Near optimal controls for partially observed stochastic linear quadratic problems

TL;DR

This paper develops a method to find near optimal controls for complex stochastic linear quadratic problems with partial observations, addressing challenges like control in diffusion and unbounded drifts.

Contribution

It introduces a novel approach combining control restriction, Girsanov theorem, and a non-standard variation method to handle intricate partial observation control problems.

Findings

Characterized a near optimal control in the weak formulation.

Addressed control in the diffusion term and unbounded drifts.

Established a limit-based approach for near optimal control approximation.

Abstract

In this article, we consider a stochastic linear quadratic control problem with partial observation. A near optimal control in the weak formulation is characterized. The main features of this paper are the presence of the control in the diffusion term of the state equation, the circular dependence between the control process and the filtration generated by the observation, and the observation process contains an unbounded drift term. We address these difficulties by first restricting the control to a smaller domain, which enables us to apply the Girsanov theorem using a conditional argument and thereby break the circular dependence. Subsequently, we study the restricted problem using a non-standard variation method. The desired near optimal control is then obtained by taking the limit of an approximating sequence.