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 $\alpha = 1$.
Author: | Peter Imkeller, Ilya Pavlyukevich, Thorsten Wetzel |
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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 |