A type of globally solvable BSDEs with triangularly quadratic generators

TL;DR

This paper investigates the existence and uniqueness of solutions for a class of backward stochastic differential equations with triangularly quadratic generators, extending local solutions to global ones under certain conditions.

Contribution

It introduces new methods for establishing well-posedness of triangularly quadratic BSDEs, including local and global solutions, with considerations for path-dependent generators.

Findings

Unique local solutions exist for bounded terminal conditions.

Global solutions can be constructed by stitching local solutions.

Solvability extends to generators with path dependence.

Abstract

The present paper is devoted to the study of the well-posedness of a type of BSDEs with triangularly quadratic generators. This work is motivated by the recent results obtained by Hu and Tang [14] and Xing and \v{Z}itkovi\'{c} [28]. By the contraction mapping argument, we first prove that this type of triangularly quadratic BSDEs admits a unique local solution on a small time interval whenever the terminal value is bounded. Under additional assumptions, we build the global solution on the whole time interval by stitching local solutions. Finally, we give solvability results when the generators have path dependence in value process.