Stochastic Control Problems Of Delay Systems And Robust Duality In Constrained Utility Maximization

In this thesis,we mainly study the stochastic optimal control problem of delay systems.Firstly,we study the near-optimal control problem of stochastic delay systems.Secondly,we study the indefinite stochastic linear quadratic control problem with delay and related forward-backward stochastic cdifferential equations.In addition,based on the stochastic maximum principle,we also study a robust utility maximization problem with random endowments.In reality,there exist many phenomena depending on the past time,which means the behaviors of states at time t depend on not only the current situations,but al-so their past histories,such as communication delay in many communication systems.Near-optimal controls,as an alternative to the "exact" optimal ones,have received con-siderable attention in recent years due to its nice structure and broad-range availability,feasibility as well as flexibility.On one hand,near-optimal controls are more available than optimal ones.Stochastic linear quadratic(LQ,for short)optimal control problems have been widely applied in many modern engineering and financial areas.The reasons of LQ problem could be widely used are twofold.Firstly,many real problems can be modeled using LQ problems.And secondly,it is a common approach to approximate nonlinear systems with according linear systems.In addition,in financial mathematics and mathematical economics,utility maximization problem is a widely studied optimal decision problem.Investors need to consider how to construct investment strategies to maximize utility.Therefore,it is important to study control problems in time delay systems,as well as robust utility maximization problems.These problems have high application value,especially in financial mathematics.In Chapter 1,we briefly introduce the historical backgrounds,motivations and the-oretical tools,as well as the main contributions of the following chapters.In Chapter 2,we study a class of near-optimization problems with controlled sys-tems described by the stochastic differential equations with delay.Thanks to the Eke-land's variational principle,we establish both necessary and sufficient conditions for the near-optimality of this problem by several delicate estimates for the state and adjoint processes.It is also shown that the error estimate for the near-optimality is of the order?1/2 exactly.Moreover,we consider a production and consumption choice problem with delay to illustrate the application of our theoretical results.In Chapter 3,we focus on an indefinte stochastic linear quadratic optimal control problem,where the controlled system is described by a stochastic differential equation with delay.By introducing the relaxed compensator as a novel method,we obtain the well-posedness of this linear quadratic problem for indefinite case.And then,we discuss the uniqueness and existence of the solutions for a kind of anticipated forward-backward stochastic differential delayed equations.Based on this,we derive the solvability of the corresponding stochastic Hamiltonian systems,and give the explicit representation of the optimal control for the linear quadratic problem with delay in an open-loop form.The theoretical results are validated as well on the control problems of engineering and economics under indefinite condition.In Chapter 4,we discuss a robust utility maximization problem with random endow-ments following,when the portfolio is constrained to take values in a given closed,convex subset of RN.The primal problem with model uncertainty is introduced firstly,and then be reduced to a problem in calculus of variations.Following that,the dual problem is formulated in terms of the point-wise convex conjugate transforms.After formulating the primal and dual problems,we derive that the necessary and sufficient conditions for both the primal and dual problems can be written in terms of FBSDEs plus additional conditions.Based on this,we prove in particular that the optimal adjoint process for the dual problem coincides with the optimal wealth process in a dynamic version and vice versa.Moreover,we explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs and vice versa.Finally,we solve two robust constrained utility maximization problems,which show explicitly the simplicity of the duality approach we propose.