Multidimensional BSDEs with uniformly continuous coefficients: the general result

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.