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Persistence of some iterated processes

Baumgarten C.

Probability and Mathematical Statistics

2014

Vol. 34, Fasc. 2

293--316

We study the asymptotic behaviour of the probability that a stochastic process (Zt)t ≥ 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 (Xt)t ϵ I and (Yt)t ≥ 0. To be precise, we consider (Zt)t ≥ 0 defined by Zt = X ◦ |Yt| if I = [0, ∞) and Zt = X ◦ Yt if I = R. For continuous self-similar processes (Yt)t ≥ 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.