Reflected Advanced Backward Stochastic Differential Equations with Default
aa r X i v : . [ m a t h . O C ] M a r Reflected Advanced Backward Stochastic DifferentialEquations with Default
N. Agram , , S. Labed , B. Mansouri & M. A. Saouli
20 March 2018
Abstract
We are interested on reflected advanced backward stochastic differential equations(RABSDE) with default. By the predictable representation property and for a Lipschitzdriver, we show that the RABSDE with default has a unique solution in the enlargedfiltration. A comparison theorem for such type of equations is proved. Finally, we givea connection between RABSDE and optimal stopping.
Keywords:
Reflected Advanced Backward Stochastic Differential Equations, Single Jump,Progressive Enlargement of Filtration.
Reflected advanced backward stochastic differential equations (RABSDE) appear in theirlinear form as the adjoint equation when dealing with the stochastic maximum principle tostudy optimal singular control for delayed systems, we refer for example to Øksendal andSulem [10] and also to Agram et al [1] for more general case. This is a natural model in pop-ulation growth, but also in finance, where people’s memory plays a role in the price dynamics.After the economic crises in 2008, researchers started to include default in banks as apart of their financial modelling. This is why we are interested on RABSDE also in thecontext of enlargement of filtration. In order to be more precise, let us consider a random Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N–0316 Oslo, Norway.Email: [email protected].
This research was carried out with support of the Norwegian Research Council, within the researchproject Challenges in Stochastic Control, Information and Applications (STOCONINF), project number250768/F20. Department of Mathematics, University of Biskra, Algeria.Emails: [email protected], [email protected], [email protected]. τ which is neither an F -stopping time nor F T -measurable. Examples of such randomtimes are default times, where the reason for the default comes from outside the Brownianmodel. We denote H t = τ ≤ t , t ∈ [0 , T ] , and consider the filtration G obtained by enlargingprogressively the filtration F by the process H , i.e., G is the smallest filtration satisfying theusual assumptions of completeness and right-continuity, which contains the filtration F andhas H as an adapted process. The RABSDE related with, we want to study is the following: Y t = ξ + R Tt f ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) ds − R Tt Z s dW s − R Tt U s dH s + K T − K t , t ∈ [0 , T ] ,Y t = ξ, t ≥ T,Z t = U t = 0 , t > T. By saying that the RBSDE is advanced we mean that driver at the present time s maydepend not only on present values of the solution processes ( Y, Z, K ), but also on the futurevalues s + δ for some δ >
0. To make the system adapted, we take the conditional expectationof the advanced terms.We will see that by using the predictable representation property (PRP) the above systemis equivalent to a RABSDE driven by a martingale, consisting of the Brownian motion W and the martingale M associated to the jump process H , as follows: Y t = ξ + R Tt F ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) ds − R Tt Z s dW s − R Tt U s dM s + K T − K t , t ∈ [0 , T ] ,Y t = ξ, t ≥ T,Z t = U t = 0 , t > T. Our aim in this paper is not to find solutions in the Brownian filtration by using thedecomposition approach, as it has been done in Kharroubi and Lim [6] for BSDE and inJeanblanc et al [9] for ABSDE. However, we want to find solutions under the enlargedfiltration rather than the Brownian one as in the previous works.In Dumitrescu et al [3], [4], [5], the authors consider directly BSDE and RBSDE drivenby general filtration generated by the pair (
W, M ).We will extend the recent woks by Dumitrescu et al [3], [4], [5] , to the anticipated caseand we will explain how such an equations appear by using the PRP.We will extend also the comparison theorem for ABSDE in Peng and Yang [14] to RAB-SDE with default and finally, we give a link between RABSDE with default and optimalstopping as it has been done in El Karoui et al [6] and Øksendal and Zhang [13].For more details about ABSDE with jumps coming from the compensated Poisson ran-dom measure which is independent of the Brownian motion, we refer to Øksendal et al [12],[11]. For RBSDE with jumps, we refer to Quenez and Sulem [15] and for more details aboutenlargement progressive of filtration, we refer to Song [16] .2 Framework
Let (Ω , G , P ) be a complete probability space. We assume that this space is equipped witha one-dimensional standard Brownian motion W and we denote by F := ( F t ) t ≥ the rightcontinuous complete filtration generated by W . We also consider on this space a randomtime τ , which represents for example a default time in credit risk or in counterparty risk,or a death time in actuarial issues. The random time τ is not assumed to be an F -stoppingtime. We therefore use in the sequel the standard approach of filtration enlargement byconsidering G the smallest right continuous extension of F that turns τ into a G -stoppingtime (see e.g. Chapter 4 in [2]). More precisely G := ( G t ) t ≥ is defined by G t := \ ε> ˜ G t + ε , for all t ≥
0, where ˜ G s := F s ∨ σ ( τ ≤ u , u ∈ [0 , s ]), for all s ≥ P ( G ) the σ -algebra of G -predictable subsets of Ω × [0 , T ], i.e. the σ -algebragenerated by the left-continuous G -adapted processes.We then impose the following assumptions, which are classical in the filtration enlarge-ment theory.( H ) The process W is a G -Brownian motion. We observe that, since the filtration F is gener-ated by the Brownian motion W , this is equivalent with the fact that all F -martingalesare also G -martingales. Moreover, it also follows that the stochastic integral R t X s dW s is well defined for all P ( G )-measurable processes X such that R t | X s | ds < ∞ , for all t ≥ • The process M defined by M t = H t − R t ∧ τ λ s ds, t ≥ , is a G -martingale with single jump time τ and the process λ is F -adapted, called the F -intensity of τ . • We assume that the process λ is upper bounded by a constant. • Under ( H ) any square integrable G martingale Y admits a representation as Y t = y + R t ϕ s dW s + R t γ s dM s , where M is the compensated martingale of H , and ϕ, γ are square-integrable G -predictable processes. (See Theorem 3.15 in [2]).Throughout this section, we introduce some basic notations and spaces.3 S G is the subset of R -valued G -adapted c`adl`ag processes ( Y t ) t ∈ [0 ,T ] , such that k Y k S := E [ sup t ∈ [0 ,T ] | Y t | ] < ∞ . • K is a set of real-valued nondecreasing processes K with K − = 0 and E [ K t ] < ∞ . • H G is the subset of R -valued P ( G )-measurable processes ( Z t ) t ∈ [0 ,T ] , such that k Z k H := E [ R T | Z t | dt ] < ∞ . • L ( λ ) is the subset of R -valued P ( G )-measurable processes ( U t ) t ∈ [0 ,T ] , such that k U k L ( λ ) := E [ R T ∧ τ λ t | U t | dt ] < ∞ . We study the RABSDE with default Y t = ξ + R Tt f ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) ds − R Tt Z s dW s − R Tt U s dH s + K T − K t , t ∈ [0 , T ] ,Y t = ξ, t ≥ T,Z t = U t = 0 , t > T, (3.1)where f is G t ⊗B ([0 , T ]) ⊗B ( R )-measurable, and the terminal condition ξ is G T -measurable.Moreover • Y t ≥ S t , for each t ≥ • K t is c`adl`ag, increasing and G -adapted process with K − = 0 . • R T ( Y t − S t ) dK ct = 0 and △ K dt = −△ Y t { Y t − = S t − } , where denote the continuous anddiscontinuous parts of K respectively. • ( S t ) t ≥ is the obstacle which is a c`adl`ag, increasing and G -adapted process.We call the quadruplet ( Y, Z, U, K ) solution of the RABSDE (3.1).Let us impose the following set of assumptions.( i) Assumption on the terminal condition: • ξ ∈ L (Ω , G T ).(ii) Assumptions on the generator function f : Ω × [0 , T ] × R → R is such that4 G -predictable and satisfies the integrability condition, such that E [ R T | f ( t, , , , , | dt ] <
0, (3.2)for all t ∈ [0 , T ] . • Lipschitz in the sense that, there exists
C > , such that | f ( t, y, z, µ, π, u ) − f ( t, y ′ , z ′ , µ ′ , π ′ , u ′ ) |≤ C ( | y − y ′ | + | z − z ′ | + | π − π ′ | + | µ − µ ′ | + λ t | u − u ′ | ) , (3.3)for all t ∈ [0 , T ] and all y, y ′ , z, z ′ , µ, µ ′ , π, π ′ , u, u ′ ∈ R . We give the existence of the solution to a RABSDE in the enlarged filtration G . Theexistence follows from the PRP as we can also say, the property of martingale representation(PMR), and a standard approach like any classical RBSDE.Under our assumptions we know that equation (3.1) is equivalent to Y t = ξ + R Tt F ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) ds − R Tt Z s dW s − R Tt U s dM s + K T − K t , t ∈ [0 , T ] ,Y t = ξ, t ≥ T,Z t = U t = 0 , t > T, (3.4)with dH s = dM s + λ s s<τ ds , and F ( s, y, z, µ, π, u ) := f ( s, y, z, µ, π ′ , u ) − λ s ( − H s ) u. By assumption, the process λ is bounded.In order to get existence and uniqueness for the RABSDE (3.4), let us check that thegenerator F satisfies the same assumption as f : The function F : Ω × [0 , T ] × R → R issuch that (i) G -predictable and integrable in the sense that, for all t ∈ [0 , T ], by inequality (3.2), wehave E [ R T | F ( t, , , , , | dt ] = E [ R T | f ( t, , , , , | dt ] < . (ii) Lipschitz in the sense that there exists a constant C ′ >
0, such that | F ( t, y, z, µ, π, u ) − F ( t, y ′ , z ′ , µ ′ , π ′ , u ′ ) | = | f ( t, y, z, µ, π, u ) − f ( t, y ′ , z ′ , µ ′ , π ′ , u ′ ) − λ t ( − H t )( u − u ′ ) |≤ | f ( t, y, z, µ, π, u ) − f ( t, y ′ , z ′ , µ ′ , π ′ , u ′ ) | + λ t ( − H t ) | u − u ′ |≤ C ( | y − y ′ | + | z − z ′ | + | µ − µ ′ | + | π − π ′ | + λ t ( − H t ) | u − u ′ | ) + λ t ( − H t ) | u − u ′ |≤ C ′ ( | y − y ′ | + | z − z ′ | + | µ − µ ′ | + | π − π ′ | + λ t | u − u ′ | ) , t ∈ [0 , T ] and all y, z, u, , π, µ, y ′ , z ′ , u ′ , π ′ , µ ′ ∈ R , where we have used the Lips-chitzianity of f (3.3). (iii) The terminal value: ξ ∈ L (Ω , G T ). Theorem 3.1
Under the above assumptions (i)-(iii), the RABSDE (3 . admits a uniquesolution ( Y, Z, U, K ) ∈ S G × H G × L ( λ ) × K . Proof. We define the mappingΦ : H G × H G × L ( λ ) → H G × H G × L ( λ ) , for which we will show that it is contracting under a suitable norm. For this we note thatfor any ( Y, Z, U, K ) ∈ H G × H G × L ( λ ) × K there exists a unique quadruple ( ˆ Y , ˆ Z, ˆ U , ˆ K ) ∈ S G × H G × L ( λ ) × K , such thatˆ Y t = ξ + R Tt F ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) ds − R Tt ˆ Z s dW s − R Tt ˆ U s dM s − R Tt d ˆ K s , t ∈ [0 , T ] , (3.5)Let Φ( Y, Z, U ) := ( ˆ
Y , ˆ Z, ˆ U ). For given ( Y i , Z i , U i ) ∈ H F × H F × L ( λ ) , for i = 1 ,
2, we usethe simplified notations:( ˆ Y i , ˆ Z i , ˆ U i ) := Φ( Y i , Z i , U i ) , ( ˜ Y , ˜ Z, ˜ U ) := ( ˆ Y , ˆ Z , ˆ U ) − ( ˆ Y , ˆ Z , ˆ U ) , ( ¯ Y , ¯ Z, ¯ U ) := ( Y , Z , U ) − ( Y , Z , U ) . The triplet of processes (cid:16) ˜ Y , ˜ Z, ˜ U (cid:17) satisfies the equation˜ Y t = R Tt { F ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) − F ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) } ds − R Tt ˜ Z s dW s − R Tt ˜ U s dM s − R Tt d ˜ K s , t ∈ [0 , T ] . We have that M t = H t − R t λ s ds which is a pure jump martingale. Then,[ M ] t = P ≤ s ≤ t ( △ M s ) = P ≤ s ≤ t ( △ H s ) = H t , and h M i t = R t λ s ds, R Tt | ˜ U s | d h M i s = R Tt λ s | ˜ U s | ds. Applying Itˆo’s formula to e βt | ˜ Y t | , taking conditional expectation and using the Lipschitzcondition, we get 6 [ R T e βs ( β | ˜ Y s | + | ˜ Z s | + λ s | ˜ U s | ) ds ] ≤ ρC E [ R T e βs (cid:12)(cid:12) ¯ Y s (cid:12)(cid:12) ds ] + ρ E [ R T e βs { (cid:12)(cid:12) ¯ Z s (cid:12)(cid:12) + λ s (cid:12)(cid:12) ¯ U s (cid:12)(cid:12) } ds ] , where we have used that˜ Y s dK ,cs = ( Y s − S s ) dK ,cs − ( Y s − S s ) dK ,cs = − ( Y s − S s ) dK ,cs ≤ Y s dK ,cs ≥ Y s dK ,ds = ( Y s − S s ) dK ,ds − ( Y s − S s ) dK ,ds = − ( Y s − S s ) dK ,ds ≤ Y s dK ,ds ≥ λ is bounded, we get that λ ≤ kλ and by choosing β = 1 + 10 ρC we obtain || ( ˜ Y , ˜ Z, ˜ U ) || ≤ ρ || ( ¯ Y , ¯ Z, ¯ U ) || which means for ρ ≥
1, there exists a unique fixed point that is a solution for our RABSDE(3 . . (cid:3) In this section we are interested in a subclass of RABSDE where the driver only depend onfuture values of Y and is not allowed to depend on future values of Z , as follows: Y t = ξ + R Tt g ( s, Y s , Z s , E [ Y s + δ |G s ] , U s ) ds − R Tt Z s dW s − R Tt U s dM s + K T − K t , t ∈ [0 , T ] ,Y t = ξ, t ≥ T,Z t = U t = 0 , t > T, such that • Y t ≥ S t , for each t ≥ • K t is c`adl`ag, increasing and G -adapted process with K − = 0 . • R T ( Y t − S t ) dK ct = 0 and △ K dt = −△ Y t { Y t − = S t − } , where denote the continuous anddiscontinuous parts of K respectively. • ( S t ) t ≥ is the obstacle which is a c`adl`ag, increasing and G -adapted process.7e impose the following set of assumptions. (a) The driver g : Ω × [0 , T ] × R → R is G -predictable, and satisfies E [ R T | g ( t, , , , | dt ] < | g ( t, y, z, µ, u ) − g ( t, y ′ , z ′ , µ ′ , u ′ ) |≤ C ( | y − y ′ | + | z − z ′ | + | µ − µ ′ | + λ t | u − u ′ | ) , for all t ∈ [0 , T ] and all y, y ′ , z, z ′ , µ, µ ′ , u, u ′ ∈ R . (b) T he terminal condition: ξ ∈ L (Ω , G T ).Let us first state the comparison theorem for RBSDE with default which relies on thecomparison theorem for BSDE with default done by Dumitrescu et al [4], Theorem 2.17. Theorem 4.1
Let g , g : Ω × [0 , T ] × R → R , ξ , ξ ∈ L (Ω , G T ) and let the quadruplet ( Y j , Z j , U j , K j ) j =1 , be the solution of the RBSDE with default Y jt = ξ j + R Tt g j ( s, Y js , Z js , U js ) ds − R Tt Z js dW s − R Tt U js dM s + R Tt dK js , t ∈ [0 , T ] ,Y jt = ξ j , t ≥ T,Z jt = U jt = 0 , t > T. The drivers ( g j ) j =1 , satisfies assumptions (a)-(b). Suppose that there exists a predictableprocess ( θ t ) t ≥ with θ t λ t bounded and θ t ≥ − dt ⊗ dP a.s. such that g ( t, y, z, u ) − g ( t, y, z, u ′ ) ≥ θ t ( u − u ′ ) λ t . Moreover, suppose that • ξ ≥ ξ , a.s. • For any t ∈ [0 , T ] , S t ≥ S t , a.s. • g ( t, y, z, u ) ≥ g ( t, y, z, u ) , ∀ t ∈ [0 , T ] , y, z, u ∈ R . Then Y t ≥ Y t , ∀ t ∈ [0 , T ] .8 heorem 4.2 Let g , g : Ω × [0 , T ] × R → R , ξ , ξ ∈ L (Ω , G T ) and let the quadruplet ( Y j , Z j , U j , K j ) j =1 , be the solution of the RABSDE Y jt = ξ j + R Tt g j ( s, Y js , Z js , E [ Y js + δ |G s ] , U js ) ds − R Tt Z js dW s − R Tt U js dM s + R Tt dK js , t ∈ [0 , T ] ,Y jt = ξ j , t ≥ T,Z jt = U jt = 0 , t > T. The drivers ( g j ) j =1 , satisfies assumptions (a)-(b). Moreover, suppose that: (i) For all t ∈ [0 , T ] , y, z, u ∈ R , g ( t, y, z, · , u ) is increasing with respect to Y t + δ in thesense that g ( t, y, z, Y t + δ , u ) ≥ g (cid:0) t, y, z, Y ′ t + δ , u (cid:1) , for all Y t + δ ≥ Y ′ t + δ . (ii) ξ ≥ ξ , a.s .(iii) For each t ∈ [0 , T ] , S t ≥ S t , a.s.(iv) Suppose that there exists a predictable process ( θ t ) t ≥ with θ t λ t bounded and θ t ≥ − dt ⊗ dP a.s., such that g ( t, y, z, Y t + δ , u ) − g ( t, y, z, Y t + δ , u ′ ) ≥ θ t ( u − u ′ ) λ t . (v) g ( t, y, z, Y t + δ , u ) ≥ g ( t, y, z, Y t + δ , u ) , ∀ t ∈ [0 , T ] , y, z, Y t + δ , u ∈ R . Then, we have Y t ≥ Y t , a.e.,a.s.Proof. Consider the following RABSDE Y t = ξ + R Tt g ( s, Y s , Z s , E [ Y s + δ |G s ] , U s ) ds − R Tt Z s dW s − R Tt U s dM s + R Tt dK s , t ∈ [0 , T ] ,Y t = ξ , t ≥ T,Z t = U t = 0 , t > T. From Proposition 3.2 in Dumitrescu et al [5], we know there exists a unique quadruplet of G -adapted processes ( Y , Z , U , K ) ∈ S G × H G × L ( λ ) × K satisfies the above RBSDEsince the advanced term is considered as a parameter.Now we have by assumptions (iii)-(v) and Theorem 4.1, that Y t ≥ Y t , for all t , a.s.Set Y t = ξ + R Tt g ( s, Y s , Z s , E [ Y s + δ |G s ] , U s ) ds − R Tt Z s dW s − R Tt U s dM s + R Tt dK s , t ∈ [0 , T ] ,Y t = ξ , t ≥ T,Z t = U t = 0 , t > T.
9y the same arguments, we get Y t ≥ Y t , a.e.,a.s.For n = 5 , , ..., we consider the following RABSDE Y nt = ξ + R Tt g ( s, Y ns , Z ns , E [ Y n − s + δ |G t ] , U ns ) ds − R Tt Z ns dW s − R Tt U ns dM s + R Tt dK ns , t ∈ [0 , T ] ,Y nt = ξ , t ≥ T,Z nt = U nt = 0 , t > T. We may remark that it is clear that Y n − s + δ is considered to be knowing on the above RABSDE.By induction on n >
4, we get Y t ≥ Y t ≥ Y t ≥ · · · ≥ Y nt ≥ · · · , a.s.If we denote by¯ Y = Y n − Y n − , ¯ Z = Z n − Z n − , ¯ U = U n − U n − , ¯ K = K n − K n − . By similar estimations as in the proof of Theorem 3.1, we can find that ( Y n , Z n , U n , K n )converges to ( Y n − , Z n − , U n − , K n − ) as n → ∞ . Iterating with respect to n , we obtain when n → ∞ , that ( Y n , Z n , U n , K n ) converges to( Y, Z, U, K ) ∈ S G × H G × L ( λ ) × K , such that Y t = ξ + R Tt g ( s, Y s , Z s , E [ Y s + δ |G s ] , U s ) ds − R Tt Z s dW s − R Tt U s dM s + R Tt dK s , t ∈ [0 , T ] ,Y t = ξ , t ≥ T,Z t = U t = 0 , t > T. By the uniqueness of the solution (Theorem 3.1), we have that Y t = Y t , a.s.Since for all t, Y t ≥ Y t ≥ Y t ≥ ... ≥ Y t , a.s. it hold immediately for a.a. tY t ≥ Y t , a.s. (cid:3) We recall here a connection between RABSDE and optimal stopping problems. The followingresult is essentially due to El Karoui et al [6] under the Brownian filtration and to Øksendaland Zhang [13]:
Definition 5.1 • Let F : Ω × [0 , T ] × R → R , be a given function such that: • F is G -adapted and E [ R T | F ( t, , , , , | dt ] < . Let S t be a given G -adapted continuous process such that E [ sup t ∈ [0 ,T ] S t ] < ∞ . • The terminal value ξ ∈ L (Ω , G T ) is such that ξ ≥ S T a.s.We say that a G - adapted triplet ( Y, Z, K ) is a solution of the reflected ABSDE with driver F , terminal value ξ and the reflecting barrier S t under the filtration G , if the following hold:1. E [ R T | F ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) | dt ] < ∞ ,2. Y t = ξ + R Tt F ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) ds − R Tt dK s − R Tt Z s dW s − R Tt U s dM s , t ∈ [0 , T ] , or, equivalently, Y t = E [ ξ + R Tt F ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) ds − R Tt dK s |G t ] , t ∈ [0 , T ] , K t is nondecreasing, G -adapted, c`adl`ag process with R T ( Y t − S t ) dK ct = 0 and △ K dt = −△ Y t { Y t − = S t − } , where denote the continuous and discontinuous parts of K respectively , Y t ≥ S t a.s., t ∈ [0 , T ] . Theorem 5.2
For t ∈ [0 , T ] let T [ t,T ] denote the set of all G -stopping times τ : Ω [ t, T ] . Suppose ( Y, Z, U, K ) is a solution of the RABSDE above. (i) Then Y t is the solution of the optimal stopping problem Y t = ess sup τ ∈T [ t,T ] { E [ R τt F ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) ds + S τ τ 0) and an optimal stopping time ˆ τ t is given byˆ τ t : = inf { s ∈ [ t, T ] , Y s ≤ S s } ∧ T = inf { s ∈ [ t, T ] , K s > K t } ∧ T. (iii) In particular, if we choose t = 0 we get thatˆ τ : = inf { s ∈ [0 , T ] , Y s ≤ S s } ∧ T = inf { s ∈ [0 , T ] , K s > } ∧ T solves the optimal stopping problem Y = sup τ ∈T [0 ,T ] E [ R τ F ( s, Y s , Z s , E [ Y s + δ |G s ] , E [ Z s + δ |G s ] , U s ) ds + S τ τ