqueness of solutionfor RBSDE (1 . . First, We prove the existence and uniqueness (when, the generator f i doesnot depend on all Y i ) respectively, by the penalization method and verification method. Andthen, using the contraction method, we obtain the existence and uniqueness result for RBSDE (1 . .The paper is organized as follows: In section , we state some notations and assumptions.In section , we prove the existence and uniqueness of solution when the generator does dependsonly on Y i . Finally, in Section , we state and prove the main result concerning the existenceand uniqueness of solutions to the system of RBSDE (1 . . Throughout this paper, we have a probability space (Ω , F , P ) endowed with a d -dimensionalBrownian motion B = ( B t ) t ≥ . {F t , t ≥ } is the natural filtration of the Brownian motionaugmented by P -null sets of F , and F ∞ = W t ≥ F t . All the measurability notion will refer to2his filtration. Let | . | denote the Euclidean norm for vectors.Let us introduce the following spaces:- P be the σ -algebra on [0 , + ∞ ] × Ω of F -progressively measurable sets.- M is the set of R d -valued, progressively measurable processes ( Z t ) t ≥ such that E (cid:20)Z + ∞ | Z s | ds (cid:21) < + ∞ . - S is the set of R -valued adapted and càdlàg processes ( Y t ) t ≥ such that E (cid:20) sup t ≥ | Y t | (cid:21) < + ∞ . - K is the subset of non-decreasing processes ( K t ) t ≥ ∈ S , starting from K = 0 .- L is the set of F ∞ -measurable random variable ξ satisfying E [ | ξ | ] .Let us now consider the following functions: for i, j ∈ I , f i ( t, x, y , ..., y m , z i ) : [0 , + ∞ ] × R k × R m × R d R , and g ij ( x ) : R k R ; ( i = j ) . Next let consider a function u ( t ) : [0 , + ∞ ] R + which is deterministic and bounded suchthat: R + ∞ u ( t ) dt < + ∞ and R + ∞ u ( t ) dt < + ∞ . In what follows we take for simplicity thesame function u ( t ) in the inequalities below, in addition we denote ~y = ( y , ..., y m ) . Now, we make the following assumptions:• [ H1 ] : For i ∈ I , f i : [0 , + ∞ ] × R k × R m × R d R is B ([0 , + ∞ ]) ⊗ B ( R k ) -measurableand satisfies: ( i ) for any i ∈ I and x ∈ R k , f i ( t, x, , belongs to M . ( ii ) f i ( t, x, ~y, z ) is Lipschitz continuous in ( ~y, z ) , i.e., for all ( t, x, ~y j , z j ) ∈ [0 , + ∞ ] × R k × R m × R d , j = 1 , we have | f i ( t, x, ~y , z ) − f i ( t, x, ~y , z ) | ≤ u ( t )( | ~y − ~y | + | z − z | ) . (2.1)• [ H2 ] : For any ( i, j ) ∈ I and x ∈ R k , g ij ( x ) is a continuous bounded function. Moreoverit satisfies the following: ( i ) g ii ( x ) = 0;( ii ) g ij ( x ) > , for i = j ; ( iii ) for any ( i, j, l ) ∈ I , such that i = j and j = l , we have g ij ( x ) + g jl ( x ) ≥ g il ( x ) . • [ H3 ] : For any i ∈ I , ξ i := lim t → + ∞ Y it is belongs to L and satisfies ξ i ≥ max j ∈I − i ( ξ j − g ij ( X ∞ )); where g ij ( X ∞ ) := lim t → + ∞ g ij ( X t ) . (1 . , we first provide its existence and uniqueness when, for any i ∈ I , the function f i does not depend on all Y j , ∀ j ∈ I − i , that is, P -a.s., f i ( t, x, ~y, z i ) := f i ( t, x, y i , z i ) , for any t, x, y i , and z i , and consider the following RBSDE, ∀ t ≥ e − rt Y it = R + ∞ t e − rs f i ( s, X s , Y is , Z is ) ds + K i ∞ − K it − R + ∞ t e − rs Z is dB s , ∀ t ≥ , e − rt Y it ≥ e − rt max j ∈I − i ( Y jt − g ij ( X t )) , R + ∞ e − rs { Y is − max j ∈I − i ( Y js − g ij ( X s )) } dK is = 0 . (2.2) (2 . In this subsection, we shall prove an existence result of the solution of RBSDE (2 . . Proposition 3.1.
Under [ H1 ] , [ H2 ] and [ H3 ] , the RBSDE (2 . has at least one solution.Proof. The proof will be divided into four steps.
Step 1 : The penalized BSDE.For any i ∈ I and n ≥ let us consider ( Y i,n , Z i,n ) ∈ S × M the unique solution of thefollowing BSDE: e − rt Y i,nt = Z + ∞ t e − rs f ni ( s, X s , Y i,ns , Z i,ns ) ds − Z + ∞ t e − rs Z i,ns dB s ; (3.1)where f ni is defined on [0 , + ∞ ] × R k × R × R d by f ni : ( s, x, y i , z i ) f i ( s, x, y i , z i ) + n m X j =1 ( y i − y j + g ij ( x )) − . Indeed, thanks to the result by Chen [ ] , this solution exists and is unique. Step 2 : A priori estimate.In this step, We derive two lemmas on the a priori estimation of the penalized BSDE (3 . ,which will play a primordial role in the proof of Proposition (3 . . Lemma 3.1.
There exist a positive constant C u which depends on u , such that the followinghold true: for any i, j ∈ I and n ≥ . E (cid:20) sup t ≥ [( Y i,nt − Y j,nt + g ij ( X t )) − ] + n Z + ∞ [( Y i,ns − Y j,ns + g ij ( X s )) − ] ds (cid:21) ≤ E (cid:20)Z + ∞ C u L ij,n ( s )[1 + | f i ( s, X s , , | + | f j ( s, X s , , | + | Y i,ns | + | Z i,ns | ] ds (cid:21) , (3.2) where L ij,n := { ( s, ω ) ∈ [0 , + ∞ ] × Ω , such that Y i,ns − Y j,ns + g ij ( X s ) < } . roof. For given i, j ∈ I , set Y ij,nt := Y i,nt − Y j,nt + g ij ( X t ) , t ∈ [0 , + ∞ ] . (3.3)Next, rewrite equation (3 . in differential form d ( e − rt Y i,nt ) = − e − rt f ni ( t, X t , Y i,nt , Z i,nt ) dt + e − rt Z i,nt dB t . (3.4)So for t ≥ , n ≥ and i, l ∈ I , equation (3 . is equivalent to Y i,nt = ξ i + Z + ∞ t [ f i ( s, X s , Y i,ns , Z i,ns ) + n m X l =1 ( Y il,ns ) − − rY i,ns ] ds − Z + ∞ t Z i,ns dB s ; (3.5)where ξ i := lim t → + ∞ Y i,nt , ∀ i ∈ I .By an application of Itô-Tanaka’s formula, for every t ≥ , we obtain d ([( Y ij,nt ) − ] ) = − Y ij,nt L ij,n ( t ) d Y ij,nt + 12 Y ij,nt dL t ( Y ij,n )+ 1 L ij,n ( t )( Z i,nt − Z j,nt ) dt ; (3.6)where L ( Y ij,n ) denotes the local-time at zero of the semi-martingale Y ij,n .We now want to find a convenient expression for d Y ij,nt . In the definition of Y ij,nt (cf. (3 . )we may express Y i,n and Y j,n in terms of their associated BSDE (3 . . This gives, for any t ∈ [0 , + ∞ ] Y ij,nt = ξ i − ξ j + g ij ( X ∞ ) + Z + ∞ t [ f i ( s, X s , Y i,ns , Z i,ns ) − f j ( s, X s , Y j,ns , Z j,ns ) − r ( Y i,ns − Y j,ns )] ds + n Z + ∞ t m X l =1 ( Y il,ns ) − ds − n Z + ∞ t m X l =1 ( Y jl,ns ) − ds − Z + ∞ t ( Z i,ns − Z j,ns ) dB s . (3.7)Then, d Y ij,nt = − [ f i ( t, X t , Y i,nt , Z i,nt ) − f j ( t, X t , Y j,nt , Z j,nt ) − r ( Y i,nt − Y j,nt )] dt − n m X l =1 ( Y il,nt ) − dt + n m X l =1 ( Y jl,nt ) − dt + ( Z i,nt − Z j,nt ) dB t . (3.8)Noticing that the integral with respect to the local-time L ( Y ij,n ) is zero and by Assumption [ H3 ] we have that ( ξ i − ξ j + g ij ( X ∞ )) − = 0 , we obtain from (3 . that for every t ∈ [0 , + ∞ ][( Y ij,nt ) − ] + 2 n Z + ∞ t [( Y ij,ns ) − ] ds + Z + ∞ t L ij,n ( s )( Z i,ns − Z j,ns ) ds = 2 Z + ∞ t ( Y ij,ns ) − [ f j ( s, X s , Y j,ns , Z j,ns ) − f i ( s, X s , Y i,ns , Z i,ns ) + r ( Y i,ns − Y j,ns )] ds + 2 Z + ∞ t ( Y ij,ns ) − ( Z i,ns − Z j,ns ) dB s − n Z + ∞ t ( Y ij,ns ) − ( Y ji,ns ) − ds + 2 n X l = i,j Z + ∞ t ( Y ij,ns ) − [( Y jl,ns ) − − ( Y il,ns ) − ] ds. (3.9)5ut, by Assumption [ H2 ] − ( iii ) we have that g ij ( X s ) + g ji ( X s ) ≥ g ii ( X s ) = 0 . Thus, we obtainthat, for every s ∈ [0 , + ∞ ] { y ∈ R m , y is − y js + g ij ( X s ) < } ∩ { y ∈ R m , y js − y is + g ji ( X s ) < } = ∅ . from which we deduce that ( Y ij,ns ) − ( Y ji,ns ) − = 0 , i, j ∈ I . Relying next on the elementary inequality x − − x − ≤ ( x − x ) − and by Assumption [ H2 ] − ( iii ) ,we get ( Y ij,ns ) − [( Y jl,ns ) − − ( Y il,ns ) − ] ≤ ( Y i,ns − Y j,ns + g ij ( X s )) − ( Y j,ns − Y i,ns + g jl ( X s ) − g il ( X s )) − = ( Y i,ns − Y j,ns + g ij ( X s )) − ( Y i,ns − Y j,ns + g il ( X s ) − g jl ( X s )) + ≤ ( Y i,ns − Y j,ns + g ij ( X s )) − ( Y i,ns − Y j,ns + g ij ( X s )) + = 0 . Then, taking expectation on both sides of (3 . , we obtain E (cid:20) [( Y ij,nt ) − ] (cid:21) + 2 n E (cid:20)Z + ∞ t [( Y ij,ns ) − ] ds (cid:21) + E (cid:20)Z + ∞ t L ij,n ( s ) | Z i,ns − Z j,ns | ds (cid:21) ≤ E (cid:20)Z + ∞ t ( Y ij,ns ) − {| f j ( s, X s , Y j,ns , Z j,ns ) − f i ( s, X s , Y i,ns , Z i,ns ) | + r | Y i,ns − Y j,ns |} ds (cid:21) . (3.10)Noting that, | f j ( s, X s , Y j,ns , Z j,ns ) − f i ( s, X s , Y i,ns , Z i,ns ) | + r | Y i,ns − Y j,ns |≤ | f i ( s, X s , Y i,ns , Z i,ns ) − f j ( s, X s , Y i,ns , Z i,ns ) | + | f j ( s, X s , Y i,ns , Z i,ns ) − f j ( s, X s , Y j,ns , Z j,ns ) | + r | Y i,ns − Y j,ns | , ≤ | f i ( s, X s , Y i,ns , Z i,ns ) | + | f j ( s, X s , Y i,ns , Z i,ns ) | + | f j ( s, X s , Y i,ns , Z i,ns ) − f j ( s, X s , Y j,ns , Z j,ns ) | + r | Y i,ns − Y j,ns | , ≤ | f i ( s, X s , , | + | f j ( s, X s , , | + ( r + u ( s )) | Y i,ns − Y j,ns | + u ( s )[2 | Y i,ns | + 2 | Z i,ns | + | Z i,ns − Z j,ns | ] , ≤ C u [1 + | f i ( s, X s , , | + | f j ( s, X s , , | + | Y i,ns | + | Z i,ns | + |Y ij,ns | + | Z i,ns − Z j,ns | ]; (3.11)where C u is a constant depending on u ( s ) , independent of n and which might hereafter varyfrom line to line.Now going back to (3 . and using (3 . , we get that E (cid:20) [( Y ij,nt ) − ] (cid:21) + 2 n E (cid:20)Z + ∞ t [( Y ij,ns ) − ] ds (cid:21) + E (cid:20)Z + ∞ t L ij,n ( s ) | Z i,ns − Z j,ns | ds (cid:21) ≤ E (cid:20)Z + ∞ t C u ( Y ij,ns ) − [1 + | f i ( s, X s , , | + | f j ( s, X s , , | + | Y i,ns | + | Z i,ns | + |Y ij,ns | + | Z i,ns − Z j,ns | ] ds (cid:21) , ≤ E (cid:20)Z + ∞ t C u [( Y ij,ns ) − ] ds (cid:21) + 12 E (cid:20)Z + ∞ t L ij,n ( s )[1 + | f i ( s, X s , , | + | f j ( s, X s , , | + | Y i,ns | + | Z i,ns | + | ( Y ij,ns ) − | + | Z i,ns − Z j,ns | ] ds (cid:21) . (3.12)6pplying Gronwall’s inequality, it follows that E (cid:20) [( Y ij,nt ) − ] (cid:21) ≤ E (cid:20)Z + ∞ C u L ij,n ( s )[1+ | f i ( s, X s , , | + | f j ( s, X s , , | + | Y i,ns | + | Z i,ns | ] ds (cid:21) , (3.13)and n E (cid:20) Z + ∞ [( Y ij,ns ) − ] ds (cid:21) + E (cid:20)Z + ∞ L ij,n ( s ) | Z i,ns − Z j,ns | ds (cid:21) ≤ E (cid:20)Z + ∞ C u L ij,n ( s )[1 + | f i ( s, X s , , | + | f j ( s, X s , , | + | Y i,ns | + | Z i,ns | ] ds (cid:21) . (3.14)Going back to (3 . and applying Burkholder-Davis-Gundy’s inequality, we obtain E (cid:20) sup t ≥ [( Y ij,nt ) − ] (cid:21) ≤ E (cid:20)Z + ∞ C u L ij,n ( s )[1 + | f i ( s, X s , , | + | f j ( s, X s , , | + | Y i,ns | + | Z i,ns | ] ds (cid:21) . (3.15)On the other hand, from (3 . , we deduce that, n E (cid:20)Z + ∞ [( Y ij,ns ) − ] ds (cid:21) ≤ E (cid:20)Z + ∞ ( n + C u )[( Y ij,ns ) − ] ds (cid:21) + 1 n E (cid:20)Z + ∞ C u L ij,n ( s )[1 + | f i ( s, X s , , | + | f j ( s, X s , , | + | Y i,ns | + | Z i,ns | ] ds (cid:21) . For n large enough we finally deduce that n E (cid:20)Z + ∞ [( Y ij,ns ) − ] ds (cid:21) ≤ E (cid:20)Z + ∞ C u L ij,n ( s )[1 + | f i ( s, X s , , | + | f j ( s, X s , , | + | Y i,ns | + | Z i,ns | ] ds (cid:21) . (3.16)Thanks to Lemma (3 . we are able to prove the next uniform estimate on the solution ofthe penalized problem. Lemma 3.2.
For any i, j ∈ I , n ≥ . There exist a positive constant C independent of n suchthat, E (cid:20) sup t ≥ | Y i,nt | + Z + ∞ | Z i,ns | ds + n Z + ∞ m X j =1 [( Y i,ns − Y j,ns + g ij ( X s )) − ] ds (cid:21) ≤ C. (3.17) Proof.
Applying Itô’s formula with | Y i,nt | and recalling (3 . we obtain that for any t ∈ [0 , + ∞ ] | Y i,nt | + Z + ∞ t [2 r | Y i,ns | + | Z i,ns | ] ds = 2 Z + ∞ t Y i,ns [ f i ( s, X s , Y i,ns , Z i,ns )+ n m X j =1 ( Y i,ns − Y j,ns + g ij ( X s )) − ] ds − Z + ∞ t Y i,ns Z i,ns dB s . (3.18)7aking expectations and using the classical inequality ab ≤ ǫ a + ǫb , for any a, b ∈ R and ǫ > , we obtain E [ | Y i,nt | ] + E (cid:20)Z + ∞ t (2 r | Y i,ns | + | Z i,ns | ) ds (cid:21) ≤ E (cid:20)Z + ∞ t | Y i,ns | [ | f i ( s, X s , Y i,ns , Z i,ns ) | + n m X j =1 ( Y i,ns − Y j,ns + g ij ( X s )) − ] ds (cid:21) , ≤ E (cid:20)Z + ∞ t | Y i,ns | [ | f i ( s, X s , , | + u ( s )( | Y i,ns | + | Z i,ns | )+ n m X j =1 ( Y i,ns − Y j,ns + g ij ( X s )) − ] ds (cid:21) , ≤ E (cid:20)Z + ∞ t [2( u ( s ) + u ( s ) + ǫ ) | Y i,ns | + 12 | Z i,ns | + ǫ − | f i ( s, X s , , | ] ds (cid:21) + ǫ − n E (cid:20)Z + ∞ t m X j =1 [( Y i,ns − Y j,ns + g ij ( X s )) − ] ds (cid:21) . In view of (3 . , if we choose ǫ = 4 C u and for r ≥ u ( s ) + u ( s ) + 4 C u + , we obtain E [ | Y i,nt | ] + 14 E (cid:20)Z + ∞ t | Z i,ns | ds (cid:21) ≤ E (cid:20)Z + ∞ t C u [1 + | f i ( s, X s , , | + | f j ( s, X s , , | ] ds (cid:21) . (3.19)Therfore, for t = 0 we have E [ Z + ∞ | Z i,ns | ds ] ≤ E (cid:20)Z + ∞ C u [1 + | f i ( s, X s , , | + | f j ( s, X s , , | ] ds (cid:21) . (3.20)Using again equation (3 . and the Burkholder-Davis-Gundy’s inequality, for some finite uni-versal constant c , we obtain that E [sup t ≥ | Y i,nt | ] ≤ E (cid:20)Z + ∞ t C u [1 + | f i ( s, X s , , | + | f j ( s, X s , , | ] ds (cid:21) + c E (cid:20) ( Z + ∞ | Y i,ns | | Z i,ns | ds ) (cid:21) ; (3.21)with, c E (cid:20)(cid:0) Z + ∞ | Y i,ns | | Z i,ns | ds (cid:1) (cid:21) ≤ E (cid:20) sup t ≥ | Y i,nt | (cid:21) + c E (cid:20)Z + ∞ | Z i,ns | ds (cid:21) . (3.22)Then, E [sup t ≥ | Y i,nt | ] ≤ E (cid:20)Z + ∞ t C u [1 + | f i ( s, X s , , | + | f j ( s, X s , , | ] ds (cid:21) + c E [ Z + ∞ | Z i,ns | ds ] . (3.23)Therefore, E [sup t ≥ | Y i,nt | ] + E [ Z + ∞ | Z i,ns | ds ] ≤ C. (3.24)8rom (3 . , we obtain n E (cid:20)Z + ∞ [( Y i,ns − Y j,ns + g ij ( X s )) − ] ds (cid:21) ≤ C. Taking the summation over all j ∈ I , we obtain n E (cid:20)Z + ∞ m X j =1 [( Y i,ns − Y j,ns + g ij ( X s )) − ] ds (cid:21) ≤ C. (3.25)The proof of Lemma (3 . is now complete. Step 3 : Convergence of the sequence.In order to show that the sequence ( Y i,nt ) n ≥ is non-decreasing and convergent for any i ∈ I weuse a comparison theorem for infinite horizon BSDE presented in [ ] . Since f ni ≤ f n +1 i then,by the comparison Theorem, we have, for any n ≥ Y i,nt ≤ Y i,n +1 t , for all i ∈ I and t ∈ [0 , + ∞ ] . Then, Y i,nt admits a limit denoted by Y it . Moreover, from the a priory estimate (3 . andFatou’s Lemma, we have E [sup t ≥ | Y it | ] ≤ C. (3.26)Then, applying Lebesgue’s dominated converge theorem, we get lim n → + ∞ E (cid:20)Z + ∞ | Y i,nt − Y it | ds (cid:21) = 0 . (3.27)Now, we prove that ( Y i,n , Z i,n ) is a Cauchy sequence. To do so we apply Itô’s formula to | Y i,nt − Y i,pt | , we get | Y i,nt − Y i,pt | + Z + ∞ t (2 r | Y i,ns − Y i,ps | + | Z i,ns − Z i,ps | ) ds = 2 Z + ∞ t ( Y i,ns − Y i,ps )( f i ( s, X s , Y i,ns , Z i,ns ) − f i ( s, X s , Y i,ps , Z i,ps )) ds + 2 n m X j =1 Z + ∞ t ( Y i,ns − Y i,ps )( Y i,ns − Y j,ns + g ij ( X s )) − ds − p m X j =1 Z + ∞ t ( Y i,ns − Y i,ps )( Y i,ps − Y j,ps + g ij ( X s )) − ds − Z + ∞ t ( Y i,ns − Y i,ps )( Z i,ns − Z i,ps ) dB s . (3.28)9aking expectation in both sides of the last equality yields E [ | Y i,nt − Y i,pt | ] + E (cid:20)Z + ∞ t (2 r | Y i,ns − Y i,ps | + | Z i,ns − Z i,ps | ) ds (cid:21) ≤ E (cid:20)Z + ∞ t | Y i,ns − Y i,ps || f i ( s, X s , Y i,ns , Z i,ns ) − f i ( s, X s , Y i,ps , Z i,ps ) | ds (cid:21) + 2 n m X j =1 E (cid:20)Z + ∞ t | Y i,ns − Y i,ps | ( Y i,ns − Y j,ns + g ij ( X s )) − ds (cid:21) + 2 p m X j =1 E (cid:20)Z + ∞ t | Y i,ns − Y i,ps | ( Y i,ps − Y j,ps + g ij ( X s )) − ds (cid:21) , ≤ E (cid:20)Z + ∞ t u ( s ) + u ( s )) | Y i,ns − Y i,ps | ds (cid:21) + 12 E (cid:20)Z + ∞ t | Z i,ns − Z i,ps | ds (cid:21) + 2 E (cid:20)Z + ∞ t | Y i,ns − Y i,ps | ds (cid:21) m X j =1 (cid:18) n E (cid:20)Z + ∞ t [( Y i,ns − Y j,ns + g ij ( X s )) − ] ds (cid:21)(cid:19) + 2 E (cid:20)Z + ∞ t | Y i,ns − Y i,ps | ds (cid:21) m X j =1 (cid:18) p E (cid:20)Z + ∞ t [( Y i,ps − Y j,ps + g ij ( X s )) − ] ds (cid:21)(cid:19) . Setting t = 0 , and choosing r ≥ u ( s ) + u ( s ) + 1 , from (3 . and (3 . , we have that lim n, p → + ∞ E (cid:20)Z + ∞ | Z i,ns − Z i,ps | d s (cid:21) = 0 . (3.29)Consequently, the sequence ( Z i,n ) n ≥ converges in M to a process which we denote Z i .Now, going back to (3 . , we deduce that E (cid:20) sup t ≥ | Y i,nt − Y i,pt | (cid:21) ≤ E (cid:20)Z + ∞ t | Y i,ns − Y i,ps || f i ( s, X s , Y i,ns , Z i,ns ) − f i ( s, X s , Y i,ps , Z i,ps ) | ds (cid:21) + 2 n m X l =1 E (cid:20)Z + ∞ t | Y i,ns − Y i,ps | ( Y i,ns − Y l,ns + g il ( X s )) − ds (cid:21) + 2 p m X l =1 E (cid:20)Z + ∞ t | Y i,ns − Y i,ps | ( Y i,ps − Y l,ps + g il ( X s )) − ds (cid:21) + 2 E (cid:20) sup t ≥ (cid:12)(cid:12)(cid:12)(cid:12) Z + ∞ t ( Y i,ns − Y i,ps )( Z i,ns − Z i,ps ) dB s (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) . (3.30)Then, by applying the Burkholder-Davis-Gundy’s inequality to the last term of the right handside of inequality (3 . , we obtain E (cid:20) sup t ≥ (cid:12)(cid:12)(cid:12)(cid:12) Z + ∞ t ( Y i,ns − Y i,ps )( Z i,ns − Z i,ps ) dB s (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) ≤ C E (cid:20)Z + ∞ t | ( Y i,ns − Y i,ps )( Z i,ns − Z i,ps ) | dB s (cid:21) , ≤ E (cid:20) sup t ≥ | Y i,nt − Y i,pt | (cid:21) + C E (cid:20) Z + ∞ t | Z i,ns − Z i,ps | ds (cid:21) . (3.31)10ombining (3 . and (3 . and taking into consideration (3 . and (3 . , we get lim n, p → + ∞ E (cid:20) sup t ≥ | Y i,nt − Y i,pt | (cid:21) = 0 , (3.32)which means that ( Y i,n ) n ≥ is a Cauchy sequence in S . Consequently Y it is continuous.Now we define K i,n as follows: K i,nt := n Z t m X l =1 e − rs ( Y i,ns − Y l,ns + g il ( X s )) − ds, ∀ t ∈ [0 , + ∞ ] and i ∈ I . (3.33)From the penalized BSDE (3 . , we have K i,nt = e − rt Y i,nt − Y i,n + Z t e − rs f i ( s, X s , Y i,ns , Z i,ns ) d s − Z + ∞ t e − rs Z i,ns d B s , we set K it = e − rt Y it − Y i + Z t e − rs f i ( s, X s , Y is , Z is ) d s − Z + ∞ t e − rs Z is d B s . Then we deduce immediately by (3 . and (3 . that ( K i,n ) n ≥ converges to K i in S . So K i is an increasing process, moreover it is continuous, then K i ∈ K .Therefore, ∀ i ∈ I , ( Y i , Z i , K i ) satisfies the first relation in RBSDE (1 . . Finally, by the apriori estimate (3 . , we have E (cid:20)Z + ∞ (cid:2) ( Y i,ns − Y j,ns + g ij ( X s )) − (cid:3) ds (cid:21) ≤ Cn , ∀ i, j ∈ I . Letting n −→ + ∞ , we deduce E (cid:20)Z + ∞ (cid:2) ( Y is − Y js + g ij ( X s )) − (cid:3) ds (cid:21) = 0 , ∀ i, j ∈ I . Hence, Y is ≥ Y js − g ij ( X s ) , ∀ i, j ∈ I . (3.34) Step 4 : The minimal boundary condition.At last we need to prove that R + ∞ e − rs [ Y is − max j ∈I − i ( Y js − g ij ( X s ))] d K is = 0 , ∀ i ∈ I . To do so,we should show before that R + ∞ e − rs [ Y i,ns − max j ∈I − i ( Y j,ns − g ij ( X s ))] + d K i,ns = 0 , ∀ i ∈ I . Firstwe remark that since K i,n is increasing then Z + ∞ e − rs [ Y i,ns − max j ∈I − i ( Y j,ns − g ij ( X s ))] + d K i,ns ≥ . (3.35)Actually we have from (3 . that, for i ∈ I Z + ∞ e − rs [ Y i,ns − max j ∈I − i ( Y j,ns − g ij ( X s ))] + d K i,ns = n m X l =1 Z + ∞ e − rs [ Y i,ns − max j ∈I − i ( Y j,ns − g ij ( X s ))] + [ Y i,ns − Y l,ns + g il ( X s )] − ds. l = i [ Y i,ns − Y j,ns + g ij ( X s )] + [ Y i,ns − Y l,ns + g il ( X s )] − = 0 . On the other hand for j = i, l = i we have [ Y i,ns − max j ∈I − i ( Y j,ns − g ij ( X s ))] + [ Y i,ns − Y l,ns + g il ( X s )] − ≤ [ Y i,ns − Y l,ns + g il ( X s )] + [ Y i,ns − Y l,ns + g il ( X s )] − = 0 . Therefore, we deduce that Z + ∞ e − rs [ Y i,ns − max j ∈I − i ( Y j,ns − g ij ( X s ))] + d K i,ns ≤ . (3.36)From (3 . and (3 . we obtain that Z + ∞ e − rs [ Y i,ns − max j ∈I − i ( Y j,ns − g ij ( X s ))] + d K i,ns = 0 . Now, by applying Lemma . in [ ] , the sequence ( R + ∞ e − rs [ Y i,ns − max j ∈I − i ( Y j,ns − g ij ( X s ))] + d K i,ns ) n ≥ converges to R + ∞ e − rs [ Y is − max j ∈I − i ( Y js − g ij ( X s ))] + d K is . Hence, Z + ∞ e − rs [ Y is − max j ∈I − i ( Y js − g ij ( X s ))] + d K is = 0 . Finally, from (3 . , we conclude that Z + ∞ e − rs [ Y is − max j ∈I − i ( Y js − g ij ( X s ))] d K is = 0 . (3.37)The proof of Proposition (3 . is now complete. In this subsection, we prove the uniqueness of (2 . by a verification method. As in Theorem . in [ ] , we give a switching representation property for the solution Y i of (2 . , which rep-resents the relationship between this solution and the optimal switching problem.In order to state this representation result, first we introduce some notations.Let a := ( τ n , ζ n ) n ≥ be an admissible strategy of switching, i.e.,- ( τ n ) n ≥ is an increasing sequence of stopping times such that P ( τ n < + ∞ , ∀ n ≥
0) = 0 . - ∀ n ≥ , ζ n is a random variable with values in I and F τ n -measurable.- If ( A at ) t ≥ is the non-decreasing, F -adapted and càdlàg process defined by ∀ t ∈ [0 , + ∞ ) , A at := X n ≥ e − rτ n g ζ n − ζ n ( x τ n )1 [ τ n ≤ t ] and A a + ∞ = lim t → + ∞ A at , P − a.s., (3.38)then E [( A a + ∞ ) ] < + ∞ . The quantity A a + ∞ stands for the switching cost at infinity whenthe strategy a is implemented. 12ext, with an admissible strategy a := ( τ n , ζ n ) n ≥ we associate a state process ( a t ) t ≥ definedby a t := ζ { τ } + X n ≥ ζ n ] τ n − ,τ n ] , ∀ t ∈ [0 , + ∞ ] . (3.39)Finally, for ( i, t ) ∈ I × [0 , + ∞ ] , we also define A it the subset of admissible strategies restrictedto start in state i at time t .Now, for any a := ( τ n , ζ n ) n ≥ which belongs to A it , let us define the pair of processes ( U a , V a ) which belongs to S × M and which solves the following switched BSDE: e − rt U at = Z + ∞ t e − rs f a s ( s, X s , U as , V as ) ds − ( A a ∞ − A at ) − Z + ∞ t e − rs V as dB s , ≤ t ≤ + ∞ . (3.40)Actually, in setting up ˜ U a := e − r. U a − A a and ˜ V a := e − r. V a we remark that BSDE (3 . isequivalent to the following one: ˜ U at = − A a ∞ + Z + ∞ t e − rs ˜ f a s ( s, X s , ˜ U as , ˜ V as ) ds − Z + ∞ t ˜ V as dB s , ≤ t ≤ + ∞ , (3.41)where the driver ˜ f a s given by ˜ f a s ( s, X s , ˜ U as , ˜ V as ) := f a s ( s, X s , e rs ( ˜ U as + A as ) , e rs ˜ V as ) . Now, since a is admissible and then E [( A a + ∞ ) ] < + ∞ . Therefore, from the result of Chen [ ] ,the solution of BSDE (3 . exists and is unique.Hence we deduce that BSDE (3 . has a solution in S × M denoted by ( U a , V a ) .The assumptions required for the uniqueness will be slightly stronger than those needed forexistence. We keep the same assumption on f i , and we assume the following for the switchingcosts [ H2 ′ ] : For any ( i, j ) ∈ I and x ∈ R k , g ij ( x ) is a continuous bounded function. Moreover itsatisfies the following: ( i ) g ii ( x ) ≥ ii ) g ij ( x ) > , for i = j ; ( iii ) for any ( i, j, l ) ∈ I , such that i = j and j = l , we have g ij ( x ) + g jl ( x ) > g il ( x ) . Furthermore, as in Theorem . in [ ] , we give in the Proposition below a switching repre-sentation property for the solution Y i of (2 . , which represents the relationship between thissolution and the optimal switching problem where the aim is to maximize U at subject to a ∈ A it .The following Proposition implies the uniqueness of the solution to RBSDE (2 . . Proposition 3.2.
Under [ H1 ] , [ H2 ′ ] , there exists a ∗ ∈ A it such that Y it = U a ∗ t = ess sup a ∈A it U at , ∀ ( i, t ) ∈ I × [0 , + ∞ ] . Proof.
The proof of Proposition (3 . is omitted, since it follows from the same reasoning as inthe proof of Theorem . in [ ] , where the only difference is that in our framework the horizonis infinite unlike to [ ] . 13 The main result
Now we give the main result of this paper.
Theorem 4.1.
Assume that [ H1 ] and [ H2 ′ ] are satisfied. Then the reflected multi-dimensionalBSDE (1 . has a unique solution ( Y i , Z i , K i ) i ∈I .Proof. We suppose that, for i ∈ I , the i -th component of the random function f depends on ~y .Next, let fix ~ Γ := (Γ , ..., Γ m ) in [ S ] m and introduce the operator φ : [ S ] m [ S ] m , ~ Γ ~Y := φ ( ~ Γ) , where ( ~Y , ~Z, ~K ) := ( Y i , Z i , K i ) i ∈I ∈ [ S × M × K ] m is the solution to the followingRBSDE, ∀ i ∈ I and t ∈ [0 , + ∞ ] , e − rt Y it = R + ∞ t e − rs f i ( s, X s , ~ Γ s , Z is ) d s + K i + ∞ − K it − R + ∞ t e − rs Z is d B s ;lim t → + ∞ e − rt Y it = 0 , ∀ t ≥ , e − rt Y it ≥ e − rt max j ∈I − i ( Y jt − g ij ( X t )) , R + ∞ e − rs ( Y is − max j ∈I − i ( Y js − g ij ( X s ))) d K is = 0 , (4.1)which exists and is unique thanks to Proposition (3 . and (3 . . Then φ is well defined and isobviously valued in [ S ] m .Now our objective is to show that φ is a contraction on [ S ] m when endowed with anappropriate equivalent norm. Proposition 4.1.
The operator φ is a contraction on the Banach space [ S ] m endowed withthe norm k . k ,r defined by: k Y k ,r := E (cid:20) sup t ≥ e − rt | Y t | (cid:21) . Proof.
In the same spirit of the proof of Proposition . in [ ] we provide the proof of Propo-sition (4 . only for the sake of completeness, since the proof remains the same even if in ourframework we consider an infinite horizon.We consider two processes ~ Γ , ~ ˙Γ ∈ [ S ] m such that ~Y := φ ( ~ Γ) and ~ ˙ Y := φ ( ~ ˙Γ) , where ( Y i , Z i , K i ) i ∈I (respectively, ( ˙ Y i , ˙ Z i , ˙ K i ) i ∈I ) is the solution of the RBSDE (4 . .Next, Let us introduce the following RBSDE: e − rt ˆ Y it = R + ∞ t e − rs ˆ f i ( s, X s , ˆ Z is ) d s + ˆ K i + ∞ − ˆ K it − R + ∞ t e − rs ˆ Z is d B s ;lim t → + ∞ e − rt ˆ Y it = 0 , ∀ t ≥ , e − rt ˆ Y it ≥ e − rt max j ∈I − i ( ˆ Y jt − g ij ( X t )) , R + ∞ e − rs { ˆ Y is − max j ∈I − i ( ˆ Y js − g ij ( X s )) } d ˆ K is = 0 , (4.2)where ˆ f i : ( t, x, z ) f i ( t, x, ~ Γ , z ) ∨ f i ( t, x, ~ ˙Γ , z ) , ∀ ( i, t, x, z ) ∈ I × [0 , + ∞ ] × R k × R d . Once again, by Proposition (3 . and (3 . , there exists a unique solution to (4 . . FromProposition (3 . , for ( i, t ) ∈ I × [0 , + ∞ ] and for any a ∈ A it , ˆ Y , Y i and ˙ Y i have the followingswitching representation property: ˆ Y it = ˆ U ˆ at := ess sup a ∈A it ˆ U at , Y it = ess sup a ∈A it U at , and ˙ Y it = ess sup a ∈A it ˙ U at ; (4.3)14here ˆ a ∈ A it and ( ˆ U a , ˆ V a ) , ( U a , V a ) and ( ˙ U a , ˙ V a ) are respectively solutions of the followingBSDE: e − rt ˆ U at = Z + ∞ t e − rs ˆ f a s ( s, X s , ˆ V as ) ds − ( A a ∞ − A at ) − Z + ∞ t e − rs ˆ V as dB s , ≤ t ≤ + ∞ ,e − rt U at = Z + ∞ t e − rs f a s ( s, X s , ~ Γ s , V as ) ds − ( A a ∞ − A at ) − Z + ∞ t e − rs V as dB s , ≤ t ≤ + ∞ , and e − rt ˙ U at = Z + ∞ t e − rs f a s ( s, X s , ~ ˙Γ s , ˙ V as ) ds − ( A a ∞ − A at ) − Z + ∞ t e − rs ˙ V as dB s , ≤ t ≤ + ∞ . Now since, ∀ ( i, t, x, z ) ∈ I × [0 , + ∞ ] × R k × R d we have that ˆ f i ( t, x, z ) ≥ f i ( t, x, ~ Γ , z ) and ˆ f i ( t, x, z ) ≥ f i ( t, x, ~ ˙Γ , z ) , then, by the classical comparison theorem of BSDE, for any a ∈ A it U at ≤ ˆ U at and ˙ U at ≤ ˆ U at , ∀ t ≥ . Next, combining this estimates with the representation (4 . , implies that Y it ≤ ˆ Y it and ˙ Y it ≤ ˆ Y it , ∀ t ≥ . Since ˆ a is an admissible strategy for the representation (4 . of Y i and ˙ Y i , we deduce that U ˆ at ≤ Y it ≤ ˆ U ˆ at and ˙ U ˆ at ≤ ˙ Y it ≤ ˆ U ˆ at , ∀ t ≥ . (4.4)Then, we obtain that | Y it − ˙ Y it | ≤ | ˆ U ˆ at − U ˆ at | + | ˆ U ˆ at − ˙ U ˆ at | , ∀ t ≥ . (4.5)We first control the first term on the right hand side of (4 . . Using Itô’s formula to e − rt | ˆ U ˆ at − U ˆ at | , we get e − rt | ˆ U ˆ at − U ˆ at | + Z + ∞ t e − rs (2 r | ˆ U ˆ as − U ˆ as | + | ˆ V ˆ as − V ˆ as | ) ds = 2 Z + ∞ t e − rs ( ˆ U ˆ as − U ˆ as )( ˆ f ˆ a s ( s, x, ˆ V ˆ as ) − f ˆ a s ( s, x, ~ Γ s , V ˆ as )) ds − Z + ∞ t e − rs ( ˆ U ˆ as − U ˆ as )( ˆ V ˆ as − V ˆ as ) dB s . (4.6)Making use of inequality | x ∨ y − y | ≤ | x − y | together with the Lipschits property of f . , weobtain | ˆ f ˆ a s ( s, x, ˆ V ˆ as ) − f ˆ a s ( s, x, ~ Γ s , V ˆ as ) | ≤ u ( s )( | ~ Γ s − ~ ˙Γ s | + | ˆ V ˆ as − V ˆ as | ) . (4.7)Then, e − rt | ˆ U ˆ at − U ˆ at | + Z + ∞ t e − rs ( r | ˆ U ˆ as − U ˆ as | + | ˆ V ˆ as − V ˆ as | ) ds ≤ Z + ∞ t u ( s ) e − rs | ˆ U ˆ as − U ˆ as | ( | ~ Γ s − ~ ˙Γ s | + | ˆ V ˆ as − V ˆ as | ) ds − Z + ∞ t e − rs ( ˆ U ˆ as − U ˆ as )( ˆ V ˆ as − V ˆ as ) dB s . xy ≤ x ǫ + ǫy for any x, y ∈ R and ǫ > , we get that e − rt | ˆ U ˆ at − U ˆ at | + Z + ∞ t e − rs ( r | ˆ U ˆ as − U ˆ as | + | ˆ V ˆ as − V ˆ as | ) ds ≤ Z + ∞ t (2 u ( s ) + ǫ − ) e − rs | ˆ U ˆ as − U ˆ as | ds + ǫ Z + ∞ t u ( s ) e − rs | ~ Γ s − ~ ˙Γ s | ds + 12 Z + ∞ t e − rs | ˆ V ˆ as − V ˆ as | ds − Z + ∞ t e − rs ( ˆ U ˆ as − U ˆ as )( ˆ V ˆ as − V ˆ as ) dB s . (4.8)Choosing r ≥ u ( s ) + ǫ − and taking expectation at t = 0 , we have E (cid:20)Z + ∞ e − rs | ˆ V ˆ as − V ˆ as | ds (cid:21) ≤ ǫ E (cid:20)Z + ∞ u ( s ) e − rs | ~ Γ s − ~ ˙Γ s | ds (cid:21) . (4.9)Going back to (4 . and applying Burkholder-Davis-Gundy’s inequality, we obtain E (cid:20) sup t ≥ e − rt | ˆ U ˆ at − U ˆ at | (cid:21) ≤ ǫ E (cid:20)Z + ∞ u ( s ) e − rs | ~ Γ s − ~ ˙Γ s | ds (cid:21) + 12 E (cid:20) sup t ≥ e − rt | ˆ U ˆ at − U ˆ at | (cid:21) + C E (cid:20)Z + ∞ e − rs | ˆ V ˆ as − V ˆ as | ds (cid:21) . Taking into account inequality (4 . , we get E (cid:20) sup t ≥ e − rt | ˆ U ˆ at − U ˆ at | (cid:21) ≤ C ǫ E (cid:20)Z + ∞ u ( s ) e − rs | ~ Γ s − ~ ˙Γ s | ds (cid:21) , (4.10)where C ǫ denotes a constant which depends on ǫ and may vary from line to line.Following the same method, we can get a similar estimate for E [sup t ≥ e − rt | ˆ U ˆ at − ˙ U ˆ at | ] .Next going back to (4 . and taking in consideration the estimates obtained above, we deduce E (cid:20) sup t ≥ e − rt | Y it − ˙ Y it | (cid:21) ≤ C ǫ E (cid:20)Z + ∞ u ( s ) e − rs | ~ Γ s − ~ ˙Γ s | ds (cid:21) . (4.11)Since the last inequality holds true for an ( i, t ) ∈ I × [0 , + ∞ ] , we have k φ ( ~ Γ) − φ ( ~ ˙Γ) k ,r ≤ C ǫ Z + ∞ u ( s ) ds k ~ Γ − ~ ˙Γ k ,r From R + ∞ u ( s ) ds < C − ǫ , φ is a contraction on [ S ] m , which concludes the proof of theproposition.As a consequence, there exists a unique fixed point in [ S ] m for φ , which is the uniquesolution of RBSDE (1.1). References [1] J.-F. Chassagneux, R. Elie, I. Kharroubi,
A note on existence and uniqueness for solutionsof multi-dimensional RBSDE . Electron. Commun. Probab. vol. , pp. - , .162] Z. Chen, Existence and uniqueness for BSDE’s with stopping time,
Chinese Science Bulletin, , p. − , . [3] Z. Chen, B. Wang, Infinite time interval BSDE and the convergence of g-martingales,
J.Aust. Math. Soc. Ser. A, , p. − , . [4] C. Dellacherie and P. A. Meyer, Probabilities and Potential , chap. I-IV. Hermann, . [5] C. Dellacherie and P. A. Meyer, Probabilités et Potentiel , chap. V-VIII, Hermann, Paris, . [6] B. El Asri, Optimal multi-modes switching problem in infinite horizon.
Stochastics andDynamics, Vol. , No. , − , .[7] B. El Asri, S. Hamadène, The finite horizon optimal multi-modes switching problem: theviscosity solution approach.
Appl. Math. Optim. , − , .[8] N. El-Karoui, Les aspects probabilistes du contrôle stochastique , In Ecole d’Etè de Saint-Flour IX. Lecture Notes in Mathematics , − . Springer Verlag Berlin, . [9] N. El-Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions ofbackward SDE’s, and related obstacle problems for PDE’s . The Annals of Probability .Vol. , No. , − , . [10] N. El-Karoui, S. Peng, M. C. Quenez, Backward Stochastic Differential Equation in Fi-nance , Math. Finance, , − , . [11] A. Gegout-Petit and E. Pardoux, Equations différentielles rétrogrades réflèchies dans unconvexe , Stochastics and Stochastics Reports, vol . , pp . − , .[12] S. Hamadène, M.-A. Morlais, Viscosity solutions of systems of PDEs with interconnectedobstacles and switching problem,
Appl. Math. Optim.
67 (2) 163 − , . [13] S. Hamadène and M. Jeanblanc, On the starting and stopping problem: Application inreversible investments,
Math. Oper. Res., , pp. − , . [14] S. Hamadène, L. Lepeltier and Z. Wu, Infinite horizon reflected backward stochastic dif-ferential equations and applications in mixed control and game problems,
Robability andMathematical Statistics, , pp. − , . [15] S. Hamadène and J. Zhang, Switching problem and related system of reflected backwardSDEs,
Stochastic Process. Appl., pp. − , . [16] Y. Hu, S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching .Probab. Theory Relat. Fields − , − , .[17] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus.
Second Edition,Springer-Verlag, New York, . [18] G. Liang, Stochastic control representations for penalized backward stochastic differentialequations,
SIAM J. Control Optm. Vol. , No. , pp. − , . [19] P.A. Meyer, Un cours sur les intégrales stochastiques,
Séminaire de probabilités (Stras-bourg), tome , p- - , . 1720] S. Ramasubramanian. Reflected backward stochastic differential equations in an orthant,
Proc. Indian Acad. Sci. Math. Sci, − , . [21] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion , Springer-Verlag,Berlin, . [22] Y. Shi and L. Zhang, Comparison Theorems of Infinite Horizon Forward-BackwardStochastic Differential Equations, http://arxiv.org/abs/1005.4139. ..