Skip to main content Accessibility help
×
Home
Hostname: page-component-558cb97cc8-6jfzc Total loading time: 1.051 Render date: 2022-10-07T20:29:40.746Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

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

Published online by Cambridge University Press:  21 September 2021

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)

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.

Type
Research Article
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anh, V. and Inoue, A.. Financial markets with memory I: dynamic models. Stoch. Anal. Appl. 23 (2005), 275300.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), 5384.CrossRefGoogle Scholar
Ascione, G., Mishura, Y. and Pirozzi, E.. Time-changed fractional Ornstein–Uhlenbeck process. Fract. Calc. Appl. Anal. 23 (2020), 450483.CrossRefGoogle Scholar
Ascione, G. and Pirozzi, E.. On a stochastic neuronal model integrating correlated inputs. Math. Biosci. Eng 16 (2019), 52065225.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), 11301163.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), 2957.CrossRefGoogle Scholar
Caputo, M.. Linear models of dissipation whose $Q$ is almost frequency independent–II. Geophys. J. Int. 13 (1967), 529539.CrossRefGoogle Scholar
Cheridito, P., Kawaguchi, H. and Maejima, M.. Fractional Ornstein–Uhlenbeck processes. Electron. J. Probab. 8 (2003), 314.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), 487501.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), 251260.Google Scholar
Gerstein, G. L. and Mandelbrot, B.. Random walk models for the spike activity of a single neuron. Biophys. J. 4 (1964), 4168.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), 150164.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), 583600.CrossRefGoogle Scholar
Leonenko, N. N., Meerschaert, M. M. and Sikorskii, A.. Correlation structure of fractional Pearson diffusions. Comput. Math. Appl. 66 (2013), 737745.CrossRefGoogle ScholarPubMed
Leonenko, N. N., Meerschaert, M. M. and Sikorskii, A.. Fractional Pearson diffusions. J. Math. Anal. Appl. 403 (2013), 532546.CrossRefGoogle ScholarPubMed
Luchko, Y.. Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351 (2009), 218223.CrossRefGoogle Scholar
Meerschaert, M. M., Nane, E. and Vellaisamy, P.. Distributed-order fractional diffusions on bounded domains. J. Math. Anal. Appl. 379 (2011), 216228.CrossRefGoogle Scholar
Meerschaert, M. M. and Scheffler, H.-P.. Triangular array limits for continuous time random walks. Stoch. Process. Their Appl. 118 (2008), 16061633.CrossRefGoogle Scholar
Meerschaert, M. M. and Straka, P.. Inverse stable subordinators. Math. Model. Nat. Phenom. 8 (2013), 116.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), 167182.CrossRefGoogle Scholar
Mishura, Y. and Yurchenko-Tytarenko, A.. Fractional Cox–Ingersoll–Ross process with non-zero mean. Mod. Stoch.: Theory Appl. 5 (2018), 99111.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), 351368.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), 11811190.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), 935951.CrossRefGoogle Scholar
Toaldo, B.. Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups. Potential Anal. 42 (2015), 115140.CrossRefGoogle Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

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

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *