2,065 research outputs found
Stochastic bounds for Levy processes
Using the Wiener-Hopf factorization, it is shown that it is possible to bound
the path of an arbitrary Levy process above and below by the paths of two
random walks. These walks have the same step distribution, but different random
starting points. In principle, this allows one to deduce Levy process versions
of many known results about the large-time behavior of random walks. This is
illustrated by establishing a comprehensive theorem about Levy processes which
converge to \infty in probability.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000031
Overshoots and undershoots of L\'{e}vy processes
We obtain a new fluctuation identity for a general L\'{e}vy process giving a
quintuple law describing the time of first passage, the time of the last
maximum before first passage, the overshoot, the undershoot and the undershoot
of the last maximum. With the help of this identity, we revisit the results of
Kl\"{u}ppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766--1801]
concerning asymptotic overshoot distribution of a particular class of L\'{e}vy
processes with semi-heavy tails and refine some of their main conclusions. In
particular, we explain how different types of first passage contribute to the
form of the asymptotic overshoot distribution established in the aforementioned
paper. Applications in insurance mathematics are noted with emphasis on the
case that the underlying L\'{e}vy process is spectrally one sided.Comment: Published at http://dx.doi.org/10.1214/105051605000000647 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Passage of L\'evy Processes across Power Law Boundaries at Small Times
We wish to characterise when a L\'{e}vy process crosses boundaries like
, , in a one or two-sided sense, for small times ; thus,
we enquire when ,
and/or are almost surely (a.s.) finite or infinite. Necessary and
sufficient conditions are given for these possibilities for all values of
. Often (for many values of ), 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 . An integral test is given to
distinguish the possibilities in that case. Some results relating to other
norming sequences for , and when is centered at a nonstochastic
function, are also given
Passage of Lévy Processes across Power Law Boundaries at Small Times
We wish to characterize when a Lévy process Xt crosses boundaries like tκ, κ > 0, in a one- or two-sided sense, for small times t; thus, we inquire when lim.supt↓0 |Xt|/tκ, lim supt↓0, Xt/tκ and/or lim inft↓0 Xt/tκ are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of κ > 0. This completes and extends a line of research, going back to Blumenthal and Getoor in the 1960s. Often (for many values of κ), when the lim sups are finite a.s., they are in fact zero, but the lim sups 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, where a new kind of analogue of an iterated logarithm law with a square root boundary is derived. An integral test is given to distinguish the possibilities in that case.Supported in part by ARC Grants DP0210572 and DP0664603
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