Integral representation of generalized grey Brownian motion

Wolfgang Bock; Sascha Desmettre; José Luís da Silva

doi:10.1080/17442508.2019.1641093

Integral representation of generalized grey Brownian motion

Stochastics (Abingdon). 2019 Jul 11;92(4):552-565. doi: 10.1080/17442508.2019.1641093.

Authors

Wolfgang Bock¹, Sascha Desmettre², José Luís da Silva³

Affiliations

¹ Department of Mathematics, TU Kaiserslautern (TUK), Kaiserslautern, Germany.
² Institute for Mathematics and Scientific Computing, University of Graz, Graz, Austria.
³ CIMA, University of Madeira, Funchal, Portugal.

Abstract

In this paper, we investigate the representation of a class of non-Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular, the underlying process can be seen as a non-Gaussian extension of the Ornstein-Uhlenbeck process, hence generalizing the representation results of Muravlev, Russian Math. Surveys 66 (2), 2011 as well as Harms and Stefanovits, Stochastic Process. Appl. 129, 2019 to the non-Gaussian case.

Keywords: Generalized grey Brownian motion; fractional stochastic processes; grey OU processes; rough paths; superposition of OU processes.

Grants and funding

Financial support from FCT – Fundação para a Ciência e a Tecnologia through the project UID/MAT/04674/2019 (CIMA Universidade da Madeira) and the Deutsche Forschungsgemeinschaft (DFG-Research Training Group 1932) Stochastic Models for Innovations in the Engineering Sciences is gratefully acknowledged. S. Desmettre is also supported by the Austrian Science Fund (FWF) project F5508-N26, which is part of the Special Research Program Quasi-Monte Carlo Methods: Theory and Applications. Moreover, the authors gratefully acknowledge support from NAWI Graz.