Optimal estimation of the local time and the occupation time measure for an α-stable Lévy process

High frequency data; local time; mixed normal distribution; occupation time; stable Lévy processes; Statistics and Probability; Modeling and Simulation; Statistics, Probability and Uncertainty

Abstract :

[en] A novel theoretical result on estimation of the local time and the occupation time measure of an α-stable Lévy process with α ∈ (1, 2) is presented. The approach is based upon computing the conditional expectation of the desired quantities given high frequency data, which is an L2-optimal statistic by construction. The corresponding stable central limit theorems are proved and a statistical application is discussed. In particular, this work extends the results of [20], which investigated the case of the Brownian motion.

Disciplines :

Mathematics

Author, co-author :

AMORINO, Chiara ; University of Luxembourg > Faculty of Science, Technology and Medicine > Department of Mathematics > Team Mark PODOLSKIJ

Jaramillo, Arturo; Research in Mathematics (CIMAT), Valenciana, Mexico

PODOLSKIJ, Mark ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

External co-authors :

yes

Language :

English

Title :

Optimal estimation of the local time and the occupation time measure for an α-stable Lévy process

Publication date :

March 2024

Journal title :

Modern Stochastics: Theory and Applications

ISSN :

2351-6046

eISSN :

2351-6054

Publisher :

VTeX

Volume :

Issue :

Pages :

149 - 168

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://www.vmsta.org/journal/VMSTA/article/294/text

Funding text :

The authors gratefully acknowledge financial support of ERC Consolidator Grant 815703 “STAMFORD: Statistical Methods for High Dimensional Diffusions.”

Available on ORBilu :

since 22 January 2025

Statistics

Number of views

89 (2 by Unilu)

Number of downloads

33 (0 by Unilu)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

WoS citations^™

Bibliography

Altmeyer, R., Chorowski, J.: Estimation error for occupation time functionals of stationary Markov processes. Stoch. Process. Appl. 128(6), 1830–1848 (2018). MR3797645. https://doi.org/10.1016/j.spa.2017.08.013
Ayache, A., Xiao, Y.: Harmonizable fractional stable fields: Local nondeterminism and joint continuity of the local times. Stoch. Process. Appl. 126(1), 171–185 (2016). MR3426515. https://doi.org/10.1016/j.spa.2015.08.001
Barlow, M.T.: Continuity of local times for Lévy processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 69, 23–35 (1985). MR0775850. https://doi.org/10.1007/BF00532583
Barlow, M.T.: Necessary and sufficient conditions for the continuity of local times of Lévy processes. Ann. Probab. 16(4), 1389–1427 (1988). MR0958195
Berman, S.M.: Local times and sample function properties of stationary Gaussian pro-cesses. Trans. Am. Math. Soc. 137, 277–299 (1969). MR0239652. https://doi.org/10. 2307/1994804
Berman, S.M.: Gaussian processes with stationary increments: Local times and sample function properties. Ann. Math. Stat. 41(4), 1260–1272 (1970). MR0272035. https://doi.org/10.1214/aoms/1177696901
Borodin, A.N.: On the character of convergence to Brownian local time. I, II. Probab. Theory Relat. Fields 72(2), 231–250, 251–277 (1986). MR0836277. https://doi.org/10. 1007/BF00699106
Cohen, S.N.: A martingale representation theorem for a class of jump processes (2013). arXiv preprint. arXiv:1310.6286
Florens-Zmirou, D.: On estimating the diffusion coefficient from discrete observations. J. Appl. Probab. 30, 790–804 (1993). MR1242012. https://doi.org/10.2307/3214513
Geman, D., Horowitz, J.: Occupation densities. Ann. Probab. 8(1), 1–67 (1980). MR0556414. https://doi.org/10.1214/aop/1176994824
Getoor, R.K., Kesten, H.: Continuity of local times of Markov processes. Compos. Math. 24, 277–303 (1972). MR0310977
Hong, M., Liu, H., Xu, F.: Limit theorems for additive functionals of some self-similar Gaussian processes (2023). Preprint
Jacod, J.: On continuous conditional Gaussian martingales and stable convergence in law. Seminaire de Probabilites XXXI, vol. 232. Springer, Berlin, Heidelberg (1997). MR1478732. https://doi.org/10.1007/BFb0119308
Jacod, J.: Rates of convergence to the local time of a diffusion. Ann. Inst. Henri Poincaré B, Probab. Stat. 34(4), 505–544 (1998). MR1632849. https://doi.org/10.1016/S0246-0203(98)80026-5
Jaramillo, A., Nourdin, I., Peccati, G.: Approximation of fractional local times: Zero energy and derivatives. Ann. Appl. Probab. 31(5), 2143–2191 (2021). MR4332693. https://doi.org/10.1214/20-aap1643
Jaramillo, A., Nourdin, I., Nualart, D., Peccati, G.: Limit theorems for additive func-tionals of the fractional Brownian motion. Ann. Probab. 51(3), 1066–1111 (2023). MR4583063. https://doi.org/10.1214/22-aop1612
Jeganathan, P.: Convergence of functionals of sums of r.v.s to local times of fractional stable motions. Ann. Probab. 32(3), 1771–1795 (2004). MR2073177. https://doi.org/10. 1214/009117904000000658
Jeganathan, P.: Limit laws for the local times of fractional Brownian and stable motions (2006). Working paper, available at: https://www.isibang.ac.in/~statmath/eprints/2006/11.pdf
Jeganathan, P.: Limit theorems for functionals of sums that converge to fractional Brownian and stable motions (2008). Working paper, available at: https://cowles.yale.edu/sites/default/files/files/pub/d16/d1649.pdf
Ivanovs, J., Podolskij, M.: Optimal estimation of the supremum and occupation times of a self-similar Lévy process. Electron. J. Stat. 16(1), 892–934 (2022). MR4372664. https://doi.org/10.1214/21-ejs1928
Kesten, H.: Hitting probabilities of single points for processes with stationary independent increments. Mem. Amer. Math. Soc. 93 (1969). MR0272059
Papanicolaou, G.C., Stroock, D., Varadhan, S.R.S.: Martingale approach to some limit theorems. Papers from the Duke Turbulence Conference 6, ii+120 (1976). MR0461684
Podolskij, M., Rosenbaum, M.: Comment on: Limit of random measures associated with the increments of a Brownian semimartingale. Asymptotic behavior of local times related statistics for fractional Brownian motion. J. Financ. Econom. 16(4), 588–598 (2018). https://doi.org/10.1093/jjfinec/nbx036
Renyi, A.: On stable sequences of events. Sankhya, Ser. A 25, 293–302 (1963). MR0170385
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer (1999). MR1725357. https://doi.org/10.1007/978-3-662-06400-9
Rosen, J.: Second order limit laws for the local times of stable processes. In: Séminaire de Probabilités, XXV. Lecture Notes in Mathematics, vol. 1485, pp. 407–424. Springer, Berlin (1991). MR1187796. https://doi.org/10.1007/BFb0100872