Higher order derivative of self-intersection local time for fractional Brownian motion

TL;DR

This paper investigates the existence, regularity, and limit behavior of higher order derivatives of self-intersection local time for fractional Brownian motion, providing new conditions and confirming a conjecture for a critical case.

Contribution

It establishes existence and Hölder continuity conditions for derivatives of self-intersection local time and proves a limit theorem for the critical case, addressing a conjecture by Jung and Markowsky.

Findings

Established conditions for existence and Hölder continuity of derivatives

Proved a limit theorem for the critical case H=2/3, d=1

Confirmed a conjecture by Jung and Markowsky

Abstract

We consider the existence and H\"{o}lder continuity conditions for the $k$-th order derivatives of self-intersection local time for $d$-dimensional fractional Brownian motion, where $k=(k_1,k_2,\cdots, k_d)$. Moreover, we show a limit theorem for the critical case with $H=\frac{2}{3}$ and $d=1$, which was conjectured by Jung and Markowsky (2014).