A White Noise Approach to Stochastic Currents of Fractional Brownian Motion Herry Pribawanto Suryawan
Department of Mathematics, Sanata Dharma University, Yogyakarta, Indonesia
Abstract
By using white noise calculus, we study the stochastic currents \(\xi(x)\), \(x\in \mathbb{R}^d\), of \(d\)-dimensional fractional Brownian motion with Hurst parameter \(H\in (0,1)\). We prove that for any non-zero \(x\in \mathbb{R}^d\) the stochastic currents are Hida distributions for any Hurst parameter \(H\) and for any dimension \(d\ge 1\). For \(x=0\) \(\xi(x)\) is a Hida distribution under the condition \(dH< 1\). To handle the remaining case, that is for \(x=0\) and \(dH\ge 1\), a renormalization procedure is needed. In this case, we prove that the renormalized stochastic current \(\xi^{(N)}(x)\), \(N\in \mathbb{N}\), exists as a Hida distribution if \(2N(H-1)+dH>1\).
