Levy Processes and Stochastic Calculus

By: David Applebaum

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eText | 8 July 2004 | Edition Number 1

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L?vy 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 L?vy processes. The second part develops the stochastic calculus for L?vy 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 L?vy processes. The book introduces all the tools that are needed for the stochastic approach to option pricing, including It?'s formula, Girsanov's theorem and the martingale representation theorem.