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Lévy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. Here, the author ties these two subjects together, beginning with an introduction to the general theory of Lévy processes, then leading on to develop the stochastic calculus for Lévy processes in a direct and accessible way. This fully revised edition now features a number of new topics. These include: regular variation and subexponential distributions; necessary and sufficient conditions for Lévy processes to have finite moments; characterisation of Lévy processes with finite variation; Kunita's estimates for moments of Lévy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Lévy processes; multiple Wiener-Lévy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Lévy-driven SDEs.

Lévy processes --- Stochastic analysis --- Lévy, Processus de --- Analyse stochastique --- Lévy processes. --- Stochastic analysis. --- Lévy processes --- Lévy, Processus de --- Lévy processes. --- Stochastic integral equations. --- Integral equations --- Random walks (Mathematics) --- Analysis, Stochastic --- Mathematical analysis --- Stochastic processes --- Levy processes.

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The theory of Lévy processes in Lie groups is not merely an extension of the theory of Lévy processes in Euclidean spaces. Because of the unique structures possessed by non-commutative Lie groups, these processes exhibit certain interesting limiting properties which are not present for their counterparts in Euclidean spaces. These properties reveal a deep connection between the behaviour of the stochastic processes and the underlying algebraic and geometric structures of the Lie groups themselves. The purpose of this work is to provide an introduction to Lévy processes in general Lie groups, the limiting properties of Lévy processes in semi-simple Lie groups of non-compact type and the dynamical behavior of such processes as stochastic flows on certain homogeneous spaces. The reader is assumed to be familiar with Lie groups and stochastic analysis, but no prior knowledge of semi-simple Lie groups is required.

Lévy processes. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Random walks (Mathematics) --- Lévy processes --- Lie groups --- Levy processes.

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Stochastic processes --- Credit --- Risk management --- Lévy processes --- Gestion du risque --- Lévy, Processus de --- Management --- Mathematical models --- Modèles mathématiques --- Lévy processes. --- Mathematical models. --- -Risk management --- -Levy processes --- 658.88015195 --- Random walks (Mathematics) --- Insurance --- Borrowing --- Finance --- Money --- Loans --- -Mathematical models --- Lévy processes --- Lévy, Processus de --- Modèles mathématiques

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"This 1996 book is a comprehensive account of the theory of Lévy processes. This branch of modern probability theory has been developed over recent years and has many applications in such areas as queues, mathematical finance and risk estimation. Professor Bertoin has used the powerful interplay between the probabilistic structure (independence and stationarity of the increments) and analytic tools (especially Fourier and Laplace transforms) to give a quick and concise treatment of the core theory, with the minimum of technical requirements. Special properties of subordinators are developed and then appear as key features in the study of the local times of real-valued Lévy processes and in fluctuation theory. Lévy processes with no positive jumps receive special attention, as do stable processes. In sum, this will become the standard reference on the subject for all working probability theorists." [Back cover]

Lévy processes --- Lévy processes --- Lévy, Processus de --- Stochastic processes --- Lévy processes. --- Lévy, Processus de. --- Markov processes. --- Markov, Processus de. --- Lévy, Processus de.

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Le;vy 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 Le;vy 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 Le;vy processes and infinitely divisible distributions. All serious students of random phenomena will find that this book has much to offer.

Stochastic processes --- Lévy processes --- Distribution (Probability theory) --- Lévy, Processus de --- Distribution (Théorie des probabilités) --- 519.282 --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Random walks (Mathematics) --- Lévy processes. --- Distribution (Probability theory). --- Lévy processes --- Lévy, Processus de --- Distribution (Théorie des probabilités)