A new multifractional process with random exponent

Issue title: Fractional Calculus and Its Applications

Guest editors: Sergio Bianchi

Article type: Research Article

Authors: Ayache, Antoine^{a; *} | Esser, Céline^b | Hamonier, Julien^c

Affiliations: [a] Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, F-59000 Lille, France. E-mail: antoine.ayache@math.univ-lille1.fr | [b] Université de Liège – Institut de Mathématique, Allée de la Découverte 12, 4000 Liège, Belgium. E-mail: celine.esser@uliege.be | [c] Univ. Lille, CHU Lille, EA 2694 – Santé publique : épidémiologie et qualité des soins, F-59000 Lille, France. E-mail: julien.hamonier@univ-lille2.fr

Correspondence: [*] Corresponding author. E-mail: antoine.ayache@math.univ-lille1.fr.

Abstract: A first type of Multifractional Process with Random Exponent (MPRE) was constructed several years ago in [Publ. Mat. 49 (2005), 459–486] by replacing in a wavelet series representation of Fractional Brownian Motion (FBM) the Hurst parameter by a random variable depending on the time variable. In the present article, we propose another approach for constructing another type of MPRE. It consists in substituting to the Hurst parameter, in a stochastic integral representation of the high-frequency part of FBM, a random variable depending on the integration variable. The MPRE obtained in this way offers, among other things, the advantages to have a representation through classical Itô integral and to be less difficult to simulate than the first type of MPRE, previously introduced in [Publ. Mat. 49 (2005), 459–486]. Yet, the study of Hölder regularity of this new MPRE is a significantly more challenging problem than in the case of the previous one. Actually, it requires to develop a new methodology relying on an extensive use of the Haar basis.

Keywords: Fractional Brownian Motion, varying Hurst parameter, Haar basis, Hölder regularity, Itô integral

DOI: 10.3233/RDA-180135

Journal: Risk and Decision Analysis, vol. 7, no. 1-2, pp. 5-29, 2018