Intersection Local Times of Independent Fractional Brownian Motions as Generalized White Noise Functionals

In this work we present expansions of intersection local times of fractional Brownian motions in ℝd, for any dimension d≥1, with arbitrary Hurst coefficients in (0,1)d. The expansions are in terms of Wick powers of white noises (corresponding to multiple Wiener integrals), being well-defined in the sense of generalized white noise functionals. As an application of our approach, a sufficient condition on d for the existence of intersection local times in L2 is derived, extending the results in Nualart and Ortiz-Latorre (J. Theoret. Probab. 20(4):759–767, 2007) to different and more general Hurst coefficients.

[1]  Jürgen Potthoff,et al.  White Noise: An Infinite Dimensional Calculus , 1993 .

[2]  尾畑 伸明,et al.  White noise calculus and Fock space , 1994 .

[3]  I︠u︡. M. Berezanskiĭ,et al.  Spectral Methods in Infinite-Dimensional Analysis , 1995 .

[4]  L. Streit,et al.  Generalized Functionals in Gaussian Spaces: The Characterization Theorem Revisited☆ , 1996, math/0303054.

[5]  Bernt Øksendal,et al.  Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach , 1996 .

[6]  Christian Bender.  An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter , 2003 .

[7]  P. Vallois,et al.  Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index H>=1/4 , 2003 .

[8]  J. Rosen.  The intersection local time of fractional Brownian motion in the plane , 1987 .

[9]  B. Øksendal,et al.  Stochastic Calculus for Fractional Brownian Motion and Applications , 2008 .

[10]  Renormalized self-intersection local time for fractional Brownian motion , 2005, math/0506592.

[11]  Hui-Hsiung Kuo,et al.  White noise distribution theory , 1996 .

[12]  G. Jantzen.  1988 , 1988, The Winning Cars of the Indianapolis 500.

[13]  S. Tindel,et al.  Probabilistic models for vortex filaments based on fractional Brownian motion , 2003 .

[14]  I︠U︡lii︠a︡ S. Mishura.  Stochastic Calculus for Fractional Brownian Motion and Related Processes , 2008 .

[15]  David Nualart,et al.  Regularity of renormalized self-intersection local time for fractional Brownian motion , 2007, Commun. Inf. Syst..

[16]  M. Taqqu,et al.  Integration questions related to fractional Brownian motion , 2000 .

[17]  Intersection Local Times of Independent Brownian Motions as Generalized White Noise Functionals , 2001 .

[18]  Rui Vilela Mendes,et al.  A Data-Reconstructed Fractional Volatility Model , 2008 .

[19]  Intersection Local Time for Two Independent Fractional Brownian Motions , 2007, 0704.1259.

[20]  L. Streit,et al.  A Characterization of Hida Distributions , 1991 .

[21]  J. L. D. Silva,et al.  Intersection local times of fractional Brownian motions with H∈(0,1) as generalized white noise functionals , 2008, 0802.4367.