Backward doubly stochastic differential equations and SPDEs with quadratic growth

2022/05/11 by Ying Hu, Jiaqiang Wen, Hu, Ying +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #60H10 #60H15 #60H30 #FOS: Mathematics #Financial Risk and Volatility Modeling #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2205.05289

openalex publication_date 2022/05/11 · openalex created_date 2022/05/22 · openalex updated_date 2026/07/28

Abstract

In this paper, we initiate the study of backward doubly stochastic differential equations (BDSDEs, for short) with quadratic growth. The existence, comparison, and stability results for one-dimensional BDSDEs are proved when the generator f(t,Y,Z) grows in Z quadratically and the terminal value is bounded, by introducing some new ideas. Moreover, in this framework, we use BDSDEs to give a probabilistic representation for the solutions of semilinear stochastic partial differential equations (SPDEs, for short) in Sobolev spaces, and use it to prove the existence and uniqueness of such SPDEs, thus extending the nonlinear Feynman-Kac formula.

Citations

• Mean-field BDSDEs and associated nonlocal semi-linear backward stochastic partial differential equations

Cited by

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

Related

• Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth
• Nonlinear Feynman-Kac formulae for SPDEs with space-time noise
• Second Order Backward Stochastic Differential Equations with Quadratic Growth
• Representation theorems for backward doubly stochastic differential equations
• Anticipated backward stochastic differential equations with quadratic growth