1D nonlinear backward stochastic differential equations: a unified theory and applications

TL;DR

This paper develops a unified theoretical framework for the existence and uniqueness of solutions to one-dimensional nonlinear backward stochastic differential equations with specific growth conditions, and discusses applications and open problems.

Contribution

It introduces a comprehensive methodology for analyzing 1D BSDEs with unilateral and quadratic growth, extending existing results and providing practical applications.

Findings

Established new existence and uniqueness theorems for 1D BSDEs.

Developed a unified test function and a priori estimate approach.

Outlined open problems for future research.

Abstract

Since the celebrated paper by El Karoui, Peng and Quenez [Mathematical Finance, 7 (1997), 1--71], backward stochastic differential equations have found wide applications in stochastic control, financial technology and machine learning. In this paper, we present a comprehensive theory on the existence and uniqueness of adapted solutions to a one-dimensional nonlinear backward stochastic differential equation (1D BSDE for short), and assume that the generator $g$ has a unilateral linear or super-linear growth in the first unknown variable $y$, and has an at most quadratic growth in the second unknown variable $z$. We develop a unified methodology, featured by the test function method and the a priori estimate technique, to establish several existence theorems and comparison theorems, which immediately yield corresponding existence and uniqueness results. We also overview relevant known results and give some practical applications of our theoretical results. Finally, we list some open problems on the well-posedness of 1D BSDEs.