Survival probabilities of some iterated processes

Abstract

We study the asymptotic behaviour of the probability that a stochastic process $(Z_t)_{t \geq 0}$ does not exceed a constant barrier up to time $T$ (the so called survival probability) when Z is the composition of two independent processes $(X_t)_{t \in I}$ and $(Y_t)_{t \geq 0}$. To be precise, we consider $(Z_t)_{t \geq 0}$ defined by $Z_t = X \circ \abs{Y_t}$ when $I = [0,\infty)$ and $Z_t = X \circ Y_t$ when $I = \mathbb{R}$. For continuous self-similar processes $(Y_t)_{t \geq 0}$, 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 L\'{e}vy process (possibly two-sided if $I = \mathbb{R}$) and $Y$ being a L\'{e}vy process or random walk under suitable moments conditions.