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

TL;DR

This paper demonstrates that the decay rate of the non-exit probability, known as the survival exponent, is universal across (fractionally) integrated processes, extending known results and linking to random polynomial constants.

Contribution

It establishes the universality of the survival exponent for integrated processes and extends Sinai's results to a broader class of random walks.

Findings

Survival exponent is universal for integrated processes.

Extension of Sinai's result to processes with finite exponential moments.

Connection between survival exponent and constants in random polynomial theory.

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.