Theory of Probability and Mathematical Statistics
Convergence results for the Time-Changed fractional Ornstein--Uhlenbeck processes
G. Ascione, Yu. Mishura, E. Pirozzi
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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 c\`{a}dl\`{a}g functions. Finally, we exploit some convergence properties of mild solutions of a generalized Fokker--Planck equation associated to the aforementioned processes, as $H \to 1/2^+$.
Keywords: Fractional Brownian motion, subordinator, weak convergence in Skorokhod space, generalized Fokker-Planck equation
Bibliography: 1. V. Anh and A. Inoue, Financial markets with memory I: Dynamic models, Stoch. Anal. Appl. 23 (2005), no. 2, 275–300.
2. W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Mono. Math., Birkh\"auser, Basel, 2001.
3. G. Ascione, N. Leonenko, and E. Pirozzi, Non-local Pearson diffusions, arXiv preprint arXiv:2009.12086 (2020).
4. G. Ascione, Y. Mishura, and E. Pirozzi, Fractional Ornstein-Uhlenbeck process with stochastic forcing, and its applications, Methodol. Comput. Appl. (2019), 1–32.
5. ----, Time-changed fractional Ornstein-Uhlenbeck process, Fract. Calc. Appl. Anal. 23 (2020), no. 2, 450–483.
6. ----, The Fokker-Planck equation for the time-changed fractional Ornstein-Uhlenbeck process, P. Roy. Soc. Edinb. A (2021), Accepted.
7. G. Ascione and B. Toaldo, A semi-Markov Leaky Integrate-and-Fire model, Math. 7 (2019), no. 11, 1022.
8. J. Bertoin, L\'{e}vy Processes, vol. 121, Cambridge University Press, 1996.
9. ----, Subordinators: Examples and applications, Lectures on Probability Theory and Statistics, Springer, 1999, pp. 1–91.
10. 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.
11. M. N. Burr, Weak convergence of stochastic integrals driven by continuous-time random walks, arXiv preprint arXiv:1110.0216 (2011).
12. P. Cheridito, H. Kawaguchi, and M. Maejima, Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab. 8 (2003), 3–14.
13. E. Cinlar, Markov additive processes. I, Z. Wahrscheinlichkeit 24 (1972), no. 2, 85–93.
14. ----, Markov additive processes. II, Z. Wahrscheinlichkeit 24 (1972), no. 2, 95–121.
15. J. Gajda and A. Wylomanska, Time-changed Ornstein–Uhlenbeck process, J. Phys. A-Math. Theor. 48 (2015), no. 13, 135004.
16. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, 2014.
17. M. Hahn, J. Ryvkina, K. Kobayashi, and S. Umarov, On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations, Electron. Commun. Prob. 16 (2011), 150–164.
18. Wolfram Research, Inc., Mathematica, Version 11.2, Champaign, IL, 2017.
19. S. Jin and K. Kobayashi, Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators, arXiv preprint arXiv:2006.10926 (2020).
20. T. Kaarakka and P. Salminen, On fractional Ornstein-Uhlenbeck processes, Commun. Stoch. Anal. 5 (2011), no. 1, 8.
21. K. Kobayashi, Stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations, J. Theor. Probab. 24 (2011), no. 3, 789–820.
22. J. Lamperti, On convergence of stochastic processes, T. Am. Math. Soc. 104 (1962), no. 3, 430–435.
23. N. Leonenko, M. M. Meerschaert, and A. Sikorskii, Fractional Pearson diffusions, J. Math. Anal. Appl. 403 (2013), no. 2, 532–546.
24. N. N. Leonenko, M. M. Meerschaert, and A. Sikorskii, Correlation structure of fractional Pearson diffusions, Comput. Math. Appl. 66 (2013), no. 5, 737–745.
25. M. M. Meerschaert and H.-P. Scheffler, Triangular array limits for continuous time random walks, Stoch. Proc. Appl. 118 (2008), no. 9, 1606–1633.
26. M. M. Meerschaert and A. Sikorskii, Stochastic models for fractional calculus, vol. 43, Walter de Gruyter, 2011.
27. M. M. Meerschaert and P. Straka, Inverse stable subordinators, Math. Model. Nat. Pheno. 8 (2013), no. 2, 1–16.
28. J. B. Mijena, Correlation structure of time-changed fractional Brownian motion, arXiv preprint arXiv:1408.4502 (2014).
29. Y. Mishura, V. Piterbarg, K. Ralchenko, and A. Yurchenko-Tytarenko, Stochastic representation and path properties of a fractional Cox–Ingersoll–Ross process, Th. Probab. Math. Stat. 97 (2018), 167–182.
30. Y. Mishura and A. Yurchenko-Tytarenko, Fractional Cox–Ingersoll–Ross process with non-zero mean, Mod. Stoch. Th. Appl. 5 (2018), no. 1, 99–111.
31. Y. Sakai, S. Funahashi, and S. Shinomoto, Temporally correlated inputs to leaky integrate-and- fire models can reproduce spiking statistics of cortical neurons, Neural Networks 12 (1999), no. 7-8, 1181–1190.
32. M. Savov and B. Toaldo, Semi-Markov processes, integro-differential equations and anomalous diffusion-aggregation, Ann. I. H. Poincar\'{e}-Pr. 56 (2020), no. 4, 2640–2671.
33. E. Scalas and N. Viles, A functional limit theorem for stochastic integrals driven by a time-changed symmetric $\alpha$-stable L\'{e}vy process, Stoch. Proc. Appl. 124 (2014), no. 1, 385–410.
34. R. L. Schilling, R. Song, and Z. Vondracek, Bernstein Functions: Theory and Applications, vol. 37, Walter de Gruyter, 2012.
35. 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. 36. D. S. Silvestrov, Limit theorems for randomly stopped stochastic processes, Springer Science & Business Media, 2012.
37. B. Toaldo, Convolution-type derivatives, hitting-times of subordinators and time-changed C0-semigroups, Potential Anal. 42 (2015), no. 1, 115–140.
38. 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.
39. A. Weron and R. Weron, Computer simulation of L\'{e}vy $\alpha$-stable variables and processes, Chaos—The Interplay Between Stochastic and Deterministic Behaviour, Springer, 1995, pp. 379–392.
40. W. Whitt, Stochastic-Process Limits: an Introduction to Stochastic-Process Limits and Their Application to Queues, Springer Science & Business Media, 2002.
41. D. Williams, Probability with Martingales, Cambridge University Press, 1991.
