Turnpike Property of Stochastic Linear-Quadratic Optimal Control Problems in Large Horizons with Regime Switching I: Homogeneous Cases

TL;DR

This paper investigates the long-term behavior of stochastic linear-quadratic control problems with regime switching, establishing the turnpike property and convergence of Riccati solutions in large horizons for homogeneous cases.

Contribution

It proves the turnpike property and solution convergence for large-horizon stochastic control problems with regime switching, extending previous results and providing new analytical tools.

Findings

Established the turnpike property for homogeneous stochastic control problems.

Proved convergence of Riccati solutions to algebraic Riccati equations.

Provided refined estimates improving upon previous literature.

Abstract

This paper is concerned with optimal control problems for a linear homogeneous stochastic differential equation having regime switching with purely quadratic functional in the large time horizons. We establish the so-called turnpike properties for the optimal pairs. The key is to prove a proper convergence of the solutions to the differential Riccati equations to the algebraic Riccati equation. Even for the problems without regime switchings, our result provides a refined estimate compared to those in the previous literature, which also provides a new tool for further research.