Large Deviation Principle for Backward Stochastic Differential Equations with a stochastic Lipschitz condition on $z$

TL;DR

This paper establishes a large deviation principle for solutions of quadratic backward stochastic differential equations with stochastic Lipschitz conditions, linking probabilistic methods to PDE viscosity solutions.

Contribution

It introduces a novel large deviation analysis for quadratic BSDEs with stochastic Lipschitz conditions, extending the understanding of their asymptotic behavior.

Findings

Proves convergence of quadratic BSDE solutions as diffusion coefficients tend to zero.

Establishes a large deviation principle for these solutions.

Provides a probabilistic interpretation of viscosity solutions for related PDEs.

Abstract

In this paper, a probabilistic interpretation for the viscosity solution of a parabolic partial differential equation is obtained by virtue of the solution of a class of quadratic backward stochastic differential equations (BSDEs, for short). Furthermore, we prove the convergence and the large deviation principle for the solution of this class of quadratic BSDEs, which is associated with a family of Markov processes with the diffusion coefficients that tend to be zero.