1,296 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
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
Distributional representations and dominance of a L\'{e}vy process over its maximal jump processes
Distributional identities for a L\'evy process , its quadratic variation
process and its maximal jump processes, are derived, and used to make
"small time" (as ) asymptotic comparisons between them. The
representations are constructed using properties of the underlying Poisson
point process of the jumps of . Apart from providing insight into the
connections between , , and their maximal jump processes, they enable
investigation of a great variety of limiting behaviours. As an application, we
study "self-normalised" versions of , that is, after division by
, or by . Thus, we
obtain necessary and sufficient conditions for
and to converge in probability to 1, or to
, as , so that is either comparable to, or dominates,
its largest jump. The former situation tends to occur when the singularity at 0
of the L\'evy measure of is fairly mild (its tail is slowly varying at 0),
while the latter situation is related to the relative stability or attraction
to normality of at 0 (a steeper singularity at 0). An important component
in the analyses is the way the largest positive and negative jumps interact
with each other. Analogous "large time" (as ) versions of the
results can also be obtained.Comment: Published at http://dx.doi.org/10.3150/15-BEJ731 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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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