528 research outputs found
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
. In particular, we investigate the {\it stabilities} of the times,
\Tstarb(r) and \Tbarb(r), at which , started with , first leaves
the space-time regions (one-sided exit),
or (two-sided exit), , 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
. In many instances these are seen to be equivalent to relative stability
of the process 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, , starting from 0, both as becomes large
and as becomes small. Our main focus is on the time, , 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 behaves
linearly as u\dto 0 or . We also consider conditional stability
of when the process drifts to , 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 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
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
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