an:05553180
Zbl 1200.60001
Applebaum, David
L??vy processes and stochastic calculus. 2nd ed
EN
Cambridge Studies in Advanced Mathematics 116. Cambridge: Cambridge University Press (ISBN 978-0-521-73865-1/pbk). xxx, 460~p. (2009).
00403693
2009
b
60-02 60H10 60H05 60G44 60G55 60G57 60G52
stochastic integration for L??vy processes; stochastic ordinary differential equations for L??vy processes; regular variation; Lyapunov function for SDE
This volume is the second edition of the author's celebrated monography about ``L??vy Processes and Stochastic Calculus'' [Cambridge Studies in Advanced Mathematics 93. Cambridge: Cambridge University Press (2004; Zbl 1073.60002)], which has become one of the standard references in the field. The organization of the book in six chapters has been maintained to a large extent, nevertheless quite some material had been added.
In the introductory Chapter 1 about L??vy processes, there is a new subsection about regularly varying functions, their representations and regularly varying L??vy processes. Chapter 2 ``Martingales, stopping times and random measures'' includes new material about finite variation L??vy processes and the existence of moments. Chapter 4 on stochastic integration contains new estimates of the stochastic integral with respect to a L??vy process due to recent results by Kunita. Major extensions are provided to Chapter 5 ``Exponential martingales, change of measure and and financial applications''. The author includes the proof of the It?? and martingale representation theorem for general complex-valued \(L^2\) martingales. In the sequel this is applied to multiple (L??vy-) Wiener-It?? integrals and a proof of the Wiener chaos decomposition. In Section 5.5 the author gives a brief introduction to Mallavin's calculus for the Brownian case. A slight extension of the Black-Scholes formula with jumps is also provided. Chapter 6 has also seen several revisions. In particular the proof of Kunita's theorem on the continuous dependence on the initial data had been streamlined exploiting the before mentioned estimates on the L??vy stochastic integral. A new section on Lyapunov functions for SDE provides tools for asymptotic stability analysis of the solutions.
This textbook on L??vy processes continues to be an indispensable exposition equally fit for students and scientists to get acquainted with the enourmously rich theory of L??vy processes and their SDE along this highly efficient and elegantly written mathematical text.
Michael H??gele (Berlin)
Zbl 1073.60002