The solution of the perturbed Tanaka-equation is pathwise unique

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.