PERSISTENCE OF SOME ITERATED PROCESSES
Abstract: We study the asymptotic behaviour of the probability that a stochastic process
does not exceed a constant barrier up to time (a so-called persistence probability) when
is the composition of two independent processes and . To be precise,
we consider defined by if and if
.
For continuous self-similar processes , the rate of decay of persistence
probability for can be inferred directly from the persistence probability of and the
index of self-similarity of . As a corollary, we infer that the persistence probability for
iterated Brownian motion decays asymptotically like .
If is discontinuous, the range of possibly contains gaps, which complicates the
estimation of the persistence probability. We determine the polynomial rate of decay for
being a Lévy process (possibly two-sided if ) or a fractional Brownian motion
and being a Lévy process or random walk under suitable moment conditions.
2000 AMS Mathematics Subject Classification: Primary: 60G99; Secondary: 60G18,
60G50, 60G51, 60J65.
Keywords and phrases: Iterated process, one-sided barrier problem, one-sided exit
problem, persistence, persistence probability, small deviation probability, survival
probability.