The Skorokhod embedding problem for inhomogeneous diffusions

TL;DR

This paper extends the Skorokhod embedding problem to inhomogeneous diffusions described by SDEs with time-dependent coefficients, providing conditions for embedding arbitrary distributions via bounded stopping times.

Contribution

It introduces a method to solve the Skorokhod embedding problem for inhomogeneous diffusions using coupled forward-backward SDEs and decoupling fields, including a practical algorithm and numerical illustration.

Findings

Established sufficient conditions for embedding arbitrary measures.

Developed a construction based on coupled FBSDEs and decoupling fields.

Provided a numerical algorithm and demonstrated its effectiveness.

Abstract

We solve the Skorokhod embedding problem for a class of stochastic processes satisfying an inhomogeneous stochastic differential equation (SDE) of the form $d A_t =\mu (t, A_t) d t + \sigma(t, A_t) d W_t$. We provide sufficient conditions guaranteeing that for a given probability measure $\nu$ on $\mathbb{R}$ there exists a bounded stopping time $\tau$ and a real $a$ such that the solution $(A_t)$ of the SDE with initial value $a$ satisfies $A_\tau \sim \nu$. We hereby distinguish the cases where $(A_t)$ is a solution of the SDE in a weak or strong sense. Our construction of embedding stopping times is based on a solution of a fully coupled forward-backward SDE. We use the so-called method of decoupling fields for verifying that the FBSDE has a unique solution. Finally, we sketch an algorithm for putting our theoretical construction into practice and illustrate it with a numerical experiment.