A Tanaka formula for the derivative of intersection local time in ${\mathbb{R}}^1$

Abstract

Let $B_t$ be a one dimensional Brownian motion, and let ${\alpha }^{\prime }$ denote the derivative of the intersection local time of $B_t$ as defined in Jay Rosen's work (see references). The object of this paper is to prove the following formula $(1/2){\alpha }_t^{\prime }(x)+(1/2)sgn(x)t={\int }_0^t{L}_s^{B_s-x}d{B}_s-{\int }_0^{t}sgn(B_t-B_u-x)du$ which was given as a formal identity by Rosen without proof.