A unified approach to gradient type formulas for BSDEs and some applications

Xiliang Fan, Michael Röckner, Shao-Qin Zhang

Published 2021-03-11Version 1

In this paper we present a unified approach to establish gradient type formulas and Bismut type formulas for backward stochastic differential equations (BSDEs). This approach relies on a mix of derivative formulas with respect to the conditional probability of forward SDEs and the expression of the solution of BSDEs. Some concrete examples are given to illustrate the results. As applications, we provide representation formulas for the control solutions to McKean-Vlasov BSDEs and derive gradient estimates for related PDEs.