Probability and Mathematical Statistics: Vol. 34, Fasc. 2

PERSISTENCE OF SOME ITERATED PROCESSES

Christoph Baumgarten

Abstract: We study the asymptotic behaviour of the probability that a stochastic process $(Z ) tt≥0$ does not exceed a constant barrier up to time $T$ (a so-called persistence probability) when $Z$ is the composition of two independent processes $(X ) tt∈I$ and $(Y ) tt≥0$ . To be precise, we consider $(Z ) tt≥0$ defined by $Z = X ∘|Y | t t$ if $I = [0,∞ )$ and $Z = X ∘Y t t$ if $I = ℝ$ .

For continuous self-similar processes $(Y ) tt≥0$ , the rate of decay of persistence probability for $Z$ can be inferred directly from the persistence probability of $X$ and the index of self-similarity of $Y$ . As a corollary, we infer that the persistence probability for iterated Brownian motion decays asymptotically like $T-1∕2$ .

If $Y$ is discontinuous, the range of $Y$ possibly contains gaps, which complicates the estimation of the persistence probability. We determine the polynomial rate of decay for $X$ being a Lévy process (possibly two-sided if $I = ℝ$ ) or a fractional Brownian motion and $Y$ being a Lévy process or random walk under suitable moment conditions.

2000 AMS Mathematics Subject Classification: Primary: 60G99; Secondary: 60G18, 60G50, 60G51, 60J65.

Keywords and phrases: Iterated process, one-sided barrier problem, one-sided exit problem, persistence, persistence probability, small deviation probability, survival probability.