On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion

Abstract

The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen [Ros05] 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) was introduced in [YYL08]; however, the definition given there presents difficulties, since it is motivated by an incorrect application of Itô's formula. To rectify this, we introduce a modified DSLT for fBm and prove existence using an explicit Wiener chaos expansion. We will then argue that our modification is the natural version of the DSLT by rigorously proving the corresponding Tanaka formula. This formula corrects a formal identity given in both [Ros05] and [YYL08]. In the course of this endeavor we prove 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. As a further byproduct of our investigation, we also provide a correction to an important technical secondmoment bound for fBm given in [Hu01].

Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative

“…This process has been further studied in [Mar08b,JM12,HN10] as well as some of their references. An FBM version of (1.3) was first considered in the works of [YYL08,YLY09].…”

“…An FBM version of (1.3) was first considered in the works of [YYL08,YLY09]. Later, using a Tanaka formula as guiding intuition, [JM12] rigorously extended (1.3) to one-dimensional FBMs with H < 2/3 bỹ An open problem stated in [JM12] was to prove the joint continuity, in space and time, of (1.4). Here, we consider a different extension of (1.3) to the case of FBM which is guided by an occupation-time formula, rather than a Tanaka formula:…”

“…The proof of the above proposition is an application of Lemma 3.1 in [Hu01], and can be deduced from the arguments of [YYL08] with a slight correction given in [JM12,Appendix]. It is therefore omitted.…”

Higher-Order Derivative of Intersection Local Time for Two Independent Fractional Brownian Motions

“…Intersection local time or self-intersection local time when the two processes are the same are important subjects in probability theory and their derivatives have received much attention recently. Jung et al [2] and [3] discussed Tanaka formula and occupation-time formula for derivative self-intersection local time of fractional Brownian motions. On the other hand, several authors paid attention to the renormalized self-intersection local time of fractional Brownian motions, see e.g., Hu et al [5] and [6].…”

“…Motivated by [2] and [4], higher-order derivative of intersection local time for two independent fractional Brownian motions is studied in this paper.…”