Backward doubly stochastic differential equations with or without reflection under weak conditions

2026/03/31 by Shuxian Gao, Ying Hu, Jiaqiang Wen
Mathematics · #math.PR #msc:60H10 #msc:60H20

paper · pdf

48 pages

arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

In this paper, we study the solvability of backward doubly stochastic differential equations (BDSDEs, for short), both with and without reflection, under weak conditions on the generator. First, when the generator f is of general growth in y and linear growth in z, we establish the existence, uniqueness, comparison principle, and the existence of maximal solutions. Second, when f is of linear growth in y and quadratic growth in z with bounded terminal value, we prove the existence, uniqueness, and comparison principle. Finally, when f is of general growth in y and quadratic growth in z with bounded terminal value, we prove the existence of maximal solutions.

Citations

• Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth
• Backward doubly stochastic differential equations and SPDEs with quadratic growth
• Multi-dimensional backward stochastic differential equations of diagonally quadratic generators: the general result
• On the uniqueness of solutions to quadratic BSDEs with non-convex generators and unbounded terminal conditions
• Backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs
• Utility maximization in constrained and unbounded financial markets: Applications to indifference valuation, regime switching, consumption and Epstein-Zin recursive utility
• Multi-Dimensional Backward Stochastic Differential Equations of Diagonally Quadratic generators
• Reflected backward doubly stochastic differential equations with discontinuous generator
• Backward SDEs with superquadratic growth
• Reflected Backward SDEs with General Jumps
• Quadratic G-BSDEs with convex generators and unbounded terminal conditions
• Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps

Related

• Reflected backward doubly stochastic differential equations with discontinuous generator
• Reflected Solutions of Backward Doubly Stochastic Differential Equations
• L^p-solutions of backward doubly stochastic differential equations
• The Equivalence between Uniqueness and Continuous Dependence of Solution for BDSDEs
• Lp-solution of backward doubly stochastic differential equations