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

Aurzada, Frank; Dereich, Steffen

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 236-251.

Nous considérons le problème unilatéral de sortie - ou problème unilatéral de barrière - pour des intégrales ($\alpha$-fractionnelles) de marches aléatoires et de processus de Lévy. Notre résultat principal est l’existence d’une fonction positive, décroissante $\alpha \mapsto \theta \left(\alpha \right)$ telle que la probabilité qu’une intégrale d’un processus de Lévy $\alpha$-fractionnel quelconque (ou marche aléatoire) avec certains moments exponentiels finis reste en dessous d’un niveau fixe jusqu’à un temps $T$ se comporte comme $T^{-\theta \left(\alpha \right)+\mathrm{o}\left(1\right)}$ pour $T$ grand. Nous analysons aussi la possibilité de remplacer le niveau fixe par une barrière différente qui satisfait certaines conditions de croissance (marge mouvante). Cela, en particulier, étend le résultat de Sinai sur l’exposant de survie d’une marche aléatoire simple intégrée à des marches aléatoires générales de moment exponentiel fini.

We consider the one-sided exit problem - also called one-sided barrier problem - for ($\alpha$-fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function $\alpha \mapsto \theta \left(\alpha \right)$ such that the probability that any $\alpha$-fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^{-\theta \left(\alpha \right)+\mathrm{o}\left(1\right)}$ for large $T$. We also investigate when the fixed level can be replaced by a different barrier satisfying certain growth conditions (moving boundary). This, in particular, extends Sinai’s result on the survival exponent $\theta \left(1\right)=1/4$ for the integrated simple random walk to general random walks with some finite exponential moment.

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