文档介绍：Andreas E. Kyprianou
Introductory Lectures on
Fluctuations of L´evy Processes
with Applications
July 29, 2005
Springer
Berlin Heidelberg NewYork
Hong Kong London
Milan Paris Tokyo
Preface
In 2003 I began teaching a course entitled L´evy processes on the Amsterdam-
Utrecht masters programme in stochastics and financial mathematics. Quite
naturally, I wanted to expose to my students my own interests in L´evy pro-
cesses. That is the role that certain subtle behaviour concerning their fluctu-
ations explain different types of phenomena appearing in a number of clas-
sical models of applied probability (as a general rule that does not necessarily
include mathematical finance). Indeed, recent developments in the theory of
L´evy processes, in particular concerning path fluctuation, has offered the clar-
ity required to revisit classical applied probability models and improve on well
established and fundamental results. Results which were initially responsible
for the popularity of the models themselves.
Whilst giving the course I wrote some lecture notes which have now ma-
tured into this text. Given the audience of students, who were either engaged
in their ‘afstudeerfase’(equivalent to masters for the . or those European
countries which have engaged in that system) or just starting a ., these
lecture notes were originally written with the restriction that the mathematics
used would not surpass the level that they should in principle have reached.
That roughly means the following. Experience to the level of third year or
fourth year university courses delivered by a mathematics department on
- foundational real plex analysis,
- elementary functional analysis (specifically basic facts about Lp spaces),
- measure theory, integration theory and measure theoretic probability theory,
- elements of the classical theory of Markov processes, stopping times and the
Strong Markov Property, Poisson processes and renewal processes,
- an understand