Indefinite stochastic linear-quadratic optimal control problems with random jumps and related stochastic Riccati equations

We discuss the stochastic linear-quadratic(LQ) optimal control problem with Poisson processes under the indefinite case. Based on the wellposedness of the LQ problem, the main idea is expressed by the definition of relax compensator that extends the stochastic Hamiltonian system and stochastic Riccati equation with Poisson processes(SREP) from the positive definite case to the indefinite case. We mainly study the existence and uniqueness of the solution for the stochastic Hamiltonian system and obtain the optimal control with open-loop form. Then, we further investigate the existence and uniqueness of the solution for SREP in some special case and obtain the optimal control in close-loop form.

STOCHASTIC DIFFERENTIAL EQUATIONS AND STOCHASTIC LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM WITH LEVY PROCESSES

In this paper, tile authors first study two kinds of stochastic differential equations （SDEs） with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations （BSDEs） driven by Levy pro- cesses, the authors proceed to study a stochastic linear quadratic （LQ） optimal control problem with a Levy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.

Error Analysis of the Feedback Controls Arising in the Stochastic Linear Quadratic Control Problems

In this work,the author proposes a discretization for stochastic linear quadratic control problems(SLQ problems)subject to stochastic differential equations.The author firstly makes temporal discretization and obtains SLQ problems governed by stochastic difference equations.Then the author derives the convergence rates for this discretization relying on stochastic differential/difference Riccati equations.Finally an algorithm is presented.Compared with the existing results relying on stochastic Pontryagin-type maximum principle,the proposed scheme avoids solving backward stochastic differential equations and/or conditional expectations.

Stochastic H_2/H_∞ Control with Random Coefcients

Abstract This paper is concerned with the mixed H2/H∞ control for stochastic systems with random coefficients, which is actually a control combining the H2 optimization with the H∞ robust performance as the name of H2/H∞ reveals. Based on the classical theory of linear-quadratic （LQ, for short） optimal control, the sufficient and necessary conditions for the existence and uniqueness of the solution to the indefinite backward stochastic Riccati equation （BSRE, for short） associated with H∞ robustness are derived. Then the sufficient and necessary conditions for the existence of the H2/H∞ control are given utilizing a pair of coupled stochastic Pdccati equations.

Characterization of optimal feedback for stochastic linear quadratic control problems

One of the fundamental issues in Control Theory is to design feedback controls.It is well-known that,the purpose of introducing Riccati equations in the study of deterministic linear quadratic control problems is exactly to construct the desired feedbacks.To date,the same problem in the stochastic setting is only partially well-understood.In this paper,we establish the equivalence between the existence of optimal feedback controls for the stochastic linear quadratic control problems with random coefficients and the solvability of the corresponding backward stochastic Riccati equations in a suitable sense.We also give a counterexample showing the nonexistence of feedback controls to a solvable stochastic linear quadratic control problem.This is a new phenomenon in the stochastic setting,significantly different from its deterministic counterpart.

Constrained LQ Problem with a Random Jump and Application to Portfolio Selection

This paper deals with a constrained stochastic linear-quadratic（LQ for short）optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the Ito-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations（BSDEs for short）. The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions. The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.