Backward Stochastic Differential Equations with interaction

TL;DR

This paper introduces a novel class of backward stochastic differential equations with interaction, establishing existence and uniqueness results under Lipschitz conditions related to Wasserstein distance, and connects them to the Monge-Kantorovich problem.

Contribution

It presents the first study of BSDEs with interaction, providing existence and uniqueness proofs using Wasserstein distance and discrete measure approximation.

Findings

Established existence and uniqueness of BSDEs with interaction

Connected BSDEs with interaction to the Monge-Kantorovich problem

Developed approximation methods using discrete measures and Wasserstein distance

Abstract

In this paper backward stochastic differential equations with interaction (shorter BSDEs with interaction) are introduced. Far to our knowledge, this type of equation is not seen in the literature before. Existence and uniqueness result for BSDE with interaction is proved under version of Lipschitz condition with respect to Wasserstein distance. Such kind of BSDE arises naturally when considering Monge-Kantorovich problem. In the proof we start from discrete measures using known result of Pardoux and Peng and approximate general measure via Wasserstein distance.