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 Lévy processes and applications to Russian, American and Canadized options

We consider spectrally negative Lévy 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 Lévy 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^½ (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 $\infty$ 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 Lévy 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