The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2

2011/06/12 by Litan Yan, Chao Chen, Yan, Litan +3
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Financial Risk and Volatility Modeling #Probability and Risk Models #Stochastic processes and financial applications #math.PR #msc:60G15 #msc:60H05 #msc:60H07

paper · pdf · doi:10.48550/arxiv.1106.2302

22 pages

arxiv created 2011/06/18 · arxiv updated 2011/06/21

Abstract

Let B^H be a fractional Brownian motion with Hurst index 0<H<1/2. In this paper we study the \it generalized quadratic covariation [f(B^H),B^H]^(W) defined by [f(B^H),B^H]^(W)_t=lim_{ε\downarrow 0}\frac2Hε^2H∫₀^t\f(B^H_s+ε)-f(B^H_s)\(B^H_s+ε- B^H_s)s^2H-1ds, where the limit is uniform in probability and x↦ f(x) is a deterministic function. We construct a Banach space \mathscr H of measurable functions such that the generalized quadratic covariation exists in L² and the Bouleau-Yor identity takes the form [f(B^H),B^H]_t^(W)=-∫_\mathbb Rf(x)\mathscr L^H(dx,t) provided f∈ \mathscr H, where \mathscr L^H(x,t) is the weighted local time of B^H. This allows us to write the fractional Itô formula for absolutely continuous functions with derivative belonging to \mathscr H. These are also extended to the time-dependent case.

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