Integral representation of renormalized self-intersection local times

TL;DR

This paper derives an explicit integral representation for the renormalized self-intersection local time of fractional Brownian motion using Clark-Ocone formula, and establishes the existence of exponential moments for this variable.

Contribution

It introduces a new integral representation for the renormalized self-intersection local time of fractional Brownian motion, facilitating analysis of its properties.

Findings

Explicit integral representation derived using Clark-Ocone formula.

Existence of exponential moments for the renormalized local time.

Applicable to fractional Brownian motion with Hurst parameter in (0,1).

Abstract

In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the $d$% -dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$. As a consequence, we derive the existence of some exponential moments for this random variable.