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Contents of PMS, Vol. 40, Fasc. 2,
pages 317 - 330
DOI: 10.37190/0208-4147.40.2.7
Published online 5.8.2020
 

Distribution tails for solutions of SDE driven by an asymmetric stable Lévy process

Richard Eon
Mihai Gradinaru

Abstract: The behaviour of the tails of the invariant distribution for stochastic differential equations driven by an asymmetric stable Lévy process is obtained. We generalize a result by Samorodnitsky and Grigoriu where the stable driving noise was supposed to be symmetric.

2010 AMS Mathematics Subject Classification: Primary 60H10; Secondary 60G52, 60E07, 60F17.

Keywords and phrases: stochastic differential equation, asymmetric stable Lévy noise, tail behaviour, ergodic processes, stationary distribution.

References


[1] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge Univ. Press, 2011.

[2] R. Eon, Asymptotique des solutions d' équations différentielles de type frottement perturbées par des bruits de Lévy stables, thése de doctorat, Université de Rennes 1, 2016; https://tel.archives-ouvertes.fr/tel-01388319v1.

[3] R. Eon M. Gradinaru, Gaussian asymptotics for a non-linear Langevin type equation driven by an a-stable Lévy noise, Electron. J. Probab. 20 (2015), art. 100, 19 pp.

[4] O. Kallenberg, Foundations of Modern Probability, Springer, 2000.

[5] A. M. Kulik, Exponential ergodicity of the solutions to SDE's with a jump noise, Stochastic Process. Appl. 119 (2009), 602-632.

[6] V. Petrov, Limit Theorems of Probability Theory. Sequences of Independent Random Variables, Oxford Univ. Press, 1995.

[7] Yu. V. Prokhorov, An extremal problem in probability theory, Theory Probab. Appl. 4 (1959), 201-203.

[8] G. Samorodnitsky M. Grigoriu, Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed Lévy motions, Stochastic Process. Appl. 105 (2003), 69-97.

[9] T. Srokowski, Asymmetric Lévy flights in nonhomogeneous environments, J. Statist. Mech. Theory Experiment 2014, no. 5, art. P05024, 18 pp.

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