Persistence of some iterated processes

• Autorzy: Baumgarten C.
• Języki publikacji: EN

Abstrakty

EN

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

Słowa kluczowe

EN

iterated process; one-sided barrier problem; one-sided exit problem; persistence; persistence probability; small deviation probability; survival probability

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2014
• Tom: Vol. 34, Fasc. 2
• Strony: 293--316
• Opis fizyczny: Bibliogr. 25 poz.

Twórcy

autor

Baumgarten C.

• Institut für Mathematische Stochastik, Technische Universität Braunschweig, Pockelsstr. 14, 38106 Braunschweig, Germany

Bibliografia

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Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).