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This accessible introduction to the theory of stochastic
processes emphasizes Levy processes and Markov processes. It gives
a thorough treatment of the decomposition of paths of processes
with independent increments (the Levy-Ito decomposition). It also
contains a detailed treatment of time-homogeneous Markov processes
from the viewpoint of probability measures on path space. In
addition, 70 exercises and their complete solutions are
included."
Over the past 10-15 years, we have seen a revival of general Levy '
processes theory as well as a burst of new applications. In the
past, Brownian motion or the Poisson process have been considered
as appropriate models for most applications. Nowadays, the need for
more realistic modelling of irregular behaviour of phen- ena in
nature and society like jumps, bursts, and extremeshas led to a
renaissance of the theory of general Levy ' processes. Theoretical
and applied researchers in elds asdiverseas
quantumtheory,statistical
physics,meteorology,seismology,statistics, insurance, nance, and
telecommunication have realised the enormous exibility of Lev ' y
models in modelling jumps, tails, dependence and sample path
behaviour. L' evy processes or Levy ' driven processes feature slow
or rapid structural breaks, extremal behaviour, clustering, and
clumping of points. Toolsandtechniquesfromrelatedbut disctinct
mathematical elds, such as point processes, stochastic
integration,probability theory in abstract spaces, and differ- tial
geometry, have contributed to a better understanding of Le 'vy jump
processes. As in many other elds, the enormous power of modern
computers has also changed the view of Levy ' processes. Simulation
methods for paths of Levy ' p- cesses and realisations of their
functionals have been developed. Monte Carlo simulation makes it
possible to determine the distribution of functionals of sample
paths of Levy ' processes to a high level of accuracy.
This accessible introduction to the theory of stochastic
processes emphasizes Levy processes and Markov processes. It gives
a thorough treatment of the decomposition of paths of processes
with independent increments (the Levy-Ito decomposition). It also
contains a detailed treatment of time-homogeneous Markov processes
from the viewpoint of probability measures on path space. In
addition, 70 exercises and their complete solutions are
included."
This book deals with topics in the area of Levy processes and
infinitely divisible distributions such as Ornstein-Uhlenbeck type
processes, selfsimilar additive processes and multivariate
subordination. These topics are developed around a decreasing chain
of classes of distributions Lm, m = 0,1,..., , from the class L0 of
selfdecomposable distributions to the class L generated by stable
distributions through convolution and convergence. The book is
divided into five chapters. Chapter 1 studies basic properties of
Lm classes needed for the subsequent chapters. Chapter 2 introduces
Ornstein-Uhlenbeck type processes generated by a Levy process
through stochastic integrals based on Levy processes. Necessary and
sufficient conditions are given for a generating Levy process so
that the OU type process has a limit distribution of Lm class.
Chapter 3 establishes the correspondence between selfsimilar
additive processes and selfdecomposable distributions and makes a
close inspection of the Lamperti transformation, which transforms
selfsimilar additive processes and stationary type OU processes to
each other. Chapter 4 studies multivariate subordination of a
cone-parameter Levy process by a cone-valued Levy process. Finally,
Chapter 5 studies strictly stable and Lm properties inherited by
the subordinated process in multivariate subordination. In this
revised edition, new material is included on advances in these
topics. It is rewritten as self-contained as possible. Theorems,
lemmas, propositions, examples and remarks were reorganized; some
were deleted and others were newly added. The historical notes at
the end of each chapter were enlarged. This book is addressed to
graduate students and researchers in probability and mathematical
statistics who are interested in learning more on Levy processes
and infinitely divisible distributions.
Levy processes are rich mathematical objects and constitute perhaps
the most basic class of stochastic processes with a continuous time
parameter. This book is intended to provide the reader with
comprehensive basic knowledge of Levy processes, and at the same
time serve as an introduction to stochastic processes in general.
No specialist knowledge is assumed and proofs are given in detail.
Systematic study is made of stable and semi-stable processes, and
the author gives special emphasis to the correspondence between
Levy processes and infinitely divisible distributions. All serious
students of random phenomena will find that this book has much to
offer. Now in paperback, this corrected edition contains a brand
new supplement discussing relevant developments in the area since
the book's initial publication."
Lévy processes are rich mathematical objects and constitute perhaps the most basic class of stochastic processes with a continuous time parameter. This book provides the reader with comprehensive basic knowledge of Lévy processes, and at the same time introduces stochastic processes in general. No specialist knowledge is assumed and proofs and exercises are given in detail. The author systematically studies stable and semi-stable processes and emphasizes the correspondence between Lévy processes and infinitely divisible distributions. All serious students of random phenomena will benefit from this volume.
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