Persistence problems for additive functionals of one-dimensional Markov processes

Abstract

In this article, we consider additive functionals ${\zeta }_t={\int }_0^{t}f(X_s)\mathrm{d}s$ of a càdlàg Markov process $(X_t{)}_{t\ge 0}$ on $\mathbb{R}$. Under some general conditions on the process $(X_t{)}_{t\ge 0}$ and on the function $f$, we show that the persistence probabilities verify $\mathbb{P}({\zeta }_s<z\text{ for all }s\le t)\sim \mathcal{V}(z)\varsigma (t)t^{-\theta }$ as $t\to \infty$, for some (explicit) $\mathcal{V}(\cdot )$, some slowly varying function $\varsigma (\cdot )$ and some $\theta \in (0,1)$. This extends results in the literature, which mostly focused on the case of a self-similar process $(X_t{)}_{t\ge 0}$ (such as Brownian motion or skew-Bessel process) with a homogeneous functional $f$ (namely a pure power, possibly asymmetric). In a nutshell, we are able to deal with processes which are only asymptotically self-similar and functionals which are only asymptotically homogeneous. Our results rely on an excursion decomposition of $(X_t{)}_{t\ge 0}$, together with a Wiener--Hopf decomposition of an auxiliary (bivariate) Lévy process, with a probabilistic point of view. This provides an interpretation for the asymptotic behavior of the persistence probabilities, and in particular for the exponent $\theta$, which we write as $\theta =\rho \beta$, with $\beta$ the scaling exponent of the local time of $(X_t{)}_{t\ge 0}$ at level $0$ and $\rho$ the (asymptotic) positivity parameter of the auxiliary Lévy process.

62 pages, 2 figures

References in corpus (3)

• The lower tail problem for homogeneous functionals of stable processes with no negative jumps
• On the excursion theory for linear diffusions
• An application of Sparre Andersen's fluctuation theorem for exchangeable and sign-invariant random variables