A Class of Forward-Backward Stochastic Differential Equations Driven by L\'{e}vy Processes and Application to LQ Problems
Abstract
In this paper, our primary focus lies in the thorough investigation of a specific category of nonlinear fully coupled forward-backward stochastic differential equations involving time delays and advancements with the incorporation of L\'{e}vy processes, which we shall abbreviate as FBSDELDAs. Drawing inspiration from diverse examples of linear-quadratic (LQ) optimal control problems featuring delays and L\'{e}vy processes, we proceed to employ a set of domination-monotonicity conditions tailored to this class of FBSDELDAs. Through the application of the continuation method, we achieve the pivotal results of unique solvability and the derivation of a pair of estimates for the solutions of these FBSDELDAs. These findings, in turn, carry significant implications for a range of LQ problems. Specifically, they are relevant when stochastic Hamiltonian systems perfectly align with the FBSDELDAs that fulfill the domination-monotonicity conditions. Consequently, we are able to establish explicit expressions for the unique optimal controls by utilizing the solutions of the corresponding stochastic Hamiltonian systems.
Keywords
Cite
@article{arxiv.2310.13195,
title = {A Class of Forward-Backward Stochastic Differential Equations Driven by L\'{e}vy Processes and Application to LQ Problems},
author = {Maozhong Xu and Maoning Tang and Qingxin Meng},
journal= {arXiv preprint arXiv:2310.13195},
year = {2023}
}