Long-Time Behaviors of Stochastic Linear-Quadratic Optimal Control Problems

TL;DR

This paper explores the long-term behavior of solutions to stochastic linear-quadratic control problems, introducing a probability cell problem that links to homogenization and ergodic costs, revealing turnpike properties.

Contribution

It introduces the probability cell problem and connects it to ergodic costs, providing new insights into the asymptotic behavior of stochastic LQ control problems.

Findings

Introduction of the probability cell problem

Connection between the cell problem and ergodic cost

Revelation of turnpike properties in stochastic control

Abstract

This paper investigates the asymptotic behavior of the solution to a linear-quadratic stochastic optimal control problems. The so-called probability cell problem is introduced the first time. It serves as the probability interpretation of the well-known cell problem in the homogenization of Hamilton-Jacobi equations. By establishing a connection between this problem and the ergodic cost problem, we reveal the turnpike properties of the linear-quadratic stochastic optimal control problems from various perspectives.