arXiv subjects

Ronald A. Doney

Publications and source records attributed to Ronald A. Doney.

2 recordsLinked to original sources

arXiv2014

Asymptotic behaviour of first passage time distributions for subordinators

In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt in 1987. In the particular case of a subordinator in the domain of attraction of a stable distribution the results are the analogue of the results obtained by the authors for non-monotone Lévy processes. For subordinators an approach different to that used for non-monotone Lévy processes is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential.

math.PR↗

arXiv2008

Passage of Lévy Processes across Power Law Boundaries at Small Times

We wish to characterise when a Lévy process $X_t$ crosses boundaries like $t^κ$, $κ>0$, in a one or two-sided sense, for small times $t$; thus, we enquire when $\limsup_{t\downarrow 0}|X_t|/t^κ$, $\limsup_{t\downarrow 0}X_t/t^κ$ and/or $\liminf_{t\downarrow 0}X_t/t^κ$ are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of $κ>0$. Often (for many values of $κ$), when the limsups are finite a.s., they are in fact zero, as we show, but the limsups may in some circumstances take finite, nonzero, values, a.s. In general, the process crosses one or two-sided boundaries in quite different ways, but surprisingly this is not so for the case $κ=1/2$. An integral test is given to distinguish the possibilities in that case. Some results relating to other norming sequences for $X$, and when $X$ is centered at a nonstochastic function, are also given.

math.PR↗