# Fluctuation Theory for Lévy Processes

@inproceedings{Doney2007FluctuationTF, title={Fluctuation Theory for L{\'e}vy Processes}, author={Ronald A. Doney}, year={2007} }

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.

## 182 Citations

Lévy Processes at First Passage

- Mathematics, Biology
- 2014

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

- Mathematics
- 2009

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

- Mathematics
- 2013

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

- MathematicsJ. Lond. Math. Soc.
- 2012

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*

- Mathematics
- 2015

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…

Stochastic periodic solutions of stochastic differential equations driven by Lévy process

- Mathematics
- 2015

Suprema of Lévy processes

- Mathematics
- 2011

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

- Mathematics
- 2007

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

- Mathematics
- 2016

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

- Mathematics
- 2014

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.

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