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The Fokker–Planck equation for the time-changed fractional Ornstein–Uhlenbeck stochastic process

Part of: Partial differential equations Stochastic processes Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  21 September 2021

Giacomo Ascione Open the ORCID record for Giacomo Ascione [Opens in a new window] ,
Yuliya Mishura Open the ORCID record for Yuliya Mishura [Opens in a new window]  and
Enrica Pirozzi Open the ORCID record for Enrica Pirozzi [Opens in a new window]
Giacomo Ascione
Affiliation:
Dipartimento di Matematica e Applicazioni ‘Renato Caccioppoli’, Universita degli Studi di Napoli Federico II, 80126 Napoli, Italy (giacomo.ascione@unina.it)
Yuliya Mishura
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska 64, Kyiv 01601, Ukraine (myus@univ.kiev.ua)
Enrica Pirozzi
Affiliation:
Dipartimento di Matematica e Applicazioni ‘Renato Caccioppoli’, Universita degli Studi di Napoli Federico II, 80126 Napoli, Italy (enrica.pirozzi@unina.it)
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Abstract

In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.

Keywords

Bernstein functionsCaputo-type derivativefractional Brownian motiongeneralized Fokker–Planck equationsubordinator

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 35R11: Fractional partial differential equations 35B50: Maximum principles
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 4 , August 2022 , pp. 1032 - 1057
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Anh, V. and Inoue, A.. Financial markets with memory I: dynamic models. Stoch. Anal. Appl. 23 (2005), 275–300.CrossRefGoogle Scholar
Arendt, W., Batty, C. J. K., Hieber, M. and Neubrander, F., Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics (Basel: Birkhäuser, 2001).CrossRefGoogle Scholar
Ascione, G., Mishura, Y. and Pirozzi, E.. Fractional Ornstein–Uhlenbeck process with stochastic forcing, and its applications. Methodol. Comput. Appl. Probab. 23 (2019), 53–84.CrossRefGoogle Scholar
Ascione, G., Mishura, Y. and Pirozzi, E.. Time-changed fractional Ornstein–Uhlenbeck process. Fract. Calc. Appl. Anal. 23 (2020), 450–483.CrossRefGoogle Scholar
Ascione, G. and Pirozzi, E.. On a stochastic neuronal model integrating correlated inputs. Math. Biosci. Eng 16 (2019), 5206–5225.CrossRefGoogle ScholarPubMed
Ascione, G., Pirozzi, E. and Toaldo, B.. On the exit time from open sets of some semi-Markov processes. Ann. Appl. Probab. 30 (2020), 1130–1163.CrossRefGoogle Scholar
Ascione, G. and Toaldo, B.. A semi-Markov Leaky integrate-and-fire model. Mathematics 7 (2019), 1022.CrossRefGoogle Scholar
Bertoin, J.. Lévy processes. Vol. 121 (New York: Cambridge University Press, 1996).Google Scholar
Buonocore, A., Caputo, L., Pirozzi, E. and Ricciardi, L. M.. The first passage time problem for Gauss-diffusion processes: algorithmic approaches and applications to LIF neuronal model. Methodol. Comput. Appl. Probab. 13 (2011), 29–57.CrossRefGoogle Scholar
Caputo, M.. Linear models of dissipation whose $Q$ is almost frequency independent–II. Geophys. J. Int. 13 (1967), 529–539.CrossRefGoogle Scholar
Cheridito, P., Kawaguchi, H. and Maejima, M.. Fractional Ornstein–Uhlenbeck processes. Electron. J. Probab. 8 (2003), 3–14.CrossRefGoogle Scholar
Coddington, E. A. and Levinson, N.. Theory of ordinary differential equations (New York: Tata McGraw-Hill Education, 1955).Google Scholar
de Oliveira, E. C.. Solved exercises in fractional calculus (Cham: Springer, 2019).CrossRefGoogle Scholar
Debnath, L.. A brief historical introduction to fractional calculus. Int. J. Math. Educ. Sci. Technol. 35 (2004), 487–501.CrossRefGoogle Scholar
Dzhrbashyan, M. M.. Integral transforms and representations of functions in the complex domain (in Russian) (Moscow: Nauka, 1966).Google Scholar
Gajda, J. and Wyłomańska, A.. Time-changed Ornstein–Uhlenbeck process. J. Phys. A: Math. Theor. 48 (2015), 135004.CrossRefGoogle Scholar
Gerasimov, A. N.. Generalization of laws of the linear deformation and their application to problems of the internal friction (in Russian). Prikl. Mat. Mekh. 12 (1948), 251–260.Google Scholar
Gerstein, G. L. and Mandelbrot, B.. Random walk models for the spike activity of a single neuron. Biophys. J. 4 (1964), 41–68.CrossRefGoogle ScholarPubMed
Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order (Berlin, Heidelberg: Springer, 2015).Google Scholar
Hahn, M., Ryvkina, J., Kobayashi, K., Umarov, S., et al. On time-changed Gaussian processes and their associated Fokker–Planck–Kolmogorov equations. Electron. Commun. Probab. 16 (2011), 150–164.CrossRefGoogle Scholar
Kaarakka, T. and Salminen, P.. On fractional Ornstein–Uhlenbeck processes. Commun. Stoch. Anal. 5 (2011), 8.Google Scholar
Kim, P., Song, R. and Vondraček, Z.. Potential theory of subordinate Brownian motions revisited. In Stochastic analysis and applications to finance: Essays in honour of Jia-an Yan, pp. 243–290 (Singapore, Hackensack N. J.: World Scientific, 2012).Google Scholar
Kochubei, A. N.. General fractional calculus, evolution equations, and renewal processes. Integr. Equ. Oper. Theory 71 (2011), 583–600.CrossRefGoogle Scholar
Leonenko, N. N., Meerschaert, M. M. and Sikorskii, A.. Correlation structure of fractional Pearson diffusions. Comput. Math. Appl. 66 (2013), 737–745.CrossRefGoogle ScholarPubMed
Leonenko, N. N., Meerschaert, M. M. and Sikorskii, A.. Fractional Pearson diffusions. J. Math. Anal. Appl. 403 (2013), 532–546.CrossRefGoogle ScholarPubMed
Luchko, Y.. Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351 (2009), 218–223.CrossRefGoogle Scholar
Meerschaert, M. M., Nane, E. and Vellaisamy, P.. Distributed-order fractional diffusions on bounded domains. J. Math. Anal. Appl. 379 (2011), 216–228.CrossRefGoogle Scholar
Meerschaert, M. M. and Scheffler, H.-P.. Triangular array limits for continuous time random walks. Stoch. Process. Their Appl. 118 (2008), 1606–1633.CrossRefGoogle Scholar
Meerschaert, M. M. and Straka, P.. Inverse stable subordinators. Math. Model. Nat. Phenom. 8 (2013), 1–16.CrossRefGoogle ScholarPubMed
Mishura, Y., Piterbarg, V., Ralchenko, K. and Yurchenko-Tytarenko, A.. Stochastic representation and path properties of a fractional Cox–Ingersoll–Ross process. Theory Probab. Math. Stat. 97 (2018), 167–182.CrossRefGoogle Scholar
Mishura, Y. and Yurchenko-Tytarenko, A.. Fractional Cox–Ingersoll–Ross process with non-zero mean. Mod. Stoch.: Theory Appl. 5 (2018), 99–111.CrossRefGoogle Scholar
Mitrinović, D. S., Pečarić, J. E. and Fink, A. M.. Classical and new inequalities in analysis (Dordrecht, The Netherlands: Springer, 1993).CrossRefGoogle Scholar
Pachpatte, B. G.. Inequalities for differential and integral equations (Bodmin, Cornwall, GB: Elsevier, 1997).Google Scholar
Rodieck, R. W., Kiang, N. Y. S. and Gerstein, G. L.. Some quantitative methods for the study of spontaneous activity of single neurons. Biophys. J. 2 (1962), 351–368.CrossRefGoogle Scholar
Rossikhin, Y. A.. Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Appl. Mech. Rev. 63 (2010), 010701.CrossRefGoogle Scholar
Sakai, Y., Funahashi, S. and Shinomoto, S.. Temporally correlated inputs to leaky integrate-and-fire models can reproduce spiking statistics of cortical neurons. Neural Netw. 12 (1999), 1181–1190.CrossRefGoogle ScholarPubMed
Schilling, R. L., Song, R. and Vondracek, Z.. Bernstein functions: Theory and applications. Vol. 37 (Berlin, Gottingen: Walter de Gruyter, 2012).CrossRefGoogle Scholar
Shinomoto, S., Sakai, Y. and Funahashi, S.. The Ornstein–Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex. Neural Comput. 11 (1999), 935–951.CrossRefGoogle Scholar
Toaldo, B.. Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups. Potential Anal. 42 (2015), 115–140.CrossRefGoogle Scholar