Lévy processes and stochastic calculus.

*(English)*Zbl 1073.60002
Cambridge Studies in Advanced Mathematics 93. Cambridge: Cambridge University Press (ISBN 0-521-83263-2/hbk; 0-511-20761-1/ebook). xxiv, 384 p. (2004).

The monograph under review deals with stochastic analysis for Lévy processes. Thus the monograph closes the gap between classical text books on stochastic analysis where either Brownian motion or general semimartingales are considered. Although most of the theory on stochastic integration with respect to Lévy processes is covered by the general theory the focus on Lévy processes avoids mathematical sophistication and allows to work within a simpler framework which might facilitate the studies on the general case. In contrast to classical books on Lévy processes the monograph under review aims at stochastic calculus and treats the specific theory on Lévy processes only in minor detail.

The monograph can be divided into two parts: the first one is a comprehensive introduction to probability theory, stochastic processes and Markov processes with emphasis on Lévy processes. The second part is devoted to stochastic analysis with respect to Lévy processes. As the first part serves as a review rather than an introduction of the basic concepts the reader should be familiar with probability and measure theory. Also some knowledge in functional analysis would be an advantage. A more detailed description is the following: the first chapter is a short review on probability and measure theory, including infinite divisibility, and a basic introduction of Lévy processes. The second chapter deals with semimartingales with emphasis on Lévy processes. The third chapter introduces the general theory of Markov processes, e.g., semigroups and their generators, and Dirichlet forms. Again, the results are applied to Lévy processes. The main part begins with Chapter 4: the stochastic integral with respect to Lévy processes is introduced and the Itô formula is derived. Briefly, the Stratonovitch and Marcus integrals as well as backward stochastic integrals are presented. The following chapter continues with some deeper results of stochastic calculus, e.g. exponential martingales, martingale representation and Girsanov’s theorem. These results are applied to mathematical finance, e.g., modelling asset pricing. The last chapter is concerned with stochastic differential equations driven by Lévy processes. 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.

The monograph can be divided into two parts: the first one is a comprehensive introduction to probability theory, stochastic processes and Markov processes with emphasis on Lévy processes. The second part is devoted to stochastic analysis with respect to Lévy processes. As the first part serves as a review rather than an introduction of the basic concepts the reader should be familiar with probability and measure theory. Also some knowledge in functional analysis would be an advantage. A more detailed description is the following: the first chapter is a short review on probability and measure theory, including infinite divisibility, and a basic introduction of Lévy processes. The second chapter deals with semimartingales with emphasis on Lévy processes. The third chapter introduces the general theory of Markov processes, e.g., semigroups and their generators, and Dirichlet forms. Again, the results are applied to Lévy processes. The main part begins with Chapter 4: the stochastic integral with respect to Lévy processes is introduced and the Itô formula is derived. Briefly, the Stratonovitch and Marcus integrals as well as backward stochastic integrals are presented. The following chapter continues with some deeper results of stochastic calculus, e.g. exponential martingales, martingale representation and Girsanov’s theorem. These results are applied to mathematical finance, e.g., modelling asset pricing. The last chapter is concerned with stochastic differential equations driven by Lévy processes. 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.

Reviewer: Markus Riedle (Berlin)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G44 | Martingales with continuous parameter |

60Hxx | Stochastic analysis |

60G51 | Processes with independent increments; Lévy processes |

60G52 | Stable stochastic processes |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

60G57 | Random measures |