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Lévy Processes and Stochastic Calculus (Cambridge Studies in Advanced Mathematics): David Applebaum

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. For the first time in a book, Applebaum ties the two subjects together. He begins with an introduction to the general theory of Lévy processes. The second part develops the stochastic calculus for Lévy processes in a direct and accessible way. En route, the reader is introduced to important concepts in modern probability theory, such as martingales, semimartingales, Markov and Feller processes, semigroups and generators, and the theory of Dirichlet forms. There is a careful development of stochastic integrals and stochastic differential equations driven by Lévy processes. 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.

For first time in book form, develops stochastic integrals and stochastic differential equations driven by Levy processes, including introduction to the theory of Dirichlet forms • Discussion of all the tools which are needed for the stochastic approach to option pricing, including Itô’s formula, Girsanov’s theorem and the martingale representation theorem • An introduction to option pricing with particular reference to incomplete markets

Contents

1. Introduction;

2. Lévy processes;

3. Martingales, stopping times and random measures;

4. Markov processes, semigroups and generators;

5. Stochastic integration;

6. Exponential martingales, change of measure and financial applications;

7. Stochastic differential equations;

Notation;

Bibliography;

Index.

Review

'...the monograph closes the gap between classical textbooks on stochastic analysis where either Brownian motion or general semimartingales are considered. … Besides standard results on existence and uniqueness of a solution and its Markov property, more advanced concepts are presented, such as representation of the solutions as Feller processes and as a stochastic flow.'

- Zentralblatt MATH

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