Small and Large Time Stability of the Time taken for a L\'evy Process to Cross Curved Boundaries

This paper is concerned with the small time behaviour of a L\'{e}vy process $X$. In particular, we investigate the {\it stabilities} of the times, \Tstarb(r) and \Tbarb(r), at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in\R^2: y\le rt^b, t\ge 0\}$ (one-sided exit), or $\{(t,y)\in\R^2: |y|\le rt^b, t\ge 0\}$ (two-sided exit), $0\le b<1$, as r\dto 0. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^p$. In many instances these are seen to be equivalent to relative stability of the process $X$ itself. The analogous large time problem is also discussed

Stability of the Exit Time for L\'evy Processes

This paper is concerned with the behaviour of a L\'{e}vy process when it crosses over a positive level, $u$, starting from 0, both as $u$ becomes large and as $u$ becomes small. Our main focus is on the time, $\tau_u$, it takes the process to transit above the level, and in particular, on the {\it stability} of this passage time; thus, essentially, whether or not $\tau_u$ behaves linearly as u\dto 0 or $u\to\infty$. We also consider conditional stability of $\tau_u$ when the process drifts to $-\infty$, a.s. This provides information relevant to quantities associated with the ruin of an insurance risk process, which we analyse under a Cram\'er condition

Convolution equivalent L\'{e}vy processes and first passage times

We investigate the behavior of L\'{e}vy processes with convolution equivalent L\'{e}vy measures, up to the time of first passage over a high level u. Such problems arise naturally in the context of insurance risk where u is the initial reserve. We obtain a precise asymptotic estimate on the probability of first passage occurring by time T. This result is then used to study the process conditioned on first passage by time T. The existence of a limiting process as $u\to\infty$ is demonstrated, which leads to precise estimates for the probability of other events relating to first passage, such as the overshoot. A discussion of these results, as they relate to insurance risk, is also given.Comment: Published in at http://dx.doi.org/10.1214/12-AAP879 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Pruitt\u27s Estimates in Banach Space

Pruitt\u27s estimates on the expectation and the distribution of the time taken by a random walk to exit a ball of radius r are extended to the infinite dimensional setting. It is shown that they separate into two pairs of estimates depending on whether the space is type 2 or cotype 2. It is further shown that these estimates characterize type 2 and cotype 2 spaces

Asymptotic Distributions of the Overshoot and Undershoots for the L\'evy Insurance Risk Process in the Cram\'er and Convolution Equivalent Cases

Recent models of the insurance risk process use a L\'evy process to generalise the traditional Cram\'er-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a L\'{e}vy process which drifts to $-\infty$ and satisfies a Cram\'er or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cram\'er case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding these quantities, the "medium-heavy" tailed convolution equivalent model segues into the "light-tailed" Cram\'er model in a natural way. This suggests a usefully expanded flexibility for modelling the insurance risk process. We illustrate this relationship by comparing the asymptotic distributions obtained for the overshoot and undershoots, assuming the L\'evy process belongs to the "GTSC" class