Asymptotic properties of the derivative of self-intersection local time\n of fractional Brownian motion

2015/12/22 by Arturo Jaramillo, David Nualart, Jaramillo, Arturo +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Complex Systems and Time Series Analysis #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1512.07219

openalex publication_date 2015/12/22 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let B_{t t\≥0} be a fractional Brownian motion with Hurst parameter\n\(2)/(3)<H<1. We prove that the approximation of the derivative of\nself-intersection local time, defined as \
alpha
_varepsilon amp;=\n
int₀^T
int₀^tp'
_varepsilon(B_t-B_s)
textds
textdt,\n where p_\ε(x) is the heat kernel, satisfies a central\nlimit theorem when renormalized by \ε^{\(3)/(2)-\(1)/(H)}. We\nprove as well that for q\≥2, the q-th chaotic component of\n\α_\ε converges in L² when \(2)/(3)<H<\(3)/(4),\nand satisfies a central limit theorem when renormalized by a multiplicative\nfactor \ε^1-\(3)/(4H) in the case\n\(3)/(4)<H<\(4q-3)/(4q-2).\n

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• On the exponential integrability of the derivative of intersection and self-intersection local time for fractional Brownian motion and a limit theorem related to the self-intersection local time for fractional Brownian motion

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