The Annals of Probability

First exit times for Lévy-driven diffusions with exponentially light jumps

Peter Imkeller, Ilya Pavlyukevich, and Torsten Wetzel

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Abstract

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.

Article information

Source
Ann. Probab., Volume 37, Number 2 (2009), 530-564.

Dates
First available in Project Euclid: 30 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1241099921

Digital Object Identifier
doi:10.1214/08-AOP412

Mathematical Reviews number (MathSciNet)
MR2510016

Zentralblatt MATH identifier
1184.60019

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60F10: Large deviations 60G17: Sample path properties

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

Citation

Imkeller, Peter; Pavlyukevich, Ilya; Wetzel, Torsten. First exit times for Lévy-driven diffusions with exponentially light jumps. Ann. Probab. 37 (2009), no. 2, 530--564. doi:10.1214/08-AOP412. https://projecteuclid.org/euclid.aop/1241099921


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