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In this paper, our primary focus lies in the thorough investigation of aspecific category of nonlinear fully coupled forward-backward stochasticdifferential equations involving time delays and advancements with theincorporation of L\'{e}vy processes, which we shall abbreviate as FBSDELDAs.Drawing inspiration from diverse examples of linear-quadratic (LQ) optimalcontrol problems featuring delays and L\'{e}vy processes, we proceed to employa set of domination-monotonicity conditions tailored to this class ofFBSDELDAs. Through the application of the continuation method, we achieve thepivotal results of unique solvability and the derivation of a pair of estimatesfor the solutions of these FBSDELDAs. These findings, in turn, carrysignificant implications for a range of LQ problems. Specifically, they arerelevant when stochastic Hamiltonian systems perfectly align with the FBSDELDAsthat fulfill the domination-monotonicity conditions. Consequently, we are ableto establish explicit expressions for the unique optimal controls by utilizingthe solutions of the corresponding stochastic Hamiltonian systems.

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