Integral Representation of Generalized Grey Brownian Motion

TL;DR

This paper presents a new integral representation for generalized grey Brownian motion, extending classical Gaussian process representations to non-Gaussian processes through stochastic differential equations.

Contribution

It introduces a novel integral representation for non-Gaussian generalized grey Brownian motion, generalizing previous Gaussian-based results.

Findings

Representation as a weighted integral of a non-Gaussian Ornstein-Uhlenbeck type process

Extension of existing Gaussian process representation results to non-Gaussian cases

Provides a framework for analyzing non-Gaussian stochastic processes

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 as well as Harms and Stefanovits to the non Gaussian case.

Full text

** Key words** :

Generalized grey Brownian motion; Fractional stochastic processes; Rough paths; superposition of OU processes; grey OU processes

1. Introduction

In recent years, as an extension of Brownian motion (Bm), fractional Brownian motion (fBm) has become an object of intensive study [oks], [mishura08], 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 [MandelbrotNess1968]. This idea is also 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 [CC1993, CCM2000, MU2011] or also the monograph of [mishura08]. Recently, further applications of this representation have for instance been investigated in [HS2018] with a focus on finance and in [BD2018] in the context of optimal portfolios.

One key tool box for the rigorous analysis of fBm is the Gaussian analysis or white noise analysis. White noise analysis has evolved into an infinite dimensional distribution theory, with rapid developments in mathematical structure and applications in multiple domains, see e.g. the monographs [HKPS93, O94, Kuo96, HS17]. Various characterization theorems [PS91, KLPSW96, GKS97] are proven to build up a strong analytical foundation. Almost at the same time, first attempts were made to introduce a non-Gaussian infinite dimensional analysis, by transferring properties of the Gaussian measure to the Poisson measure [I88], which could be generalized with the help of a biorthogonal generalized Appell systems [Da91, ADKS96, KSWY95]. Mittag-Leffler Analysis is established in [GJRS14] and [GJ15]. In fact, it generalizes methods from white noise calculus to the case, where in the characteristic function of the Gaussian measure the exponential function is replaced by a Mittag-Leffler function. The corresponding stochastic process is referred to as generalized grey Brownian motion (ggBm) and is in general neither a martingale nor a Markov process. Moreover, it is not possible - as in the Gaussian case - to find a proper orthonormal system of polynomials for the test and generalized functions. Here it is necessary to make use of an Appell system of biorthogonal polynomials. The grey noise measure [Schneider90, Schneider92, MM09] is included as a special case in the class of Mittag-Leffler measures, which offers the possibility to apply the Mittag-Leffler analysis to fractional differential equations, in particular to fractional diffusion equations, which carry numerous applications in science, like relaxation type differential equations or viscoelasticity. In [GJ15] also a relation between the heat kernel in this setting and the associated processes grey Brownian motion could be proven. In [BS17] Wick-type stochastic differential equations and Ornstein-Uhlenbeck processes were solved within the framework of Mittag-Leffler analysis. Indeed the underlying fractional differential equations are of interest in applications like human mobility in disease spread [SMGABS2012].

The aim of this paper is to establish a link between ggBm and generalized grey Ornstein-Uhlenbeck processes as an extension of the results in [BS17] and to extend the representation results of [MU2011] and [HS2018] to the non-gaussian case of ggBm. To this end, we make use of a representation of ggBm as a product of a positive and time-independent random variable and a fBm [Mainardi_Mura_Pagnini_2010]. Our work also shares common features with [Pagnini18].

The manuscript is organized as follows: In Section 2 we give preliminaries about generalized grey Brownian motion, defined on an abstract probability space. In Section 3 we develop a representation of ggBm using an infinite dimensional superposition of generalized grey Ornstein-Uhlenbeck processes. We thus enhance the results in [CC1993, CCM2000, MU2011, HS2018] to the setting of ggBm. The Appendix contains auxiliary results needed in the two main proofs.

2. Generalized grey Brownian motion in arbitrary dimensions

2.1. Prerequisites