Proof of a Tanaka-like formula stated by J. Rosen in Séminaire XXXVIII

Abstract

Let B t be a one dimensional Brownian motion, and let α′ denote the derivative of the intersection local time of B t as defined by J. Rosen in [2]. The object of this paper is to prove the following formula 1/2′t(x)+ 1/2 sgn(x)t= ∫t0 LBs-xs dBs - 1/2∫t0 sgn(Bt - Bu - x)du (1) which was given as a formal identity in [2] without proof. © 2008 Springer-Verlag Berlin Heidelberg.