Regularity of the Local Time for the d-dimensional Fractional Brownian Motion with N-parameters

Abstract

We give the Wiener-Itô chaotic decomposition for the local time of the d-dimensional fractional Brownian motion with N -parameters and study its smoothness in the Sobolev-Watanabe spaces. 384 Eddahbi et al. representation of the form B H t = t 0 386 Eddahbi et al.

A Stroock formula for a certain class of Lévy processes and applications to finance

“…The classical idea of approximating the Dirac distribution δ x by p ε has been used to calculate the chaotic decomposition of the local time in the case of the Brownian motion by Nualart and Vives [11] and for the fractional Brownian motion by Coutin et al [3] and Eddahbi et al [4]. Before stating precise results of this section, we prove some technical lemmas.…”

Renormalization Of The Local Time For The d-Dimensional Fractional Brownian Motion With N Parameters

“…This local time for (N, d)-fBm has been studied by Xiao and Zhang [7], Hu and Oksendal [2] and Eddahbi et al [1] between others.…”

“…Section 2 is devoted to the presentation of the problem. In particular we review from [1] the chaotic decomposition of the local time L(t, x) as a functional of the (N, d)-fBm and its regularity in terms of SobolevWatanabe norms. In Section 3 we present a list of auxiliary lemmas.…”

Multidimensional Bifractional Brownian Motion: Itô and Tanaka Formulas

“…if F is given by (10). For p > 1 and α ∈ R we introduce the Sobolev-Watanabe space D α,p as the closure of the set of polynomial random variables with respect to the norm…”

“…See the last section for the definition of the function U . Our method is based on the Wiener-Itô chaotic expansion into multiple stochastic integrals following ideas from [14] or [10]. The multidimensional Tanaka formula also involves a generalized local time.…”