Universality of the asymptotics of the one-sided exit problem for integrated processes

Abstract

We consider the one-sided exit problem for (fractionally) integrated random walks and L\'evy processes. We prove that the rate of decrease of the non-exit probability -- the so-called survival exponent -- is universal in this class of processes. In particular, the survival exponent can be inferred from the (fractionally) integrated Brownian motion. This, in particular, extends Sinai's result on the survival exponent for the integrated simple random walk to general random walks with some finite exponential moment. Further, we prove existence and monotonicity of the survival exponent of fractionally integrated processes. We show that this exponent is related to a constant appearing in the study of random polynomials.