Reflected backward stochastic differential equations with two optional barriers

Abstract

We consider reflected backward stochastic differential equations with two general optional barriers. The solutions to these equations have the so-called regulated trajectories, i.e trajectories with left and right finite limits. We prove the existence and uniqueness of L p solutions, p ≥ 1, and show that the solutions may be approximated by a modified penalization method.MSC 2000 subject classifications: Primary 60H10; secondary 60G40.

Nonlinear BSDEs with two optional Doob's class barriers satisfying weak Mokobodzki's condition and extended Dynkin games

“…(H6*) there exists a process X ∈ M loc (0, T )+V 1 F (0, T ) such that X is of class (D), L ≤ X ≤ U and f (⋅, X, 0) ∈ L 1 F (0, T ). The following result has been proven in [27].…”

“…Equations of that type with L 2 -data and Lipschitz generator were studied in [30] (Brownian filtration), in [13,14] (Brownian-Poisson filtration) and [1,2,15,25] (general filtration). RBSDEs with optional barriers and L 1 -data were considered only in [26,27], in the case of Brownian filtration. Results on optional barriers, L 1 -data and possibly infinite horizon time were presented in [25].…”

Reflected BSDEs with two optional barriers and monotone coefficient on general filtered space

“…In our framework part a) presents no difficulties and its proof proceeds analogously to the case of more regular data (see e.g. [14,26]). The problems begin when we proceed to the nonlinear case since we can not apply fixed point argumentthe reason is twofold: T may be infinite and f is assumed to be merely non-decreasing with respect to y.…”

“…Such equations with L 2 -data and Lipschitz continuous generator were studied in [30] (Brownian filtration), in [13,14,31] (Brownian-Poisson filtration) and [3,4,15] (general filtration). RBSDEs with L 1 -data and optional barriers were considered only in [25,26] in case of Brownian filtration and bounded terminal time.…”

A priori estimates for multidimensional BSDEs with integrable data

“…1). This follows at once from[3, Remark 2.1] and[1, Lemma 3.1]. Let a ∈ R. Observe that if (Y, Z) is a solution to BSDE T (ξ, f ), then ( Ȳ , Z) is a solution to BSDE T ( ξ, f ),where ( Ȳt , Zt ) ∶= (e at Y t , e at Z t ), ξ ∶= e aT ξ, f (t, y, z) ∶= e at f (t, e −at y, e −at z) − ay Clearly, if (ξ, f ) satisfies any of conditions (H1), (H3)-(H5), then ( ξ, f ) satisfies it too.…”