Derivative for the intersection local time of fractional Brownian Motions

2014/03/17 by Litan Yan, Yan, Litan · 2 citations
Economics, Econometrics and Finance · Mathematics · #Complex Systems and Time Series Analysis #Financial Risk and Volatility Modeling #Stochastic processes and financial applications #math.PR

paper · pdf · doi:10.48550/arxiv.1403.4102

34 pages

arxiv created 2014/08/20 · arxiv updated 2014/08/21

Abstract

Let B^H₁ and B^H₂ be two independent fractional Brownian motions on \mathbb R with respective indices H_i∈ (0,1) and H₁≤ H₂. In this paper, we consider their intersection local time ℓ_t(a). We show that ℓ_t(a) is differentiable in the spatial variable if \frac1H₁+\frac1H₂>3, and we introduce the so-called \it hybrid quadratic covariation [f(B^H₁-B^H₂),B^H₁]^(HC). When H₁<\frac12, we construct a Banach space \mathscr H of measurable functions such that the quadratic covariation exists in L²(Ω) for all f∈ \mathscr H, and the Bouleau-Yor type identity [f(B^H₁-B^H₂),B^H₁]^(HC)_t=-∫_{\mathbb R}f(a)ℓ_t(da) holds. When H₁≥ \frac12, we show that the quadratic covariation exists also in L²(Ω) and the above Bouleau-Yor type identity holds also for all Hölder functions f of order ν>(2H₁-1)/(H₁).

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• Derivatives of local times for some Gaussian fields II
• Higher-order derivative of intersection local time for two independent fractional Brownian motions

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