Stochastic analysis in discrete and continuous settings: With normal martingales

Nicolas Privault (auth.)3642023797, 9783642023798

This volume gives a unified presentation of stochastic analysis for continuous and discontinuous stochastic processes, in both discrete and continuous time. It is mostly self-contained and accessible to graduate students and researchers having already received a basic training in probability. The simultaneous treatment of continuous and jump processes is done in the framework of normal martingales; that includes the Brownian motion and compensated Poisson processes as specific cases. In particular, the basic tools of stochastic analysis (chaos representation, gradient, divergence, integration by parts) are presented in this general setting. Applications are given to functional and deviation inequalities and mathematical finance.

Table of contents :
Front Matter….Pages 1-7
Introduction….Pages 1-6
The Discrete Time Case….Pages 7-58
Continuous Time Normal Martingales….Pages 59-112
Gradient and Divergence Operators….Pages 113-130
Annihilation and Creation Operators….Pages 131-160
Analysis on the Wiener Space….Pages 161-194
Analysis on the Poisson Space….Pages 195-246
Local Gradients on the Poisson Space….Pages 247-280
Option Hedging in Continuous Time….Pages 281-293
Appendix….Pages 295-300
Back Matter….Pages 1-15