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

作者：Aurzada Frank^*; Dereich Steffen

来源：Annales de l Institut Henri Poincare-Probabilites et Statistiques, 2013, 49(1): 236-251.

DOI：10.1214/11-AIHP427

• Integrated Brownian motion
• Integrated Levy process
• Integrated random walk
• Lower tail probability
• Moving boundary
• One-sided barrier problem
• One-sided exit problem
• Persistence probabilities
• Survival exponent

We consider the one-sided exit problem - also called one-sided barrier problem - for (alpha-fractionally) integrated random walks and Levy processes. %26lt;br%26gt;Our main result is that there exists a positive, non-increasing function alpha %26lt;bar right arrow%26gt; theta(alpha) such that the probability that any alpha-fractionally integrated centered Levy processes (or random walk) with some finite exponential moment stays below a fixed level until time T behaves as T-theta(alpha)+o(1) for large T. We also investigate when the fixed level can be replaced by a different barrier satisfying certain growth conditions (moving boundary). %26lt;br%26gt;This, in particular, extends Sinai%26apos;s result on the survival exponent theta (1) = 1/4 for the integrated simple random walk to general random walks with some finite exponential moment.

• 出版日期2013-2