Lévy matters III. Lévy-type processes: construction, approximation and sample path properties.

*(English)*Zbl 1384.60004
Lecture Notes in Mathematics 2099. Lévy Matters. Cham: Springer (ISBN 978-3-319-02683-1/pbk; 978-3-319-02684-8/ebook). xviii, 199 p. (2013).

This is the third in a series of monographs dedicated to Lévy processes and related questions. The previous volumes are [O. E. Barndorff-Nielsen (ed.) et al., Lévy matters I. Recent progress in theory and applications: foundations, trees and numerical issues in finance. With a short biography of Paul Lévy by Jean Jacod. Berlin: Springer (2010; Zbl 1197.60001); S. Cohen et al., Lévy matters II. Recent progress in theory and applications: fractional Lévy fields, and scale functions. Berlin: Springer (2012; Zbl 1252.60001)].
This volume is dedicated to Lévy-type processes, which are Markov processes satisfying the identity
\[
\lim_{t\to 0} {1- E^x ( e^{i\xi (X_t-x)}) \over t}= q(x,\xi)
\]
for a function \( q(x,\xi) \) that is called the symbol of the process \(X\).
A process which admits a symbol behaves locally as a Lévy process and Lévy processes are the Lévy-type processes whose symbol does not depend on \(x\).
The first chapter recalls known material on Feller semi-groups and Feller processes, among which are the Hille-Yosida-Ray theorem and the connexion with martingale problems. Chapter 2 is devoted to the study of the generator of Feller semi-groups and to the particular convolution semi-groups, which are the ones of Lévy processes. The connexion to negative definite functions is explained. The Courrège theorem, which relates Feller generators to a Lévy-Khintchin type formula, is stated and proved and the parameters of the Lévy-Khintchin formula (“the characteristic triplet”) are related to Itô formula coefficients. In Chapter 3, sufficient conditions for a function \(q\) to be the symbol of a Markov process are given. Several approaches are proposed using different tools from probability theory, notably: the Hille-Yosida construction, stochastic differential equations, Dirichlet forms and martingale problems. Chapter 4 is short and summarizes different transformations of Feller processes: the so-called \(h\)-processes, subordination in the sense of Bochner and the Feynman-Kac semigroups. Chapter 5 is dedicated to path properties of Lévy-type processes: the authors give estimates on the distribution function, for small and large values, of the exit time from a ball of a Feller process. The Hausdorff dimension of its range in terms of behavior of the variation of the symbol function is also studied. Some comparison theorems of the sample paths with power functions, together with their strong variations for small times and for large times, are given. In Chapter 6, the coupling property for Feller processes as well as the recurrence and transience of a Lévy-type process are introduced. In Chapter 7, the authors give some results on the approximation of the semi-group, or its generator, with a view towards computer simulations. At the end of the book, one can find a very rich bibliography and a complete index.
In conclusion, this monograph is nice to read and is a successful compromise between a survey and an exhaustive treatment.

Reviewer: Sonia Fourati (MR3156646)

##### MSC:

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

60G17 | Sample path properties |

60G51 | Processes with independent increments; Lévy processes |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60J25 | Continuous-time Markov processes on general state spaces |

60J35 | Transition functions, generators and resolvents |

60J75 | Jump processes (MSC2010) |