Backward doubly stochastic differential equations and SPDEs with quadratic growth

TL;DR

This paper studies backward doubly stochastic differential equations with quadratic growth, establishing existence, comparison, and stability results, and uses them to represent solutions of semilinear SPDEs, extending the nonlinear Feynman-Kac formula.

Contribution

It introduces the first analysis of BDSDEs with quadratic growth and connects them to solutions of semilinear SPDEs in Sobolev spaces.

Findings

Proved existence and stability of 1D BDSDEs with quadratic growth.

Provided a probabilistic representation for solutions of semilinear SPDEs.

Extended the nonlinear Feynman-Kac formula to this setting.

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.