First exit times of solutions of non-linear stochastic differential equations driven by symmetric Levy processes with alpha-stable components

Details First exit times of solutions of non-linear stochastic differential equations driven by symmetric Levy processes with alpha-stable components First exit times of solutions of non-linear stochastic differential equations driven by symmetric Levy processes with alpha-stable components

2004-09-15

arxiv.org/abs/math/0409246v2

Mathematics

Probability

Slightly revised version. To appear in Stochastic Processes and their Applications. 24 pages, 2 figures

Scientific paper

We study the exit problem of solutions of the stochastic differential equation dX(t)=-U'(X(t))dt+epsilon dL(t) from bounded or unbounded intervals which contain the unique asymptotically stable critical point of the deterministic dynamical system dY=-U'(Y) dt. The process L is composed of a standard Brownian motion and a symmetric alpha-stable Levy process. Using probabilistic estimates we show that in the small noise limit epsilon->0, the exit time of X from an interval is an exponentially distributed random variable and determine its expected value. Due to the heavy-tail nature of the alpha-stable component of L, the results differ strongly from the well known case in which the deterministic dynamical system undergoes purely Gaussian perturbations.

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