English

The solution of the perturbed Tanaka-equation is pathwise unique

Probability 2013-07-12 v3

Abstract

The Tanaka equation dXt=sign(Xt)dBtdX_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.

Cite

@article{arxiv.1104.0740,
  title  = {The solution of the perturbed Tanaka-equation is pathwise unique},
  author = {Vilmos Prokaj},
  journal= {arXiv preprint arXiv:1104.0740},
  year   = {2013}
}

Comments

Published in at http://dx.doi.org/10.1214/11-AOP716 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

R2 v1 2026-06-21T17:49:29.085Z