Advances in Applied Probability

Generalized fractional Lévy processes with fractional Brownian motion limit

Claudia Klüppelberg and Muneya Matsui

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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.

Article information

Adv. in Appl. Probab., Volume 47, Number 4 (2015), 1108-1131.

First available in Project Euclid: 11 December 2015

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Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion 60G51: Processes with independent increments; Lévy processes 60F17: Functional limit theorems; invariance principles
Secondary: 91B24: Price theory and market structure 91B28 62P20: Applications to economics [See also 91Bxx]

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


Klüppelberg, Claudia; Matsui, Muneya. Generalized fractional Lévy processes with fractional Brownian motion limit. Adv. in Appl. Probab. 47 (2015), no. 4, 1108--1131. doi:10.1239/aap/1449859802.

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