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

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.