Persistence of integrated stable processes

Abstract

We compute the persistence exponent of the integral of a stable Lévy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Shi (Lower tails of some integrated processes. In: Small deviations and related topics (problem panel 2003). Along the way, we investigate the law of the stable process L evaluated at the first time its integral X hits zero, when the bivariate process (X, L) starts from a coordinate axis. This extends classical formulae by McKean (J Math Kyoto Univ 2:227-235, 1963) and Gor'kov (Soviet. Math. Dokl. 16:904-908, 1975) for integrated Brownian motion.

Cramér’s estimate for stable processes with power drift

“…Theorem C is an extension of Theorem A in [12] which dealt with the case µ = 0. In this respect, we should mention that the condition γα > 1 on the drift power is optimal: in the Cauchy case α = γ = 1, the same Theorem A in [12] shows that the lower tail probability exponent depends on µ. Our argument relies in an essential way on the strong Markov property of the bidimensional process {(L (1) t , L t ), t ≥ 0} and is hence specific to the case β = 1.…”

“…Following the notation of [12], we will set θ = ρ α(1 − ρ) + 1 once and for all. The upper bound follows easily from…”

Persistence and Exit Times for Some Additive Functionals of Skew Bessel Processes

“…where H(δ, η, γ, c) is a positive constant, and it remains to prove that we may exchange the integral and the expectation on the left hand-side of (2.18). To do so, it is sufficient by Lemma 1 in [19] to prove that E |X 1 | Integrating twice by parts, we deduce that…”

“…The upper bound is much more involved, and we shall follow the idea of Profeta-Simon [19]. Applying the Markov property and Fubini's theorem, we have for λ > 0 and taking r ∈ 0, δ 2+γ :…”

“…The power decay of Theorem 1 is typical of self-similar processes, but computing the value of θ is generally a difficult task. Other examples for which θ is explicitly known are for instance the fractional Brownian motion ([2, Section 3.3]), stable Lévy processes ([2, Section 2.2]) as well as their integrals [19]. Conversely, it still remains an open problem for instance to find the value of the persistence exponent for the integrated fractional Brownian motion, or for the twice integrated Brownian motion.…”