Reflected BSDEs on filtered probability spaces

Abstract

We study the problem of existence and uniqueness of solutions of backward stochastic differential equations with two reflecting irregular barriers, L p data and generators satisfying weak integrability conditions. We deal with equations on general filtered probability spaces. In case the generator does not depend on the z variable, we first consider the case p = 1 and we only assume that the underlying filtration satisfies the usual conditions of right-continuity and completeness. Additional integrability properties of solutions are established if p ∈ (1, 2] and the filtration is quasi-continuous. In case the generator depends on z, we assume that p = 2, the filtration satisfies the usual conditions and additionally that it is separable. Our results apply for instance to Markov-type reflected backward equations driven by general Hunt processes.

Systems of BSDES with oblique reflection and related optimal switching problems

“…However, because of the general filtration and weak assumption on the data, our proof is more involved. Also note that in our proof we use in an essential way some results on one-dimensional reflected BSDEs with general filtration proved in [11,20]. We are not able to prove that the solution to (1.1) is unique for general H. However, we show that the uniqueness for (1.1) holds true if f j (t, y) does not depend on y 1 , .…”

“…Clearly, f S also satisfies (H3) and (H4). Note that X − S is a difference of supermartingales of class D such that E T 0 |f S (r, X r − S r )| dr < ∞ and, by Remark 2.1, S admits decomposition (2.2) with C ∈ V 1 0 and N ∈ M. Therefore, by [11,Theorem 2.13]…”

“…The following assumptions will be needed throughout this section. Similar assumptions were considered in [11,20].…”

“…Let d = 2, T = 2 and 2], we see that (2, 0, 0, 0) is a solution of RBSDE(ξ 1 , 0, Y 2 −1, 2). By uniqueness (see [11,Corollary 3.2]), (2, 0, 0, 0) = (Y 1 , M 1 , K 1 , A 1 ).…”

“…, d (see Section 3 for detailes). In the case where d = 1, (1.1) reduces to the one-dimensional reflected BSDE with upper barrier U , which was thoroughly investigated in Klimsiak [11]. Therefore, in the present paper, we consider the case where d ≥ 2.…”