The solution of the perturbed Tanaka-equation is pathwise unique

TL;DR

This paper demonstrates that adding a sufficiently strong independent noise to the Tanaka equation ensures pathwise uniqueness of its solutions, which is not possible in the original form.

Contribution

It introduces a modified version of the Tanaka equation with an independent noise component that guarantees pathwise uniqueness.

Findings

Pathwise uniqueness holds for the modified equation.

Strong additive noise ensures solution uniqueness.

Original Tanaka equation lacks pathwise uniqueness.

Abstract

The Tanaka equation $dX_t={\operatorname{sign}}(X_t)\,dB_t$ is an example of a stochastic differential equation (SDE) without strong solution. Hence pathwise uniqueness does not hold for this equation. In this note we prove that if we modify the right-hand side of the equation, roughly speaking, with a strong enough additive noise, independent of the Brownian motion B, then the solution of the obtained equation is pathwise unique.