On the local time of Gaussian and Lévy processes

Zineb Boudebane; Anis Rezgui

doi:10.1515/rose-2023-2017

Published by De Gruyter May 23, 2023

On the local time of Gaussian and Lévy processes

Zineb Boudebane and Anis Rezgui

From the journal Random Operators and Stochastic Equations

https://doi.org/10.1515/rose-2023-2017

Showing a limited preview of this publication:

Abstract

The local time (LT) of a given stochastic process { X t : t ≥ 0 } is defined informally as

L X ⁢ ( t , x ) = ∫ 0 t δ x ⁢ ( X s ) ⁢ d s ,

where δ x denotes the Dirac function; actually, it counts the duration of the process’s stay at 𝑥 up to time 𝑡. Using an approximation approach, we study the existence and the regularity of the LT process for two kinds of stochastic processes. The first type is the stochastic process defined by the indefinite Wiener integral X t := ∫ 0 t f ⁢ ( u ) ⁢ d B u for a given deterministic function f ∈ L 2 ( [ 0 , + ∞ [ ) , and secondly, for Lévy type processes, i.e. ones that are stationary and with independent increments.

Keywords: Local time; Wiener process; Lévy process

MSC 2010: 60H40; 60J65

Communicated by: Nikolai Leonenko

References

[1] R. J. Adler, The Geometry of Random Fields, Wiley Ser. Probab. Math. Statist., John Wiley & Sons, Chichester, 1981. Search in Google Scholar

[2] M. T. Barlow, Necessary and sufficient conditions for the continuity of local time of Lévy processes, Ann. Probab. 16 (1988), no. 4, 1389–1427. 10.1214/aop/1176991576Search in Google Scholar

[3] M. T. Barlow, E. A. Perkins and S. J. Taylor, The behaviour and construction of local times for Lévy processes, Seminar on Stochastic Processes 1984, Progr. Probab. Statist. 9, Birkhäuser, Boston (1986), 23–54. 10.1007/978-1-4684-6745-1_2Search in Google Scholar

[4] S. M. Berman, Local times and sample function properties of stationary Gaussian processes, Trans. Amer. Math. Soc. 137 (1969), 277–299. 10.1090/S0002-9947-1969-0239652-5Search in Google Scholar

[5] S. M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J. 23 (1973/74), 69–94. 10.1512/iumj.1974.23.23006Search in Google Scholar

[6] D. Geman and J. Horowitz, Occupation densities, Ann. Probab. 8 (1980), no. 1, 1–67. 10.1214/aop/1176994824Search in Google Scholar

[7] T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, White Noise. An Infinite-Dimentional Calculus, Math. Appl. 253, Kluwer Academic, Dordrecht, 1993. 10.1007/978-94-017-3680-0Search in Google Scholar

[8] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Grad. Texts in Math. 113, Springer, New York, 1997. 10.1007/978-1-4612-0949-2Search in Google Scholar

[9] D. Khoshnevisan, Y. Xiao and Y. Zhong, Local times of additive Lévy processes, Stochastic Process. Appl. 104 (2003), no. 2, 193–216. 10.1016/S0304-4149(02)00237-5Search in Google Scholar

[10] P. Lévy, Processus stochastiques et mouvement brownien, Gauthier-Villars & Cie, Paris, 1948. Search in Google Scholar

[11] M. B. Marcus and J. Rosen, Gaussian processes and local times of symmetric Lévy processes, Lévy Processes, Birkhäuser, Boston (2001), 67–88. 10.1007/978-1-4612-0197-7_4Search in Google Scholar

[12] M. B. Marcus and J. Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge Stud. Adv. Math. 100, Cambridge University, Cambridge, 2006. 10.1017/CBO9780511617997Search in Google Scholar

[13] P. Mörters and Y. Peres, Brownian Motion, Camb. Ser. Stat. Probab. Math. 30, Cambridge University, Cambridge, 2010. Search in Google Scholar

[14] D. Nualart and J. Vives, Chaos expansions and local times, Publ. Mat. 36 (1992), no. 2B, 827–836. 10.5565/PUBLMAT_362B92_07Search in Google Scholar

[15] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Stud. Adv. Math. 68, Cambridge University, Cambridge, 1999. Search in Google Scholar

[16] D. Wu and Y. Xiao, Continuity in the Hurst index of the local times of anisotropic Gaussian random fields, Stochastic Process. Appl. 119 (2009), no. 6, 1823–1844. 10.1016/j.spa.2008.09.001Search in Google Scholar

[17] Y. Xiao, Local times and related properties of multidimensional iterated Brownian motion, J. Theoret. Probab. 11 (1998), no. 2, 383–408. Search in Google Scholar

Received: 2022-04-08

Accepted: 2022-12-04

Published Online: 2023-05-23

Published in Print: 2023-12-01

On the local time of Gaussian and Lévy processes

Abstract

References

Journal and Issue

Articles in the same Issue