Competition between Lévy 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 Lévy-law of index α. We concentrate on the case 0<α<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 α<1 and is exemplified by an exact solution for the special case . The splitting probabilities are also discussed.