Driven By Levy Process Limited Time Zone And The Infinite Time Zones Of Stochastic Optimal Control And Optimal Stopping Time Of Hybrid Control

This thesis is concerned with finite horizon and infinite horizon linear-quadratic optimal stochastic control problem with Levy processes and finite horizon mixed optimal stochastic control and stopping problem. It is organized as follow.In Chapter1, a brief introduction is given for the background of linear-quadratic optimal stochastic control problems and mixed optimal stochastic con-trol and stopping problem, historical developments and present research situation. Moreover, the main results of this thesis are also presented.In Chapter2, we study a linear quadratic optimal control problem for stochas-tic systems driven by Levy 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 the (?)-martingale the-ory, we prove the existence and uniqueness of the solutions to the adjoint equation in finite horizon. In the case of infinite horizon, assuming some stabilizability, we prove 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, we synthesize the optimal control.In Chapter3, we discuss the stochastic linear-quadratic optimal control prob-lem for the linear stochastic system driven by both a Brownian motion and Levy processes where the cost functional takes conditional expectation with respect to the σ-algebras generated by Levy processes. We obtain the new multidimensional backward stochastic Riccati differential equation driven by Levy processes and prove the existence of solutions of the corresponding stochastic Riccati equation using Bellman’s quasilinear principle and a method of monotone convergence.In Chapter4, We study the parabolic variational inequality associated with the combined stochastic control problem on finite horizon. Under some sufficiently smooth conditions, a Hamilton-Jacobi-Bellman (HJB) variational inequality for the value function is derived from the dynamic programming principle. It is shown that the value function is the unique viscosity solution of the HJB variational inequality without assuming the uniform ellipticity. An application to the quasi-variational inequality is given.