Competition between Levy jumps and continuous drift

Motivated by certain birth-death processes with strongly fluctuating birth rates, we consider a level-crossing problem for a random process being a superposition of a continuous drift to the left and jumps to the right. The lengths of the corresponding jumps follow a one-sided extreme Levy-law of index alpha. We concentrate on the case 0 < alpha < 1 and discuss the probability of crossing a left boundary ("extinction"). We show that this probability decays exponentially as a function of the initial distance to the boundary. Such behavior is universal for all alpha < 1 and is exemplified by an exact solution for the special case a The splitting probabilities are also discussed. (C) 2003 Elsevier B.V. All rights reserved.