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Fractional Laplace motion

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  01 July 2016

T. J. Kozubowski ,
M. M. Meerschaert  and
K. Podgórski
T. J. Kozubowski*
Affiliation:
University of Nevada at Reno
M. M. Meerschaert*
Affiliation:
University of Otago
K. Podgórski*
Affiliation:
Indiana University-Purdue University Indianapolis
*
∗ Postal address: Department of Mathematics and Statistics, Mail Stop 84, University of Nevada, Reno, NV 89557, USA. Email address: [email protected]
∗∗ Postal address: Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9001, New Zealand.
∗∗∗ Postal address: Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202, USA.
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Abstract

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Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

Keywords

Compound processfractional Brownian motionG-type distributiongamma processgeneralized gamma distributioninfinite divisibilitylong-range dependencescalingself-affinityself-similaritysubordination

MSC classification

Primary: 60G07: General theory of processes 60G18: Self-similar processes 60E07: Infinitely divisible distributions; stable distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 451 - 464
Copyright
Copyright © Applied Probability Trust 2006 

Footnotes

Partially supported by NSF grant DMS-0139927.

Partially supported by NSF grants DMS-0139927 and DMS-0417869 and the Marsden Fund administered by the Royal Society of New Zealand.

References

Amoroso, L. (1925). Ricerche intorno alla curva dei redditi. Ann. Mat. Pura Appl. Ser. 4 21, 123–159.CrossRefGoogle Scholar
Baeumer, B. and Meerschaert, M. M. (2001). Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4, 481–500.Google Scholar
Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M. (2000). Application of a fractional advection-dispersion equation. Water Resources Res. 36, 1403–1412.CrossRefGoogle Scholar
Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M. (2000). The fractional-order governing equation of Lévy motion. Water Resources Res. 36, 1413–1424.CrossRefGoogle Scholar
Benson, D. A., Schumer, R., Meerschaert, M. M. and Wheatcraft, S. W. (2001). Fractional dispersion, Lévy motions, and the MADE tracer tests. Transport Porous Media 42, 211–240.Google Scholar
Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.Google Scholar
Bibby, B. M. and Sørensen, M. (2003). Hyperbolic distributions in finance. In Handbook of Heavy Tailed Distributions in Finance, ed. Rachev, S. T., Elsevier, Amsterdam, pp. 211–248.Google Scholar
Bondesson, L. (1979). A general result on infinite divisibility. Ann. Prob. 7, 965–979.Google Scholar
Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities (Lecture Notes Statist. 76). Springer, New York.CrossRefGoogle Scholar
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods. Springer, New York.Google Scholar
Ching, E. S. and Tu, Y. (1994). Passive scalar fluctuations with and without a mean gradient: a numerical study. Phys. Rev. E 49, 1278–1282.Google Scholar
Feller, V. (1971). Introduction to the Theory of Probability and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Gnedenko, B. V. (1970). Limit theorems for sums of a random number of positive independent random variables. In Proc. 6th Berkeley Symp. Math. Statist. Probab., Vol. 2, University of California Press, Berkeley, pp. 537–549.Google Scholar
Gupta, V. K. and Waymire, E. C. (1990). Multiscaling properties of spatial rainfall in river flow distributions. J. Geophys. Res. 95, 1999–1999.Google Scholar
Heyde, C. C. (1999). A risky asset model with strong dependence through fractal activity time. J. Appl. Prob. 36, 1234–1239.Google Scholar
Heyde, C. C. (2002). On modes of long-range dependence. J. Appl. Prob. 39, 882–888.Google Scholar
Heyde, C. C. and Yang, Y. (1997). On defining long-range dependence. J. Appl. Prob. 34, 939–944.Google Scholar
Johnson, N. L., Kotz, S. and Balakrishnan, B. (1994). Continuous Univariate Distributions, Vol. 1, 2nd edn. John Wiley, New York.Google Scholar
Kelker, D. (1971). Infinite divisibility and variance mixtures of the normal distribution. Ann. Math. Statist. 42, 802–808.Google Scholar
Kotz, S., Kozubowski, T. J. and Podgórski, K. (2001). The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance. Birkhäuser, Boston, MA.CrossRefGoogle Scholar
Levin, A. and Tchernitser, A. (2003). Multifactor stochastic variance models in risk management: maximum entropy approach and Lévy processes. In Handbook of Heavy Tailed Distributions in Finance, ed. Rachev, S. T., Elsevier, Amsterdam, pp. 443–480.CrossRefGoogle Scholar
Madan, D. B., Carr, P. P. and Chang, E. C. (1998). The variance gamma process and option pricing. Europ. Fin. Rev. 2, 74–105.Google Scholar
Madan, D. B. and Seneta, E. (1990). The variance-gamma (V.G.) model for share markets returns. J. Business 63, 511–524.Google Scholar
Maejima, M. and Rosiński, J. (2001). The class of type G distributions on R d and related subclasses of infinitely divisible distributions. Demonstr. Math. 34, 251–266.Google Scholar
Maejima, M. and Rosiński, J. (2002). Type G distributions on R d . J. Theoret. Prob. 15, 323–341.CrossRefGoogle Scholar
Marcus, M. B. (1987). ξ-radial Processes and Random Fourier Series (Mem. Ser. 68). American Mathematical Society, Providence, RI.Google Scholar
Meerschaert, M. M., Kozubowski, T. J., Molz, F. J. and Lu, S. (2004). Fractional Laplace model for hydraulic conductivity. Geophys. Res. Lett. 31, L08501.Google Scholar
Metzler, R. and Klafter, J. (2000). The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77.CrossRefGoogle Scholar
Molz, F. J., Kozubowski, T. J., Podgórski, K. and Castle, J. W. (2005). A generalization of the fractal/facies model. Preprint.Google Scholar
Molz, F. J., Meerschaert, M. M., Kozubowski, T. J., and Hyden, P. D. (2005). Do heterogeneous sediment properties and turbulent velocity fluctuations have something in common? Some history and a new stochastic process. In Dynamics of Fluids and Transport in Fractured Rock (Geophys. Monogr. Ser. 162), eds Faybishenko, B., Witherspoon, P. A. and Gale, J., American Geophysical Union, Washington, DC, pp. 13–22.Google Scholar
Press, S. J. (1967). On the sample covariance from a bivariate normal distribution. Ann. Inst. Statist. Math. 19, 355–361.CrossRefGoogle Scholar
Rényi, A. (1976). A characterization of Poisson processes. In Selected Papers of Alfréd Rényi, Vol. 1, ed. Turán, P., Akadémiai Kiadó, Budapest, pp. 622–628.Google Scholar
Rosiński, J. (1991). On a class of infinitely divisible processes represented as mixtures of Gaussian processes. In Stable Processes and Related Topics, eds Cambanis, S., Samorodnitsky, G. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 27–41.Google Scholar
Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London.Google Scholar
Steutel, F. W. and van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York.Google Scholar
Taqqu, M. S. (2003). Fractional Brownian motion and long-range dependence. In Theory and Applications of Long-Range Dependence, eds Doukhan, P., Oppenheim, G. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 5–38.Google Scholar
Thorin, O. (1978). An extension of the notion of a generalized gamma convolution. Scand. Actuar. J. 1978, 141–149.CrossRefGoogle Scholar
Thorin, O. (1978). Proof of a conjecture of L. Bondesson concerning infinite divisibility of powers of a gamma variable. Scand. Actuar. J. 1978, 151–164.Google Scholar
Veneziano, D. (1999). Basic properties and characterization of stochastically self-similar processes in R d . Fractals 7, 59–78.Google Scholar
Wong, R. (1989). Asymptotic Approximations of Integrals. Academic Press, Boston, MA.Google Scholar