A representation theorem for generators of BSDEs with general growth generators in $y$ and its applications

TL;DR

This paper establishes a general representation theorem for BSDE generators with broad growth conditions in y, using localization and approximation, and applies it to derive probabilistic viscosity solutions for second-order semilinear PDEs.

Contribution

It introduces a novel representation theorem for BSDE generators with general growth in y, expanding the theoretical framework and applications to PDEs.

Findings

Proves a general representation theorem for BSDE generators with weak monotonicity and growth conditions.

Derives a probabilistic viscosity solution formula for second-order semilinear PDEs.

Provides a new tool for analyzing viscosity solutions of PDEs using BSDEs.

Abstract

In this paper we first prove a general representation theorem for generators of backward stochastic differential equations (BSDEs for short) by utilizing a localization method involved with stopping time tools and approximation techniques, where the generators only need to satisfy a weak monotonicity condition and a general growth condition in $y$ and a Lipschitz condition in $z$. This result basically solves the problem of representation theorems for generators of BSDEs with general growth generators in $y$. Then, such representation theorem is adopted to prove a probabilistic formula, in viscosity sense, of semilinear parabolic PDEs of second order. The representation theorem approach seems to be a potential tool to the research of viscosity solutions of PDEs.