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

Part of: Stochastic processes Miscellaneous topics - Partial differential equations 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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