Convergence results for the time-changed fractional Ornstein–Uhlenbeck processes
Authors:
G. Ascione, Yu. Mishura and E. Pirozzi
Journal:
Theor. Probability and Math. Statist. 104 (2021), 23-47
MSC (2020):
Primary 60G22; Secondary 60F17, 35R11
DOI:
https://doi.org/10.1090/tpms/1143
Published electronically:
September 24, 2021
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Abstract: In this paper we study some convergence results concerning the one-dimensional distribution of a time-changed fractional Ornstein–Uhlenbeck process. In particular, we establish that, despite the time change, the process admits a Gaussian limit random variable. On the other hand, we prove that the process converges towards the time-changed Ornstein–Uhlenbeck as the Hurst index $H \to 1/2^+$, with locally uniform convergence of one-dimensional distributions. Moreover, we also achieve convergence in the Skorokhod $J_1$-topology of the time-changed fractional Ornstein–Uhlenbeck process as $H \to 1/2^+$ in the space of càdlàg functions. Finally, we exploit some convergence properties of mild solutions of a generalized Fokker–Planck equation associated to the aforementioned processes, as $H \to 1/2^+$.
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- W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Mono. Math., Birkhäuser, Basel, 2001. MR 1886588
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- —, Time-changed fractional Ornstein-Uhlenbeck process, Fract. Calc. Appl. Anal. 23 (2020), no. 2, 450–483. MR 4098657
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- G. Ascione and B. Toaldo, A semi-Markov Leaky Integrate-and-Fire model, Math. 7 (2019), no. 11, 1022.
- J. Bertoin, Lévy Processes, vol. 121, Cambridge University Press, 1996. MR 1406564
- —, Subordinators: Examples and applications, Lectures on Probability Theory and Statistics, Springer, 1999, pp. 1–91.
- K. Buchak and L. Sakhno, On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators, Th. Probab. Math. Stat. 98 (2019), 91–104. MR 3824680
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- N. Leonenko, M. M. Meerschaert, and A. Sikorskii, Fractional Pearson diffusions, J. Math. Anal. Appl. 403 (2013), no. 2, 532–546. MR 3037487
- N. N. Leonenko, M. M. Meerschaert, and A. Sikorskii, Correlation structure of fractional Pearson diffusions, Comput. Math. Appl. 66 (2013), no. 5, 737–745. MR 3089382
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- M. Savov and B. Toaldo, Semi-Markov processes, integro-differential equations and anomalous diffusion-aggregation, Ann. I. H. Poincaré-Pr. 56 (2020), no. 4, 2640–2671. MR 4164851
- E. Scalas and N. Viles, A functional limit theorem for stochastic integrals driven by a time-changed symmetric $\alpha$-stable Lévy process, Stoch. Proc. Appl. 124 (2014), no. 1, 385–410. MR 3131299
- R. L. Schilling, R. Song, and Z. Vondracek, Bernstein Functions: Theory and Applications, vol. 37, Walter de Gruyter, 2012. MR 2978140
- S. Shinomoto, Y. Sakai, and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex, Neural Comput. 11 (1999), no. 4, 935–951.
- D. S. Silvestrov, Limit theorems for randomly stopped stochastic processes, Springer Science & Business Media, 2012. MR 2261597
- B. Toaldo, Convolution-type derivatives, hitting-times of subordinators and time-changed ${C}_0$-semigroups, Potential Anal. 42 (2015), no. 1, 115–140. MR 3297989
- H. Totoki, A method of construction of measures on function spaces and its applications to stochastic processes, Mem. Fac. Sci., Kyushu Univ. A Math. 15 (1962), no. 2, 178–190. MR 138127
- A. Weron and R. Weron, Computer simulation of Lévy $\alpha$-stable variables and processes, Chaos—The Interplay Between Stochastic and Deterministic Behaviour, Springer, 1995, pp. 379–392. MR 1452625
- W. Whitt, Stochastic-Process Limits: an Introduction to Stochastic-Process Limits and Their Application to Queues, Springer Science & Business Media, 2002. MR 1876437
- D. Williams, Probability with Martingales, Cambridge University Press, 1991. MR 1155402
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Additional Information
G. Ascione
Affiliation:
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Universita degli Studi di Napoli Federico II, 80126 Napoli, Italy
Email:
giacomo.ascione@unina.it
Yu. Mishura
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska 64, Kyiv 01601, Ukraine
Email:
myus@univ.kiev.ua
E. Pirozzi
Affiliation:
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Universita degli Studi di Napoli Federico II, 80126 Napoli, Italy
Email:
enrica.pirozzi@unina.it
Keywords:
Fractional Brownian motion,
subordinator,
weak convergence in Skorokhod space,
generalized Fokker–Planck equation
Received by editor(s):
October 19, 2020
Published electronically:
September 24, 2021
Additional Notes:
This research was partially supported by MIUR - PRIN 2017, project Stochastic Models for Complex Systems, no. 2017JFFHSH, by Gruppo Nazionale per il Calcolo Scientifico (GNCS-INdAM), by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA-INdAM). The 2nd author was partially supported by the project STORM: Stochastics for Time-Space Risk Models, funded by the University of Oslo and the Research Council of Norway within the ToppForsk call, N. 274410, and by the research project Exact formulas, estimates, asymptotic properties and statistical analysis of complex evolutionary systems with many degrees of freedom, funded by the Ministry of science and education of Ukraine, N. 0119U100317
Article copyright:
© Copyright 2021
Taras Shevchenko National University of Kyiv
