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

TL;DR

This paper establishes a Bouleau-Yor identity linking quadratic covariation and local time for bi-fractional Brownian motion with specific parameters, expanding stochastic calculus tools for such processes.

Contribution

It constructs a Banach space of functions for which the quadratic covariation and local time integral are well-defined, proving the Bouleau-Yor identity in this context.

Findings

Defined a Banach space ${\\mathscr H}$ for bi-fractional Brownian motion

Proved the existence of quadratic covariation and local time integral for functions in ${\mathscr H}$

Established the Bouleau-Yor identity for bi-fractional Brownian motion with $2HK=1$

Abstract

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