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March 2009 First exit times for Lévy-driven diffusions with exponentially light jumps
Peter Imkeller, Ilya Pavlyukevich, Torsten Wetzel
Ann. Probab. 37(2): 530-564 (March 2009). DOI: 10.1214/08-AOP412


We consider a dynamical system described by the differential equation t=−U'(Yt) with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity ɛ to obtain the stochastic differential equation dXtɛ=−U'(Xtɛ) dt+ɛdLt. The process L is a symmetric Lévy process whose jump measure ν has exponentially light tails, ν([u, ∞))∼exp(−uα), α>0, u→∞. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval (−1, 1). In the small noise limit ɛ→0, the law of the first exit time σx, x∈(−1, 1), has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index α=1, namely, ln Eσɛα for 0<α<1, whereas ln Eσɛ−1|ln ɛ|1−1/α for α>1.


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Peter Imkeller. Ilya Pavlyukevich. Torsten Wetzel. "First exit times for Lévy-driven diffusions with exponentially light jumps." Ann. Probab. 37 (2) 530 - 564, March 2009.


Published: March 2009
First available in Project Euclid: 30 April 2009

zbMATH: 1184.60019
MathSciNet: MR2510016
Digital Object Identifier: 10.1214/08-AOP412

Primary: 60F10 , 60G17 , 60H15

Keywords: Convex optimization , extreme events , first exit time , jump diffusion , Lévy process , regular variation , sub-exponential and super-exponential tail

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 2 • March 2009
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