On the backward stochastic differential equation with generator $f(y)|z|^2$

TL;DR

This paper studies backward stochastic differential equations with a quadratic generator involving a function of y and the square of z, establishing existence, uniqueness, comparison theorems, and linking to viscosity solutions of quadratic PDEs.

Contribution

It provides new results on existence, uniqueness, and comparison theorems for BSDEs with a specific quadratic generator, and connects these to viscosity solutions of quadratic PDEs.

Findings

Existence and uniqueness of bounded solutions.

Comparison theorems for these BSDEs.

Probabilistic interpretation of viscosity solutions of quadratic PDEs.

Abstract

In this paper, we consider the backward stochastic differential equation (BSDE) with generator $f(y)|z|^2,$ where the function $f$ is defined on an open interval $D$ and locally integrable. The existence and uniqueness of bounded solutions and $L^p(p\geq1)$ solutions of such BSDEs are obtained. Some comparison theorems and a converse comparison theorem of such BSDEs are established. As an application, we give a probabilistic interpretation of viscosity solution of quadratic PDEs.