The generalized Bouleau-Yor identity for a sub-fractional Brownian motion

Let SH be a sub-fractional Brownian motion with index 0 < H < 1/2. In this paper we study the existence of the generalized quadratic covariation [f(SH), SH](W) defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[f(S^H ),S^H ]_t^{(W)} = \mathop {\lim }\limits_{\varepsilon \downarrow 0} \tfrac{1} {{\varepsilon ^{2H} }}\int_0^t {\{ f(S_{s + \varepsilon }^H ) - f(S_s^H )\} (S_{s + \varepsilon }^H - S_s^H )ds^{2H} } , $\end{document} provided the limit exists in probability, where x ↦ f(x) is a measurable function. We construct a Banach space ℋ of measurable functions such that the generalized quadratic covariation exists in L2 provided f ∈ ℋ. Moreover, the generalized Bouleau-Yor identity takes the form [graphic not available: see fulltext] for all f ∈ ℋ, where ℒH(x, t) is the weighted local time of SH. This allows us to write the generalized Itô’s formula for absolutely continuous functions with derivative belonging to ℋ.