The Bouleau-Yor identity for a bi-fractional Brownian motion

Abstract

Let $B$ be a bi-fractional Brownian motion with indices $H\in (0,1),K\in (0,1]$, $2HK=1$ and let $\mathcal{L}(x,t)$ be its local time process. We construct a Banach space $\mathcal{H}$ of measurable functions such that the quadratic covariation $[f(B),B]$ and the integral ${\int }_{\mathbb{R}}f(x)\mathcal{L}(dx,t)$ exist provided $f\in \mathcal{H}$. Moreover, the Bouleau-Yor identity $$[f(B),B{]}_t=-2^{1-K}{\int }_{\mathbb{R}}f(x)\mathcal{L}(dx,t),t\ge 0,$$ holds for all $f\in \mathcal{H}$.

32 pages

References in corpus (3)

• Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion
• Two-parameter $p,q$-variation Paths and Integrations of Local Times
• On Itô's formula for elliptic diffusion processes