General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients

The main purpose of this paper is to discuss detailed the stochastic LQ control problem with random coefficients where the linear system is a multidimensional stochastic differential equation driven by a multidimensional Brownian motion and a Poisson random martingale measure. In the paper, we will establish the connections of the multidimensional Backward stochastic Riccati equation with jumps (BSRDEJ in short form) to the stochastic LQ problem and to the associated Hamilton systems. By the connections, we show the optimal control have the state feedback representation. Moreover, we will show the existence and uniqueness result of the multidimensional BSRDEJ for the case where the generator is bounded linear dependence with respect to the unknowns martingale term

Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability

summary:In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see \cite {chen}, for finite dimensional stochastic equations or \cite {UC}, for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see \cite {1990}, \cite {ukl}). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on ${\mathbf R}_{+}$ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known \cite {ukl} that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see \cite {1990})

Open-Loop and Closed-Loop Solvabilities for Stochastic Linear Quadratic Optimal Control Problems

This paper is concerned with a stochastic linear quadratic (LQ, for short) optimal control problem. The notions of open-loop and closed-loop solvabilities are introduced. A simple example shows that these two solvabilities are different. Closed-loop solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. Conditions ensuring the convexity of the cost functional are discussed, including the issue that how negative the control weighting matrix-valued function R(s) can be. Finiteness of the LQ problem is characterized by the convergence of the solutions to a family of Riccati equations. Then, a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Finally, an illustrative example is presented.Comment: 40 page

On tracking a deterministic growth.

Zhang Li.Thesis (M.Phil.)--Chinese University of Hong Kong, 2003.Includes bibliographical references (leaves 67-69).Abstracts in English and Chinese.Chapter 1 --- Introduction and Literature Review --- p.1Chapter 2 --- The Tracking Portfolio Models --- p.7Chapter 2.1 --- Problem Formulation --- p.8Chapter 2.2 --- Reformulation of Tracking Models --- p.12Chapter 2.3 --- A Stochastic LQ Control Approach --- p.13Chapter 3 --- Efficient Tracking: Deterministic Market Parameters --- p.16Chapter 3.1 --- Solution to Model I --- p.17Chapter 3.2 --- A Special Case of Model I --- p.23Chapter 3.3 --- Solution to Model II --- p.24Chapter 3.4 --- A Special Case of Model II: Mean-Variance Portfolio Selection --- p.32Chapter 3.5 --- Solution to Model III --- p.36Chapter 4 --- Efficient Tracking: Markov-Modulated Market Parameters --- p.41Chapter 4.1 --- Problem Formulation --- p.42Chapter 4.2 --- Solution to Model I with Regime Switching --- p.47Chapter 4.3 --- Solution to Model II with Regime Switching --- p.52Chapter 4.4 --- Solution to Model III with Regime Switching --- p.59Chapter 5 --- Conclusion --- p.64Bibliography --- p.6