Infinite Horizontal Optimal Quadratic Control for an Affine Equation Driven by Lévy Processes

摘要

The author studies a linear quadratic optimal control problem for stochastic systems driven by Lévy processes where the linear state equation has stochastic coefficients and moreover, an affine term. The adjoint equation has unbounded coefficients, and its solution is not known. Employing BMO-martingale theory, the author proves the existence and uniqueness of the solutions to the adjoint equation in a finite horizon. In the case of an infinite horizon, assuming some stabilizability, the author proves via suitable finite horizontal approximation the existence of the solutions to the backward stochastic Riccati differential equation and the adjoint backward stochastic equation. Using these solutions, the author performs the synthesis of the optimal control.