Skorokhod embeddings for two-sided Markov chains

TL;DR

This paper establishes a criterion for embedding a given distribution into a two-sided Markov chain via a stopping time, providing explicit solutions and analyzing their optimality and moment properties.

Contribution

It introduces a necessary and sufficient condition for Skorokhod embedding in two-sided Markov chains and offers explicit, optimal solutions with moment analysis.

Findings

Provides a criterion for the existence of a suitable stopping time.

Constructs explicit solutions that are also stopping times.

Shows the solution minimizes expected concave functions of the stopping time.

Abstract

Let $(X_n \colon n\in\Z)$ be a two-sided recurrent Markov chain with fixed initial state $X_0$ and let $\nu$ be a probability measure on its state space. We give a necessary and sufficient criterion for the existence of a non-randomized time $T$ such that $(X_{T+n} \colon n\in\Z)$ has the law of the same Markov chain with initial distribution $\nu$. In the case when our criterion is satisfied we give an explicit solution, which is also a stopping time, and study its moment properties. We show that this solution minimizes the expectation of $\psi(T)$ in the class of all non-negative solutions, simultaneously for all non-negative concave functions $\psi$.