Invariant representation for stochastic differential operator by BSDEs with uniformly continuous coefficients and its applications

TL;DR

This paper establishes a new representation for second order stochastic differential operators using BSDEs with uniformly continuous coefficients, extending existing results and deriving related theorems and properties.

Contribution

It generalizes the representation of stochastic differential operators via BSDEs with uniformly continuous coefficients and explores related comparison and convexity properties.

Findings

Representation of second order stochastic operators via BSDEs

Converse comparison theorem for BSDEs with uniformly continuous coefficients

New proof of g-convexity

Abstract

In this paper, we prove that a kind of second order stochastic differential operator can be represented by the limit of solutions of BSDEs with uniformly continuous coefficients. This result is a generalization of the representation for the uniformly continuous generator. With the help of this representation, we obtain the corresponding converse comparison theorem for the BSDEs with uniformly continuous coefficients, and get some equivalent relationships between the properties of the generator $g$ and the associated solutions of BSDEs. Moreover, we give a new proof about $g$-convexity.