From minimal embeddings to minimal diffusions

TL;DR

This paper explores the relationship between minimal diffusions and solutions to the Skorokhod Embedding Problem, introducing the concept of minimal diffusion linked to boundary behavior, martingales, and local time minimization.

Contribution

It introduces the novel concept of minimal diffusion, connecting boundary behavior, martingale property, and local time minimization within the framework of the SEP.

Findings

Minimal diffusions minimize expected local time at all points.

A clear link is established between minimality, boundary behavior, and martingale properties.

The approach clarifies the structure of diffusions related to the SEP.

Abstract

There is a natural connection between the class of diffusions, and a certain class of solutions to the Skorokhod Embedding Problem (SEP). We show that the important concept of minimality in the SEP leads to the new and useful concept of a minimal diffusion. Minimality is closely related to the martingale property. A diffusion is minimal if it minimises the expected local time at every point among all diffusions with a given distribution at an exponential time. Our approach makes explicit the connection between the boundary behaviour, the martingale property and the local time characteristics of time-homogeneous diffusions.