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

$$\frac{1}{2}\alpha _t^\prime (x) + \frac{1}{2}\operatorname{sgn} (x)t = \int_0^t {L_s^{B_s - x} dB_s - \frac{1}{2}\int_0^t {\operatorname{sgn} (B_t - B_u - x)du} }$$

((1))

which was given as a formal identity in [2] without proof.