An Elementary Introduction to Mathematical Finance: Options and Other Topics

This book provides a comprehensive review of generalized Poisson processes and the Cox and compound Cox processes in particular. It consists of 12 chapters. Chapter 1 gives a brief overview of probability theory, and Chapter 2 introduces the Poisson process. Chapter 3 covers asymptotic theory of a class of stochastic processes needed for later chapters. Chapter 4 presents compound Poisson distributions and their properties in great detail. Chapter 5 reviews results in the classical ruin theory, where the aggregate claims process arising from an insurance portfolio is compound Poisson. Chapters 6–10, the central part of the book, extensively discuss the Cox and compound Cox processes with applications in ruin theory. There is also a chapter (Chap. 8) on  nancial applications. Chapter 11 generalizes the classical risk process to incorporate a  uctuating premium rate, and Chapter 12 concerns statistical inference for the generalized risk process. Most of the book is nicely organized and relatively easy to read, although Chapter 4 would be better placed immediately after Chapter 1. Probabilistic properties of generalized Poisson processes and their proofs are well presented. A distinct feature of the book is that a large portion is devoted to asymptotic properties of generalized Poisson processes. The book contains numerous results in this area, including many recent results by the authors and their colleagues. From this aspect, this book is an excellent reference for researchers who are interested in generalized Poisson processes and for practitioners who need to model claims arising from an insurance portfolio. My criticism is on the applications in the book. The applications in insurance are narrowly focused on ruin probabilities, and hence the book’s title is too general. [In this regard, a title similar to that of Asmussen (2000) would be more appropriate. ] Many important aspects uniquely related to risk theory are not discussed. For instance, the surplus immediately before the time of ruin, and the de cit at the time of ruin, are completely ignored. This and other aspects have been discussed by Gerber and Shiu (1998), Lin and Willmot (2000), Rolski et al. (1999), Willmot and Dickson (2003), and others. Further, there is little discussion on how to use generalized Poisson processes to model insurance claims, arguably a much more important application in insurance. Although some results in the book may be useful for this purpose (e.g., the statistical inference in Chap. 12), many results on asymptotic expansions and approximations of risk processes often are not applicable in practice, because insurance companies are more interested in evaluating claims over a relatively short period. Furthermore, the  nancial application in Chapter 8 might not be very useful. First, the chapter does not provide any empirical evidence or references to support the use of the Cox process for stock prices; second, the purpose of modeling stock prices often is not prediction (unless one is performing a so-called “technical analysis”), but rather pricing and hedging respective derivative securities. Unfortunately, the latter is not discussed at all. Nevertheless, this book provides excellent coverage of the generalized Poisson processes. I enjoyed reading it, and learned a great deal from it.