60 research outputs found
Travelling wave solutions to the K-P-P equation at supercritical wave speeds: a parallel to Simon Harris' probabilistic analysis
Recently Harris using probabilistic methods alone has given new proofs for the known existence asymptotics and unique ness of travelling wave solutions to the KPP equation Following in this vein we outline alternative probabilistic proofs for wave speeds exceeding the critical minimal wave speed Speci
cally the analysis is con
ned to the study of additive and multiplicative martingales and the construction of size biased measures on the space of marked trees generated by the branching process This paper also acts as a prelude to its companion Kyprianou b which deals with the more dif
cult case of travelling waves at criticality The importance of these new probabilistic proofs is their generic nature which in principle can be extended to study other types of spatial branching di
usions and associated travelling wave
Potentials of stable processes
For a stable process, we give an explicit formula for the potential measure
of the process killed outside a bounded interval and the joint law of the
overshoot, undershoot and undershoot from the maximum at exit from a bounded
interval. We obtain the equivalent quantities for a stable process reflected in
its infimum. The results are obtained by exploiting a simple connection with
the Lamperti representation and exit problems of stable processes.Comment: 10 page
Exit problems for spectrally negative LĂ©vy processes and applications to Russian, American and Canadized options
We consider spectrally negative LĂ©vy process and determine the joint Laplace trans-
form of the exit time and exit position from an interval containing the origin of the
process reflected in its supremum. In the literature of fluid models, this stopping time
can be identified as the time to buffer-overflow. The Laplace transform is determined
in terms of the scale functions that appear in the two sided exit problem of the given
LĂ©vy process. The obtained results together with existing results on two sided exit
problems are applied to solving optimal stopping problems associated with the pricing
of American and Russian options and their Canadized versions
Upper and lower space-time envelopes for oscillating random walks conditioned to stay positive
We provide integral tests for functions to be upper and lower space
time envelopes for random walks conditioned to stay positive. As a result
we deduce a 'Hartman-Winter' Law of the Iterated Logarithm for random
walks conditioned to stay positive under a third moment assumption.
We also show that under a second moment assumption the conditioned
random walk grows faster than n^œ (log n)^(-(1+e)) for any e > 0. The
results are proved using three key facts about conditioned random walks.
The first is the step distribution obtained in Bertoin and Doney (1994),
the second is the pathwise construction in terms of excursions in Tanaka
(1998) and the third is a new Skorohod type embedding of the conditioned
process in a Bessel-3 process
Precautionary Measures for Credit Risk Management in Jump Models
Sustaining efficiency and stability by properly controlling the equity to
asset ratio is one of the most important and difficult challenges in bank
management. Due to unexpected and abrupt decline of asset values, a bank must
closely monitor its net worth as well as market conditions, and one of its
important concerns is when to raise more capital so as not to violate capital
adequacy requirements. In this paper, we model the tradeoff between avoiding
costs of delay and premature capital raising, and solve the corresponding
optimal stopping problem. In order to model defaults in a bank's loan/credit
business portfolios, we represent its net worth by Levy processes, and solve
explicitly for the double exponential jump diffusion process and for a general
spectrally negative Levy process.Comment: 31 pages, 4 figure
On the harmonic measure of stable processes
Using three hypergeometric identities, we evaluate the harmonic measure of a
finite interval and of its complementary for a strictly stable real L{\'e}vy
process. This gives a simple and unified proof of several results in the
literature, old and recent. We also provide a full description of the
corresponding Green functions. As a by-product, we compute the hitting
probabilities of points and describe the non-negative harmonic functions for
the stable process killed outside a finite interval
Large deviations for clocks of self-similar processes
The Lamperti correspondence gives a prominent role to two random time
changes: the exponential functional of a L\'evy process drifting to
and its inverse, the clock of the corresponding positive self-similar process.
We describe here asymptotical properties of these clocks in large time,
extending the results of Yor and Zani
Queues with LĂ©vy input and hysteretic control
We consider a (doubly) reflected Lévy process where the Lévy exponent is controlled by a hysteretic policy consisting of two stages. In each stage there is typically a different service speed, drift parameter, or arrival rate. We determine the steady-state performance, both for systems with finite and infinite capacity. Thereby, we unify and extend many existing results in the literature, focusing on the special cases of M/G/1 queues and Brownian motion. © The Author(s) 2009
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