Backward doubly SDEs with continuous and stochastic linear growth coefficients

Abstract

We study backward doubly stochastic differential equations when the coefficients are continuous with stochastic linear growth. Via an approximation and comparison theorem, the existence of minimal and maximal solutions are obtained.

General Mean-Field BDSDEs with Stochastic Linear Growth and Discontinuous Generator

“…Under the assumptions (A1), (A3), (A4) and (A6)-(A8), the general mean-field BDSDE ( 5) at least has one solution (Y, Z) ∈ M 2,β (0, T). Moreover, there is a minimal solution (Y, Z) ∈ M 2,β (0, T) of the general mean-field BDSDE (5).…”

“…Shi, Gu, and Liu [3] proved the comparison theorem of BDSDEs. Recently, Owo [4][5][6] generalized these results under a series of stochastic conditions, including the existence and uniqueness theorem of solution for BDSDEs with stochastic Lipschitz generator, the existence theorem of solutions under stochastic linear growth and continuous or discontinuous conditions, and he also proved the associated comparison theorems. Inspired by this literature, in this paper, we study a new class of BDSDEs called general mean-field BDSDEs to obtain the corresponding results, and the equation's form is as follows:…”

Penalization method for reflected BDSDEs with two-sided jumps and driven by Lévy process

“…Following this, Ren et al [37] have considered BDSDEs driven by Teugels martingales associated with a Lévy process satisfying some moment condition and an independent Brownian motion. Later, there are several works which have focused on developing the theory of BDSDEs in different direction (see for instance [4,13,14,29,30,31,38,39]). As a variation of BDSDEs, Bahlali et al [7] were introduced reflected BDSDEs (RBDSDEs in short) where an additional nondecreasing process K is added to the standard BDSDEs (1.1) in order to keep the Y-component of the solution above a certain lower continuous process, called barrier (or obstacle), and to do so in a minimal fashion.…”

Irregular barrier reflected BDSDEs with general jumps under stochastic Lipschitz and linear growth conditions

“…To this direction, several attempts have been done. Among others, we refer to [4,5,9,15,[21][22][23][24] for the case of BSDEs, and [16,25,26,30] for BDSDEs. In our paper, we use a generalization of the Doob-Meyer decomposition called the Mertens decomposition.…”

“…To this direction, several attempts have been done. Among others, we refer to [4,5,9,15,[21][22][23][24] for the case of BSDEs, and [16,25,26,30] for BDSDEs.…”