Fluctuation Theory for Lévy Processes

  title={Fluctuation Theory for L{\'e}vy Processes},
  author={Ronald A. Doney},
Recently there has been renewed interest in fluctuation theory for Levy processes. Inthis brief survey we describe several aspects of this topic, including Wiener-Hopf factorisation,the ladder processes, Spitzer’s condition, the asymptotic behaviour of Levy processes at zero and infinity, and other path properties. Some open problems are also presented. 
Lévy Processes at First Passage
This chapter is devoted to studying how the Wiener–Hopf factorisation can be used to characterise the behaviour of any Levy process at first passage over a fixed level. The case of a subordinator
Explicit identities for Lévy processes associated to symmetric stable processes
In this paper we introduce a new class of Levy processes which we call hypergeometric- stable Levy processes, because they are obtained from symmetric stable processes through several transformations
Small time Chung-type LIL for Lévy processes
We prove Chung-type laws of the iterated logarithm for general Levy processes at zero. In particular, we provide tools to translate small deviation estimates directly into laws of the iterated
A Wiener–Hopf type factorization for the exponential functional of Lévy processes
This work uses and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual MarkOV process.
Small-Maturity Digital Options in Lévy Models: An Analytic Approach*
We prove a small-time Tauberian theorem for transition probabilities of certain Lévy processes. The main assumption is a condition on the asymptotic behavior of the characteristic function. This
Suprema of Lévy processes
In this paper we study the supremum functional Mt=sup0≤s≤tXs, where Xt, t≥0, is a one-dimensional Levy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative
Fluctuation theory and stochastic games for spectrally negative Lévy processes
Levy processes have stationary, independent increments. This seemingly unassuming (defining) property leads to a surprisingly rich class of processes which appear in a large number of applications
On a Small-Time Limit Behaviorof the Probability That a Lévy Process Stays Positive
In this paper, we find analytically the upper and lower limits (as the time parameter tends to zero) of the probability that a Lévy process starting at 0 stays positive. We confine ourselves to the
Exit Problems for Spectrally Negative Processes
This chapter devotees its time to gathering facts about spectrally negative processes, and then to an ensemble of fluctuation identities which are semi-explicit in terms of a class of functions known as scale functions, whose properties the authors shall also explore.


Fluctuation identities for Levy processes and splitting at the maximum
It6's notion of a Poisson point process of excursions is used to give a unified approach to a number of results in the fluctuation theory of LUvy processes, including identities of Pecherskii,
Increase of a lévy process with no positive jumps
We extend a result due to Dvoretzky, Erdos and Kakutani for the Brownian motion, by specifying the class of Levy processes with no positive jumps which possess increase times. Our approach relies on
Increase of Lévy processes
A rather complicated condition is shown to be necessary and sufficient for a Levy process to have points of increase. A much simpler condition is then shown to be sufficient in the general case, and
Wiener – hopf factorization revisited and some applications
A reformulation of the classical Wiener-Hopf factorization for random walks is given; this is applied to the study of the asymptotic behaviour of the ladder variables, the distribution of the maximum
Regularity of the half-line for Lévy processes
Consider a real-valued Levy process X started at 0. One says that 0 is regular for (0, ∞) if X enters (0, ∞) immediately. ROGOZIN proved that 0 is regular for (0, ∞) if X has unbounded variation, and
Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity
We prove some limiting results for a Lévy process Xt as t↓0 or t→∞, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are
Points de croissance des processus de Lévy et théorie générale des processus
Summary. We prove a conjecture of J. Bertoin: a Lévy process has increase times if and only if the integral is finite, where G and H are the distribution functions of the minimum and the maximum of
Increase of stable processes
One says thatt>0 is an increase time for a real-valued path ω if ω stays above the level ω(t) immediately after timet, and below ω(t) immediately before timet. Dvoretzkyet al.,(10) proved that
Hitting probabilities of single points for processes with stationary independent increments
Our purpose is to determine when h(r) is strictly positive, respectively zero. An old and obvious result is that h(r) >0 for all r if Xt is Brownian motion. Somewhat more difficult is the behavior of