Finite, integrable and bounded time embeddings for diffusions

TL;DR

This paper addresses the Skorokhod embedding problem for general diffusions, providing a unified construction method and characterizing conditions for finite, integrable, and bounded solutions.

Contribution

It introduces a martingale-based construction for the SEP in diffusions and characterizes when solutions are finite, integrable, or bounded.

Findings

Provides conditions for finite time solutions.

Characterizes distributions embeddable with integrable stopping times.

Derives necessary and sufficient conditions for bounded embeddings.

Abstract

We solve the Skorokhod embedding problem (SEP) for a general time-homogeneous diffusion $X$: given a distribution $\rho$, we construct a stopping time $\tau$ such that the stopped process $X_{\tau}$ has the distribution $\rho$. Our solution method makes use of martingale representations (in a similar way to Bass (In Seminar on Probability XVII. Lecture Notes in Math. 784 (1983) 221-224 Springer) who solves the SEP for Brownian motion) and draws on law uniqueness of weak solutions of SDEs. Then we ask if there exist solutions of the SEP which are respectively finite almost surely, integrable or bounded, and when does our proposed construction have these properties. We provide conditions that guarantee existence of finite time solutions. Then, we fully characterize the distributions that can be embedded with integrable stopping times. Finally, we derive necessary, respectively sufficient, conditions under which there exists a bounded embedding.