Renormalized self-intersection local time for fractional Brownian motion

By Yaozhong Hu, David Nualart. 2005

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Let B H t be a d-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1). Assume d ≥ 2. We prove that the renormalized self-intersection local time l=∫ T 0 ∫ t 0 δ(B H t - B H s )ds dt - E(∫ T 0 ∫ t 0 δ(B H t -B H s )ds dt) exists in L 2 if and only if H H ≥ 3 2d, r(e)l e converges in distribution to a normal law N(0, Tσ 2 ), as e tends to zero, where l e is an approximation of l, defined through (2), and r(e) = |loge| -1 if H = 3/(2d), and r(e) = e d-3/(2H) if 3/(2d) < H.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ace30c4d57eb7f6ce20dd0b01bb27c5cb7200a49

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Citations in the corpus

Stein’s method on Wiener chaos (2007, background): “3 and its generalizations have been applied to a variety of frameworks, such as: p-variations of stochastic integrals with respect to Gaussian processes [2,12], quadratic functionals of bivariate Gaussian processes [14], selfintersection local times of fractional Brownian motion [24], approximation ”
Stochastic heat equation driven by fractional noise and local time (2007, unknown): “2 of [6].”
Power variation of some integral fractional processes (2006, methodology): “L (B, V ) is proved in Proposition 10 of the Appendix, by means of a direct argument based on a recent central limit theorem for stochastic integrals (see Nualart and Peccati 2005; Peccati and Tudor 2005; Hu and Nualart 2005).”
Central limit theorems for multiple stochastic integrals and Malliavin calculus (2007, background): “…(0, 1) law if and only if one of the following two equivalent conditions hold: i) limk→∞ E(F 4k ) = 3, ii) limk→∞ fk ⊗l fk = 0, for all 1 ≤ l ≤ n − 1, where fk is the square integrable kernel associated with the random variable Fk, and fk ⊗l fk denotes the contraction of l indices of both kernels.”
Variations and estimators for selfsimilarity parameters via Malliavin calculus (2009, methodology): “Our main tools are the Malliavin calculus, the Wiener–Itô chaos expansions and recent results on the convergence of multiple stochastic integrals proved in [10, 22, 23] and [24].”
Central Limit Theorems for Multiple Skorokhod Integrals (2010, background): “[5]), self-intersection local times of fractional Brownian motion (Hu and Nualart [7]), approximation schemes for scalar fractional differential equations (Neuenkirch and Nourdin [12]), high-frequency CLTs for random fields on homogeneous spaces (Marinucci and Peccati [10, 11] and Peccati [23]), nee”
Bipower Variation for Gaussian Processes with Stationary Increments (2009, unknown): “The latter is based on the theory for multiple stochastic integrals developed in [17], [24], and [26].”