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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.

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