2011/06/15 by Baumgarten, Christoph
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1106.2999
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 (the so called survival probability) when Z is the composition of two independent processes (Xt)t ∈ I and (Yt)t ≥ 0. To be precise, we consider (Zt)t ≥ 0 defined by Zt = X ∘ \absYt when I = [0,∞) and Zt = X ∘ Yt when I = ℝ. For continuous self-similar processes (Yt)t ≥ 0, the rate of decay of survival probability for Z can be inferred directly from the survival probability of X and the index of self-similarity of Y. As a corollary, we obtain that the survival 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 survival probability. We determine the polynomial rate of decay for X being a Lévy process (possibly two-sided if I = ℝ) and Y being a Lévy process or random walk under suitable moments conditions.