Multidimensional BSDEs with uniformly continuous coefficients: the general result

TL;DR

This paper proves the unique solvability of multidimensional backward stochastic differential equations with uniformly continuous coefficients by introducing a new envelope concept, solving an open problem in the field.

Contribution

It introduces a novel envelope notion and demonstrates the existence and uniqueness of solutions for multidimensional BSDEs with general uniformly continuous coefficients.

Findings

Established unique solvability of multidimensional BSDEs with uniformly continuous coefficients.

Developed a new envelope concept for stochastic processes.

Solved an open problem in the theory of multidimensional BSDEs.

Abstract

In this paper, by introducing a new notion of envelope of the stochastic process, we construct a family of random differential equations whose solutions can be viewed as solutions of a family of ordinary differential equations and prove that the multidimensional backward stochastic differential equations (BSDEs for short) with the general uniformly continuous coefficients are uniquely solvable. As a result, we solve the open problem of multidimensional BSDEs with uniformly continuous coefficients.