Stochastic LQ Optimal Control Problem With Random Jumps And Its Applications

The optimal control problem asks the controllers to maximize(minimize)some cost functional under some certain constraint condition(state equation)over the admis-sible control set for seeking the optimal control regulator and its optimal value.The linear quadratic(LQ for short)optimal control problem has been studied in detail by many scholars due to its linear property of controlled system state and the quadratic feature of cost functional,especially due to their nice structures and successful applica-tions in the fields of finance and engineering,more over,the optimal feedback regulator and optimal value can be expressed explicitly by the solution of associate Riccati equa-tions.The random jump diffusion system driven by a Brownian motion and a Poisson process contains all the information of the continuous and discontinuous jump process,and can depict more complex and realistic stochastic models,this thesis is to further develop and perfect stochastic optimal control theory,especially,the stochastic LQ op-timal control theory and application are further promoted which the controlled system is driven by a Brownian motion and a Poisson jump process.Besides,the stochastic jump diffusion model is dynamic with time varying,characterized by Ito type stochas-tic differential equation.The thesis is divided into six chapters as follows:In Chapter 1,we introduce some related research backgrounds,significance and remarks and also the main results of this dissertationIn the second chapter,in view of the finite interval[0,T],the initial data parameter-ized LQ stochastic optimal control problems with random coefficients and random Pois-son jumps are studied,optimal control can be proved to be unique.A stochastic Riccati equation is rigorous derived from the stochastic Hamilton system(Fordward-Backward Stochastic Differential Equation with Poisson Jumps)by a decoupling technique,which provides an optimal feedback control and optimal valueIn chapter 3,based on the intrinsic properties of a stochastic system,one kind of approach,which we called "equivalent cost functional method" is applied to get a solv-ability of a stochastic LQ optimal control problem with Poisson jumps and indefinite control weight costs in a finite time horizon.Our analysis is featured by some equiva-lent cost functionals which enable us to transform the indefinite stochastic LQ problems to positive-definite case,it is remarkable that the solvability of the former is rather com-plicated than the latter.Finally,an explicit state feedback representation is given by the solution of associated indefinite stochastic Riccati equationIn Chapter 4,a mean-field stochastic LQ optimal control problem with distur-bances and random jumps in a finite horizon is considered.The mean field model can describe the robustness and reflect the anti-interference ability of a system,which means that the considered model could characterize more general problems.The well-posedness result on our mean-field stochastic differential equation is obtained by a clas-sical fixed point theorem,we also show the solvability for our mean-field stochastic optimal control problem.The introduction of an adjoint variable,classic variational calculus and some dual representation enable us to derive an optimal condition.Be-cause of the adjoint process is an auxiliary function,so in order to link the optimal control and the state directly,two Riccati differential equations and two perturbation equations are obtained by a decoupling technique,which provide an optimal feedback regulator.Meanwhile,the relationship between the two Riccati equations and the so-called mean-field stochastic Hamilton system is established.At last,the optimal value is characterized.In Chapter 5,we study the open-loop and closed-loop solvability for indefinite mean-field stochastic LQ optimal control problem,where the controlled stochastic sys-tem is driven by a Brownian motion and a Poisson random martingale measure and also disturbed by some stochastic processes.The intrinsic property of stochastic system re-sults in the inequivalence of these two solvability,which is different from deterministic system,actually,the closed-loop solvability implies the open-loop solvability,but the opposite is not true.It is shown that the uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem.By a matrix minimum principle,the necessity condition of regular solvability for a decoupled Riccati equation is estab-lished.And then the closed-loop solvability is turned out to be equivalent to the regular solvability of Riccati equation with the some constraints on the solution of disturbances equations,moreover,the optimal closed-loop strategy is characterized by the regular solution of Riccati equation and the adapted solution of perturbation equationsIn chapter 6,we study two portfolio optimization problems by using stochastic optimal control theory in order to seek a best allocation of wealth among a market of securities.In the section 1,we study a maximization utility functional with the price of stocks is characterized by a random jumps diffusion process.By the Pontryagin maxi-mum principle,dual parameters are introduced to build the Hamilton system,based on a priori assumptions and a decoupling technique,the optimal portfolio selection strategy and marginal strategy is derived.In the section 2,we discuss a mean-variance portfolio selection problem on an insurance company with jump diffusion liabilities and surplus,we firstly transform the multi-objective problem to the parameterized single objective problem,and then by constructing mean-field stochastic LQ optimal control problem,and using the result presented in the preceding chapter 4,the precisely expression of optimal portfolio selection strategy for the insurance company is obtained.It is worth noticing that,traditionally,due to the square of expectations of terminal wealth come into the cost functional,the usually method is to embed the problem into a tractable auxiliary problem,but in this section,we only need to use the associate result of the mean-field stochastic LQ optimal control to obtain the optimal portfolio strategy direct-ly,which provide a new and more convenient approach.