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

Please always quote using this URN:urn:nbn:de:0296-matheon-10446
  • We consider a dynamical system described by the differential equation $\dot{Y}_t = -U^'(Y_t)$ with a unique stable point at the origin. We perturb the system by L\'evy noise of intensity $\varepsilon$, to obtain the stochastic differential equation $dX^\varepsilon_t = -U^'(X^\varepsilon_{t-})dt + \varepsilon dL_t}. The process $L$ is a symmetric L\'evy process whose jump measure $\nu$ has exponentially light tails, $\nu([u, \infty))\sim exp(-u^\alpha), \alpha > 0, u \to\infty$. 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 $\varepsilon\to 0$ we determine the law and the mean value of the first exit time, to discover an intriguing phase transition at the critical index $\alpha = 1$.

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Metadaten
Author:Peter Imkeller, Ilya Pavlyukevich, Thorsten Wetzel
URN:urn:nbn:de:0296-matheon-10446
Referee:Dirk Becherer
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2012/01/31
Release Date:2012/01/31
Institute:Humboldt-Universität zu Berlin
MSC-Classification:60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Jxx Markov processes / 60J60 Diffusion processes [See also 58J65]
Preprint Number:908
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