The Lévy-Itô decomposition of sample paths of Lévy processes with values in the space of probability measures
Kouji YAMAMURO
Source: J. Math. Soc. Japan Volume 61, Number 1
(2009), 263-289.
Abstract
A definition of Lévy processes with values in the space of probability measures was introduced by Shiga and Tanaka (Electronic J. Prob. 11 (2006)). It is shown that the Lévy process with values in the space of probability measures in law has a modification satisfying a certain condition. The modification is a Lévy process in the sense of Shiga and Tanaka. The Lévy-Itô decomposition of sample paths of the Lévy process satisfying the condition is derived.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1234189036
Digital Object Identifier: doi:10.2969/jmsj/06110263
Mathematical Reviews number (MathSciNet): MR2272879
References
P. Billingsley, Probability and Measure, 3rd ed., Wiley, New York, 1995.
Mathematical Reviews (MathSciNet): MR1324786
E. B. Dynkin, Criteria of continuity and absence of discontinuity of the second kind for trajectories of a Markov process, Izv. Akad. Nauk SSSR Ser. Mat., 16 (1952), 563–572.
Mathematical Reviews (MathSciNet): MR52055
K. Itô, On stochastic processes, I (Infinitely divisible laws of probability), Japan J. Math., 18 (1942), 261–301.
Mathematical Reviews (MathSciNet): MR14629
Zentralblatt MATH: 0060.28908
K. Itô, Stochastic processes, Lectures given at Aarhus University, Reprint of the 1969 original, Edited by Ole E. Barndorff-Nielsen and Ken-iti Sato, Springer-Verlag, 2004.
Mathematical Reviews (MathSciNet): MR2053326
J. R. Kinney, Continuity properties of sample functions of Markov processes, Trans. Amer. Math. Soc., 74 (1953), 280–302.
P. Lévy, Sur les intégrales dont les éléments sont des variables aléatoires indépendantes, Ann. Scuola Norm. Sup. Pisa (2), 3 (1934), 337–366; 4 (1934), 217–218.
P. Lévy, Théorie de l'Addition des Variables Aléatoires, Gauthier-Villars, Paris, 1937.
K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.
Mathematical Reviews (MathSciNet): MR1739520
T. Shiga and H. Tanaka, Infinitely divisible random probability distributions with an application to a random motion in a random environment, Electronic J. Prob., 11 (2006), 1144–1183.
Mathematical Reviews (MathSciNet): MR2268541
Zentralblatt MATH: 1127.60051
Journal of the Mathematical Society of Japan
