A Class of Forward-Backward Stochastic Differential Equations Driven by L\'evy 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\'evy processes, which we shall abbreviate as FBSDELDAs. Drawing inspiration from diverse examples of linear-quadratic (LQ) optimal control problems featuring delays and L\'evy 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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