L^p-SOLUTIONS OF BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS

Abstract

The goal of this paper is to solve backward doubly stochastic differential equations (BDSDEs, in short) under weak assumptions on the data. The first part is devoted to the development of some new technical aspects of stochastic calculus related to this BDSDEs. Then we derive a priori estimates and prove existence and uniqueness of solution in L p , p ∈ (1, 2), extending the work of Pardoux and Peng (see Probab. Theory Related Fields 98 (1994), no. 2).MSC 2000: 60H05, 60H20.

Stochastic partial differential equations with singular terminal condition

“…First we have to prove existence and uniqueness of the solution of a BDSDE with monotone generator f . To our best knowlegde the closest result on this topic is in Aman [1]. Nevertheless we think that there is a lack in this paper (precisely Proposition 4.2).…”

“…Thus in order to prove existence of a solution for (1), one can not directly follow the scheme of [34]. And thus Proposition 4.2 in [1] is not proved. The first part of this paper is devoted to the existence of a solution for a monotone BDSDE (see Section 2) in the space E 2 (see Definition 1).…”

“…Proposition 3 Assume that BDSDE(1) with data (f 1 , g, ξ1 ) and (f 2 , g, ξ 2 ) have solutions(Y 1 , Z 1 ) and (Y 2 , Z 2 ) in E 2 (0, T ), respectively. The coefficient g satisfies (A4).…”

Lp - estimates of solutions of backward doubly stochastic differential equations

“…Regarding that C > 0 and α ∈ 0, 1/2(p − 1) are the known constants, we can determine positive numbers 1 and 2 such that c 3 > 0. Applying Jensen's inequality [16] on the concave function ρ, we find from (16) that…”

“…Since then, many authors tried to weaken the conditions for the functions f and and to give more general results. For example, Lin [14], Aman [1], Aman and Owo [2], N'zi and Owo [18,19], Boufoussi, Casteren and Mrhardy [4], Ren, Lin and Hu [26], Hu and Ren [12]. All previous papers deal with BDSDEs which have only the process z in the forward integral, i.e.…”