Asymptotic Properties of the Derivative of Self-Intersection Local Time of Multidimensional Fractional Brownian Motion

2025/11/18 by Gu, Jiazhen, Jiang, Jinchi, Yu, Qian
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Stochastic processes and financial applications #Fractional Differential Equations Solutions #stochastic dynamics and bifurcation

paper · doi:10.48550/arxiv.2511.14464

Abstract

Let \B_t^H,t≥0\ be a d-dimensional fractional Brownian motion. We prove that the approximation of the first-order derivative of self-intersection local time, defined as α_ε,t⁽¹⁾(0)=-∫₀^t∫₀^sp_ε⁽¹⁾(B_s^H-B_r^H)\d r\d s, where p_ε⁽¹⁾(x₁,⋯,x_d):=∂ _x1p(x₁,⋯,x_d) and p_ε(x)=(2πε)^-d/2e^|x|2/2ε,x∈ℝ^d, d≥2 is the heat kernel, exits in L² sense if and only if H<(3)/(2(1+d)) and satisfies three different central limit theorems when normalized by ε^{\frac d2+1-\frac1H} for H>\frac12 and d≥2, normalized by ε^\frac d2+\frac12-\frac 34H for (3)/(2(1+d))

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