Mou-Hsiung Chang

Optimal control of stochastic functional differential equations with a bounded memory

This paper treats a finite time horizon optimal control problem in which the controlled state dynamics are governed by a general system of stochastic functional differential equations with a bounded memory. An infinite dimensional Hamilton–Jacobi–Bellman (HJB) equation is derived using a Bellman-type dynamic programming principle. It is shown that the value function is the unique viscosity solution of the HJB equation.

A Stochastic Portfolio Optimization Model with Bounded Memory

This paper considers a portfolio management problem of Mertons type in which the risky asset return is related to the return history. The problem is modeled by a stochastic system with delay. The investors goal is to choose the investment control as well as the consumption control to maximize his total expected, discounted utility. Under certain situations, we derive the explicit solutions in a finite dimensional space.

Optimal Stopping Problem for Stochastic Differential Equations with Random Coefficients

An optimal stopping problem for stochastic differential equations with random coefficients is considered. The dynamic programming principle leads to a Hamiltion-Jacobi-Bellman equation, which, for the current case, is a backward stochastic partial differential variational inequality (BSPDVI, for short) for the value function. Well-posedness of such a BSPDVI is established, and a verification theorem is proved.

Finite Difference Approximations for Stochastic Control Systems with Delay

Abstract This article considers the computation issues of the infinite dimensional HJB equation arising from the finite horizon optimal control problem of a general system of stochastic functional differential equations with a bounded memory treated in [2]. The finite difference scheme, using the result in [1], is obtained to approximate the viscosity solution of the infinite dimensional HJB equation. The convergence of the scheme is proved using the Banach fixed point theorem. The computational algorithm also is provided based on the scheme obtained.

Recurrence and Transience of Quantum Markov Semigroups

This article introduces concepts and surveys recent results on recurrence and transience of general quantum Markov semigroups (QMS) of bounded linear maps acting on a C*- or von Neumann algebra . In particular, the concept of potentials for classical Markov semigroups/processes is extended to its noncommutative counterpart. The characterization of recurrent and transient quantum Markov semigroups and classification of irreducible quantum Markov semigroups are established in terms of the potential of some subharmonic projection for the QMS. This introductory and survey work can be treated as a continuation of the closely related paper by Chang [12], which dealt with the invariance, mean ergodicity and ergodicity of QMS. Since it is intended as an introduction to large time asymptotic behavior of quantum Markov semigroups, this article is made self-contained by reviewing relevant concepts and results in quantum probability space, quantum states, and quantum Markov semigroups that are necessary for the subsequent developments and readability for nonexperts in this research areas.

A Survey on Invariance and Ergodicity of Quantum Markov Semigroups

This article surveys recent results on invariances and ergodicity of general quantum Markov semigroups of bounded linear maps acting on C*- or von Neumann algebra . In particular, we consider existence and uniqueness of invariant (stationary) quantum states as well as ergodicity and mean ergodicity of quantum states via heavy usage of the GNS representation. This survey is made self-contained by also reviewing relevant concepts and results necessary for the subsequent developments.

Viscosity Solution of Optimal Stopping Problem for Stochastic Systems with Bounded Memory

We consider a finite time horizon optimal stopping problem for a system of stochastic functional differential equations with a bounded memory. Under some sufficiently smooth conditions, a Hamilton-Jacobi-Bellman (HJB) variational inequality for the value function is derived via dynamical programming principle. It is shown that the value function is the unique viscosity solution of the HJB variational inequality.

Discrete Approximations of Controlled Stochastic Systems with Memory: A Survey

This survey article considers discrete approximations of an optimal control problem in which the controlled state equation is described by a general class of stochastic functional differential equations with a bounded memory. Specifically, three different approximation methods, namely (i) semidiscretization scheme; (ii) Markov chain approximation; and (iii) finite difference approximation, are investigated. The convergence results as well as error estimates are established for each of the approximation methods.