Stochastic Linear Quadratic Optimal Control Problems with Random Coefficients and Markovian Regime Switching System

TL;DR

This paper investigates stochastic linear-quadratic control problems with random coefficients and Markovian switching, establishing solvability conditions, deriving optimal controls, and applying results to a mean-variance portfolio problem.

Contribution

It provides new solvability results for stochastic Riccati equations and characterizes optimal controls in systems with random coefficients and regime switching.

Findings

Proves solvability of stochastic Riccati equations under uniform convexity.

Derives closed-loop optimal control representations using Riccati solutions.

Applies theoretical results to a mean-variance portfolio selection problem.

Abstract

This paper thoroughly investigates stochastic linear-quadratic optimal control problems with the Markovian regime switching system, where the coefficients of the state equation and the weighting matrices of the cost functional are random. We prove the solvability of the stochastic Riccati equation under the uniform convexity condition and obtain the closed-loop representation of the open-loop optimal control using the unique solvability of the corresponding stochastic Riccati equation. Moreover, by applying It\^{o}'s formula with jumps, we get a representation of the cost functional on a Hilbert space, characterized as the adapted solutions of some forward-backward stochastic differential equations. We show that the necessary condition of the open-loop optimal control is the convexity of the cost functional, and the sufficient condition of the open-loop optimal control is the uniform convexity of the cost functional. In addition, we study the properties of the stochastic value flow of the stochastic linear-quadratic optimal control problem. Finally, as an application, we present a continuous-time mean-variance portfolio selection problem and prove its unique solvability.