Perkins Embedding for General Starting Laws

TL;DR

This paper introduces a new optimal solution to the Skorokhod embedding problem for general starting laws, extending the Perkins embedding beyond the initial Dirac distribution case and providing new geometric insights.

Contribution

It develops the first optimal solution to the SEP for arbitrary starting distributions, generalizing the Perkins embedding and offering a new geometric interpretation.

Findings

Provides an optimal solution for the SEP with general starting laws.

Connects the new solution to the classical Perkins embedding.

Offers a clearer geometric understanding of the Perkins solution.

Abstract

The Skorokhod embedding problem (SEP) is to represent a given probability measure as a Brownian motion $B$ at a particular stopping time. In recent years particular attention has gone to solutions which exhibit additional optimality properties due to applications to martingale inequalities and robust pricing in mathematical finance. Among these solutions, the Perkins embedding sticks out through its distinct geometric properties. Moreover is the only optimal solution to the SEP which so far has been limited to the case of Brownian motion started in a dirac distribution. In this paper we provide for the first time an optimal solution to the Skorokhod embedding problem for the general SEP which leads to the Perkins solution when applied to Brownian motion with start in a dirac. This solution to the SEP also suggests a new geometric interpretation of the Perkins solution which better clarifies the relation to other optimal solutions of the SEP.