Lévy Processes and Stochastic Calculus

David Applebaum

doi:10.1017/CBO9780511809781

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.

'The book introduces all the tools that are needed for the stochastic approach to option pricing, including Itô's formula, Girsanov's theorem and the martingale representation theorem.'

Source: L’Enseignement Mathématique

'The monograph provides a good introduction to the subject, the exposition is clear and systematic, the key points and proofs are easy to follow; therefore it can be a valuable guide both as a textbook for graduate students and as a reference for researchers in the field of stochiastic calculus … This book is written with great care and precision. Due to its lucid and comprehensive style of presentation, it will make the theory of Lévy processes accessible to a broad mathematical audience.'

Source: Mathematical Reviews

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Lévy Processes and Stochastic Calculus

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Contents

Frontmatter
pp i-viii

Contents
pp ix-xii

Preface to Second Edition
pp xiii-xiv

Preface
pp xv-xx

Overview
pp xxi-xxviii

Notation
pp xxix-xxx

1 - Lévy processes
pp 1-81

2 - Martingales, stopping times and random measures
pp 82-142

3 - Markov processes, semigroups and generators
pp 143-213

4 - Stochastic integration
pp 214-279

5 - Exponential martingales, change of measure and financial applications
pp 280-353

6 - Stochastic differential equations
pp 354-430

References
pp 431-448

Index of notation
pp 449-453

Subject index
pp 454-460

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