A Class of Forward-Backward Stochastic Differential Equations Driven by L\'{e}vy Processes and Application to LQ Problems

Maozhong Xu; Maoning Tang; Qingxin Meng

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arXiv · 2310.13195

A Class of Forward-Backward Stochastic Differential Equations Driven by L\'{e}vy Processes and Application to LQ Problems

Maozhong Xu·Maoning Tang·Qingxin Meng

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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.

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Maozhong Xu, Maoning Tang, Qingxin Meng. 2023-10-19. A Class of Forward-Backward Stochastic Differential Equations Driven by L\'{e}vy Processes and Application to LQ Problems. https://arxiv.org/abs/2310.13195

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