Reflected backward doubly stochastic differential equations with discontinuous generator

Abstract

In this note, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous generator (left-or right-continuous). By a comparison theorem establish here for RBDSDEs, we provide a minimal or a maximal solution to RBDSDEs.

Reflected backward doubly stochastic differential equations with discontinuous barrier

“…Moreover,Ȳ T = ξ T a.s. Combining this with (27) gives us (1). By definition ofȲ we also haveȲ ν ≥ ξ ν a.s ∀ν ∈ T 0,T which with Proposition A.5 show thatȲ satisfies inequality (2).…”

“…Furthermore, the authors established the existence of a maximal and a minimal solution in the case of RBDSDEs with continuous generators. Other authors studied RBDS-DEs with continuous barrier [1,2,18]. Ren [26] studied RBDSDEs with a right-continuous obstacle with left limits.…”

Backward doubly stochastic equations with jumps and comparison theorems

“…Let B(s) is a n-dimensional (G (1) t )-Brownian motion and {M (dt, du) : 0 ≤ t ≤ T, u ∈ F } a (G (1) t )-Poisson random measure with intensity dtν(du), where ν(du) is a σ-finite Borel measure on the Polish spaces F . Let {G (2) t } 0≤t≤T be another filtration on (Ω, F , P) satisfying the usual hypotheses and independent from {G (1) t } 0≤t≤T . Let {W (ds, du) = (W 1 (ds, du), · · · , W n (ds, du)) T ; 0 ≤ t ≤ T, u ∈ E} be a n-dimensional (G (2) t )-Gaussian white noise constructed with n orthogonal white noises W i (ds, du) on R + × E with intensity dsπ i (du) respectively.…”

“…Here we choose b small enough such that b F |C(s, u)| 2 ν(du) ≤ 1. From condition (2) and (3) we have…”

Stochastic viscosity solutions of reflected stochastic partial differential equations with non-Lipschitz coefficients

“…Some authors have previously tried to give weak conditions for reflected BDSDEs. We can cite the work of Aman [1], Aman and Owo [3]. However, all this conditions remains insufficient to take into account all situations.…”