Persistence of some iterated processes
Słowa kluczowe:Iterated process, one-sided barrier problem, one-sided exit problem, persistence, persistence probability, small deviation probability, survival probability
We study the asymptotic behaviour of the probability that a stochastic process Ztt≥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 Xtt∈I and Ytt≥0. To be precise, we consider Ztt≥0 defined by Zt = X ◦ |Yt| if I = [0;∞ and Zt = X ◦ Yt if I = ℜ. For continuous self-similar processes Ytt≥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.