Backward doubly stochastic differential equations with non-Lipschitz coefficients

Probabilistic representation of parabolic stochastic variational inequality with Dirichlet-Neumann boundary and variational generalized backward doubly stochastic differential equations

“…Therefore we built a decreasing sequence (T p ) p≥1 such that on each [T p , T p+1 ], GBDSDE associated to the data (Y p+1 , f , g, h) admit a solution. We also prove as like in [16] that there exists a finite p such that T p = 0. Finally by the path continuity of the solution on each interval, we obtain the solution on [0, T ] and end the proof of existence.…”

“…We emphasize that when g ≡ 0, equation (2.3) becomes the classical backward doubly stochastic equations (BDSDEs, in short) introduit by Pardoux and Peng [17] under Lipschitz generator and recently relax by many authors. We can see in [16], Owo and N'zi's work and references therein.…”

“…In the evidence, it would have adjustments in the proof to take account the fact that ρ depends on t. In this case the Bihari inequality could no longer be applied and will have to be replaced by (H3 ′ )-(c). We refer the reader to the works of Owo and N'zi [16] for more details. Now we are ready to give the main result of this section.…”

“…Several kinds of BSDEs have been established under this assumption and it relaxed version (see Owo and N'zi [16], Wand and Wang [24], Wang and Huang [23]).…”

Lp - estimates of solutions of backward doubly stochastic differential equations

“…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.…”