A stochastic differential equation with a sticky point

TL;DR

This paper studies a special stochastic differential equation with a sticky point, proving weak existence and uniqueness but showing pathwise uniqueness and strong solutions do not exist.

Contribution

It introduces a degenerate SDE with a sticky point and establishes fundamental existence and uniqueness results, highlighting the absence of pathwise uniqueness.

Findings

Weak existence and weak uniqueness are proved.

Pathwise uniqueness does not hold.

No strong solution exists.

Abstract

We consider a degenerate stochastic differential equation that has a sticky point in the Markov process sense. We prove that weak existence and weak uniqueness hold, but that pathwise uniqueness does not hold nor does a strong solution exist.