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

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

2007-11-06

arxiv.org/abs/0711.0982v2

Annals of Probability 2009, Vol. 37, No. 2, 530-564

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AOP412 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/08-AOP412

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 the 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 $\varepsilon\to0$, the law of the first exit time $\sigma_x$, $x\in(-1,1)$, has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index $\alpha=1$, namely, $\ln\mathbf{E}\sigma\sim\varepsilon^{-\alpha}$ for $0<\alpha<1$, whereas $\ln\mathbf{E}\sigma\sim\varepsilon^{- 1}|\ln\varepsilon|^{1-{1}/{\alpha}}$ for $\alpha>1$.

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