Functional limit theorem for the self-intersection local time of the\n fractional Brownian motion

2017/01/18 by Arturo Jaramillo, David Nualart, Jaramillo, Arturo +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1701.05289

openalex publication_date 2017/01/18 · openalex created_date 2023/02/16 · openalex updated_date 2026/07/28

Abstract

Let B_{t t\≥0} be a d-dimensional fractional Brownian motion with\nHurst parameter 0<H<1, where d\≥2. Consider the approximation of the\nself-intersection local time of B, defined as nI_T
^varepsilon\n amp;=
int₀^T
int₀^tp
_varepsilon(B_t-B_s)dsdt, nwhere p_\ε(x) is the heat kernel. We prove that the process\n I_T^\ε-\𝔼\[I_T^\ε\] _T\≥0,\nrescaled by a suitable normalization, converges in law to a constant multiple\nof a standard Brownian motion for \(3)/(2d)<H\≤\(3)/(4) and to a\nmultiple of a sum of independent Hermite processes for \(3)/(4)<H<1, in\nthe space C[0,\∞), endowed with the topology of uniform convergence on\ncompacts.\n

Citations

• Renormalized self-intersection local time for fractional Brownian motion

Cited by

• 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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