Abstract
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.
Highlights
IntroductionAs an extension of Brownian motion (Bm), fractional Brownian motion (fBm) has become an object of intensive study [BHØZ07], [Mis08], due to its specific properties, such as short/long range dependence and self-similarity, with natural applications in different fields (e.g. mathematical finance, telecommunications engineering, etc.)
In recent years, as an extension of Brownian motion (Bm), fractional Brownian motion has become an object of intensive study [BHØZ07], [Mis08], due to its specific properties, such as short/long range dependence and self-similarity, with natural applications in different fields
In order to cast fractional Brownian motion (fBm) into the classical Bm framework, there are various representations of fBm, starting with the famous definition by Mandelbrot and van Ness [MvN68]. This idea is the starting point for a characterization of fBm using an infinite superposition of Ornstein-Uhlenbeck processes w.r.t. the standard Wiener process; compare the works of Carmona, Coutin, Montseny, and Muravlev [CC93, CCM00, Mur11] or the monograph of [Mis08]
Summary
As an extension of Brownian motion (Bm), fractional Brownian motion (fBm) has become an object of intensive study [BHØZ07], [Mis08], due to its specific properties, such as short/long range dependence and self-similarity, with natural applications in different fields (e.g. mathematical finance, telecommunications engineering, etc.). In order to cast fBm into the classical Bm framework, there are various representations of fBm, starting with the famous definition by Mandelbrot and van Ness [MvN68] This idea is the starting point for a characterization of fBm using an infinite superposition of Ornstein-Uhlenbeck processes w.r.t. the standard Wiener process; compare the works of Carmona, Coutin, Montseny, and Muravlev [CC93, CCM00, Mur11] or the monograph of [Mis08]. The corresponding stochastic process is referred to as generalized grey Brownian motion (ggBm) and is in general neither a martingale nor a Markov process. It is not possible - as in the Gaussian case - to find a proper orthonormal system of polynomials for the test and generalized functions. The Appendix contains auxiliary results needed in the two main proofs
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