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Generalized fractional Lévy processes with fractional Brownian motion limit

Part of: Stochastic processes Mathematical economics Limit theorems Applications

Published online by Cambridge University Press:  21 March 2016

Claudia Klüppelberg  and
Muneya Matsui
Claudia Klüppelberg*
Affiliation:
Technische Universität München
Muneya Matsui*
Affiliation:
Nanzan University
*
∗ Postal address: Center for Mathematical Sciences, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany. Email address: cklu@ma.tum.de
∗∗ Postal address: Department of Business Administration, Nanzan University, 18 Yamazato-cho, Showa-ku, Nagoya 466-8673, Japan. Email address: mmuneya@nanzan-u.ac.jp
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Abstract

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Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

Keywords

Shot-noise processfractional Brownian motionfractional Lévy processgeneralized fractional Lévy processfractional Ornstein-Uhlenbeck processfunctional central limit theoremregular variationstochastic volatility model

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion 60G51: Processes with independent increments; L'evy processes 60F17: Functional limit theorems; invariance principles
Secondary: 91B24: Price theory and market structure 62P20: Applications to economics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 1108 - 1131
Copyright
Copyright © Applied Probability Trust 2015 

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