Distribution dependent BSDEs driven by Gaussian processes

TL;DR

This paper studies distribution dependent backward stochastic differential equations driven by Gaussian processes, establishing existence, uniqueness, comparison theorems, new representations, and inequalities, thus advancing the theoretical understanding of these complex stochastic systems.

Contribution

It introduces a transfer principle for well-posedness, provides a novel representation for DDBSDEs driven by Gaussian processes, and derives key inequalities, enhancing the theoretical framework of distribution dependent BSDEs.

Findings

Proved existence and uniqueness of solutions.

Established a comparison theorem under Lipschitz conditions.

Derived transportation and Logarithmic-Sobolev inequalities.

Abstract

In this paper we are concerned with distribution dependent backward stochastic differential equations (DDBSDEs) driven by Gaussian processes. We first show the existence and uniqueness of solutions to this type of equations. This is done by formulating a transfer principle to transfer the well-posedness problem to an auxiliary DDBSDE driven by Brownian motion. Then, we establish a comparison theorem under Lipschitz condition and boundedness of Lions derivative imposed on the generator. Furthermore, we get a new representation for DDBSDEs driven by Gaussian processes, this representation is even new for the case of the equations driven by Brownian motion. The new obtained representation enables us to prove a converse comparison theorem. Finally, we derive transportation inequalities and Logarithmic-Sobolev inequalities via the stability of the Wasserstein distance and the relative entropy of measures under the homeomorphism condition.