Reflected BSDEs with regulated trajectories

Abstract

We consider reflected backward stochastic different equations with optional barrier and so-called regulated trajectories, i.e trajectories with left and right finite limits. We prove existence and uniqueness results. We also show that the solution may be approximated by a modified penalization method. Application to an optimal stopping problem is given. MSC 2000 subject classifications: primary 60H10; secondary 60G40.Keywords: Reflected backward stochastic differential equation, processes with regulated trajectories, modified penalization method, optimal stopping problem.M loc (resp. M) is the set of all F-martingales (resp. local martingales) M such that M 0 = 0. M p , p ≥ 1, denotes the space of all M ∈ M such that

Reflected backward stochastic differential equations with two optional barriers

“…Moreover, if the barriers are càdlàg, then condition (1.5) reduces to the minimality condition considered in [12]. In the present paper, we generalize the existence, uniqueness and approximation results proved in [17]. It is worth pointing out, however, that the proofs are essentially more complicated and in many points different from those in [17].…”

“…and let (Y 2,n , Z 2,n , K 2,n ) be a solution of RBSDE(0,0,Y 1,n−1 − U ) such that if p > 1, Y 1,n , Y 2,n ∈ S p , Z 1,n , Z 2,n ∈ H p , K 1,n , K 2,n ∈ V +,p , and if p = 1, then Y 1,n , Y 2,n are of class (D), Z 1,n , Z 2,n ∈ H q , q ∈ (0, 1), K 1,n , K 2,n ∈ V +,1 . For each n ≥ 0 the existence of the above solutions follows from [17,Theorem 3.20]. In both cases (p > 1, p = 1), by Proposition 3.5, we have…”

“…We may assume that µ = 0. By [17,Corollary A.5], for all σ, τ ∈ Γ such that σ ≤ τ , we have 3.2 Existence of solutions for f independent of z Theorem 3.6. Assume that f is independent of z.…”

“…The following definition of a solution of reflected BSDE with one optional barrier was introduced in [17]. Definition 2.3.…”

“…This condition automatically implies (1.4) with p = 2 in case f is Lipschitz continuous. The assumptions on ξ and f adopted in the present paper are the same as in our previous paper [17] devoted to equations with one lower barrier, and our definition of a solution is a counterpart to the definition introduced in [17]. For a process η, let ∆ + η t = η t+ − η t , ∆ − η t = η t − η t− , i.e.…”