Controlled Martingale Problems And Their Markov Mimics

TL;DR

This paper demonstrates that marginals of solutions to controlled martingale problems can be replicated by Markov solutions, and introduces a method using relative entropy minimization to construct such Markov mimics, with various applications.

Contribution

It establishes the existence of Markov mimics for controlled martingale problems and presents a novel approach using relative entropy minimization.

Findings

Mimicking marginals with Markov solutions is possible under certain conditions.

Relative entropy minimization can be used to construct Markov mimics.

The results are applicable to numerous examples in the field.

Abstract

In this article we prove under suitable assumptions that the marginals of any solution to a relaxed controlled martingale problem on a Polish space $E$ can be mimicked by a Markovian solution of a Markov-relaxed controlled martingale problem. We also show how such `Markov mimics' can be obtained by relative entropy minimisation. We provide many examples where the above results can be applied.