On the one-sided Tanaka equation with drift

TL;DR

This paper investigates the existence and uniqueness of solutions for a one-sided Tanaka equation with constant drift, revealing a dichotomy based on the drift parameter and demonstrating how perturbations can restore solution properties.

Contribution

It characterizes the conditions under which strong and weak solutions exist for the equation, highlighting the impact of the drift parameter and perturbations on solution uniqueness.

Findings

For mbda leq; 0, strong pathwise unique solutions exist.

For mbda > 0, only weak solutions are unique in distribution, with no strong solutions.

Brownian perturbations can restore strength and pathwise uniqueness.

Abstract

We study questions of existence and uniqueness of weak and strong solutions for a one-sided Tanaka equation with constant drift \lambda. We observe a dichotomy in terms of the values of the drift parameter: for \lambda\leq 0, there exists a strong solution which is pathwise unique, thus also unique in distribution; whereas for \lambda>0, the equation has a unique in distribution weak solution, but no strong solution (and not even a weak solution that spends zero time at the origin). We also show that strength and pathwise uniqueness are restored to the equation via suitable "Brownian perturbations".