On the Tanaka formula for the derivative of self-intersection local time of fBm

Abstract

The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen and subsequently used by others to study the $L^2$ and $L^3$ moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) has also appeared in the literature. In the present work, we further its study by proving a Tanaka-style formula for it. In the course of this endeavor we provide a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously seen in the literature. Finally, we also give an explicit Wiener chaos expansion for the DSLT of fBm