The quadratic covariation for a weighted fractional Brownian motion

2016/03/05 by Xichao Sun, XIchao Sun, Litan Yan +4
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60G15 #60G17 #60H05 #Advanced Harmonic Analysis Research #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #math.PR #msc:60G15 #msc:60G17 #msc:60H05

paper · pdf · doi:10.48550/arxiv.1603.01720

31 pages

arxiv created 2016/03/05 · openalex publication_date 2016/03/05 · arxiv updated 2016/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B^a,b be a weighted fractional Brownian motion with indices a,b satisfying a>-1,-1<b<0,|b|<1+a. In this paper, motivated by the asymptotic property E[(B^a,b_s+ε-B^a,b_s)²] =O(ε^1+b)\not∼ ε^1+a+b=E[(B^a,b_ε)²] (ε→ 0) for all s>0, we consider the generalized quadratic covariation [f(B^a,b),B^a,b]^(a,b) defined by [f(B^a,b),B^a,b]^(a,b)_t=lim_{ε\downarrow 0}\frac1+a+bε^1+b∫_ε^t+ε \f(B^a,b_s+ε) -f(B^a,b_s)\(B^a,b_s+ε-B^a,b_s)s^bds, provided the limit exists uniformly in probability. We construct a Banach space \mathscr H of measurable functions such that the generalized quadratic covariation exists in L²(Ω) and the generalized Bouleau-Yor identity [f(B^a,b),B^a,b]^(a,b)_t=-\frac1(1+b)\mathbb B(a+1,b+1) ∫_{\mathbb R}f(x)\mathscr L^a,b(dx,t) holds for all f∈ \mathscr H, where \mathscr L^a,b(x,t)=∫₀^tδ(B^a,b_s-x)ds^1+a+b is the weighted local time of B^a,b and \mathbb B(⋅,⋅) is the Beta function.

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