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

Jiaqiang Wen, Xun Li, Jie Xiong, Xin Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)

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 Itô'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.

Original languageEnglish
Pages (from-to)949-979
Number of pages31
JournalSIAM Journal on Control and Optimization
Volume61
Issue number2
DOIs
Publication statusPublished - Apr 2023

Keywords

  • Markovian regime switching
  • mean-variance portfolio selection
  • random coefficient
  • stochastic linear-quadratic optimal control
  • stochastic Riccati equation

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics

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