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December 2015 Generalized fractional Lévy processes with fractional Brownian motion limit
Claudia Klüppelberg, Muneya Matsui
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Adv. in Appl. Probab. 47(4): 1108-1131 (December 2015). DOI: 10.1239/aap/1449859802


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.


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Claudia Klüppelberg. Muneya Matsui. "Generalized fractional Lévy processes with fractional Brownian motion limit." Adv. in Appl. Probab. 47 (4) 1108 - 1131, December 2015.


Published: December 2015
First available in Project Euclid: 11 December 2015

zbMATH: 1333.60074
MathSciNet: MR3433298
Digital Object Identifier: 10.1239/aap/1449859802

Primary: 60F17 , 60G22 , 60G51
Secondary: 62P20 , 91B24 , 91B28

Keywords: fractional Brownian motion , fractional Lévy process , fractional Ornstein-Uhlenbeck process , functional central limit theorem , generalized fractional Lévy process , regular variation , Shot-noise process , stochastic volatility model

Rights: Copyright © 2015 Applied Probability Trust


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Vol.47 • No. 4 • December 2015
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