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Generalized fractional Lévy processes with fractional Brownian motion limit

Part of: Applications Limit theorems Mathematical economics Stochastic processes

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: [email protected]
∗∗ Postal address: Department of Business Administration, Nanzan University, 18 Yamazato-cho, Showa-ku, Nagoya 466-8673, Japan. Email address: [email protected]
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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 

References

Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein–Uhlenbeck type processes. Theory Prob. Appl. 45, 175–194.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. B 63, 167–241.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Bingham, N. H, Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Brockwell, P. J. and Lindner, A. (2009). Existence and uniqueness of stationary Lévy-driven CARMA processes. Stoch. Process. Appl. 119, 2660–2681.Google Scholar
Brockwell, P. and Marquardt, T. (2005). Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Statist. Sinica 15, 477–494.Google Scholar
Buchmann, B. and Klüppelberg, C. (2006). Fractional integral equations and state space transforms. Bernoulli 12, 431–456.Google Scholar
Cheridito, P., Kawaguchi, H. and Maejima, M. (2003). Fractional Ornstein–Uhlenbeck processes. Electron. J. Prob. 8, 14 pp.CrossRefGoogle Scholar
Comte, F., Coutin, L. and Renault, E. (2012). Affine fractional stochastic volatility models. Ann. Finance 8, 337-378.CrossRefGoogle Scholar
Eberlein, E. and Raible, S. (1999). Term structure models driven by general Lévy processes. Math. Finance 9, 31–53.Google Scholar
Fink, H. and Klüppelberg, C. (2011). Fractional Lévy-driven Ornstein–Uhlenbeck processes and stochastic differential equations. Bernoulli 17, 484–506.CrossRefGoogle Scholar
Gander, M. P. S. and Stephens, D. A. (2007). Simulation and inference for stochastic volatility models driven by Lévy processes. Biometrika 94, 627–646.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Klüppelberg, C. and Kühn, C. (2004). Fractional Brownian motion as a weak limit of Poisson shot noise processes—with applications to finance. Stoch. Process. Appl. 113, 333–351.Google Scholar
Klüppelberg, C. and Mikosch, T. (1995). Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1, 125–147.Google Scholar
Klüppelberg, C., Lindner, A. and Maller, R. (2004). A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour. J. Appl. Prob. 41, 601–622.Google Scholar
Lane, J. A. (1984). The central limit theorem for the Poisson shot-noise process. J. Appl. Prob. 21, 287–301.Google Scholar
Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12, 1099–1126.Google Scholar
Mikosch, T. and Norvaiša, R. (2000). Stochastic integral equations without probability. Bernoulli 6, 401–434.CrossRefGoogle Scholar
Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford University Press.Google Scholar
Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Prob. Theory Relat. Fields 118, 251–291.CrossRefGoogle Scholar
Pipiras, V. and Taqqu, M. S. (2008). Small and large scale asymptotics of some Lévy stochastic integrals. Methodology Comput. Appl. Prob. 10, 299–314.Google Scholar
Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–487.CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach, Yverdon.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282.Google Scholar
Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Prob. Theory Relat. Fields 111, 333–374.CrossRefGoogle Scholar