Simulation and Inference for Stochastic Differential Equations

Stochastic di?erential equations model stochastic evolution as time evolves. These models have a variety of applications in many disciplines and emerge naturally in the study of many phenomena. Examples of these applications are physics (see, e. g. , [176] for a review), astronomy [202], mechanics [147], economics [26], mathematical ?nance [115], geology [69], genetic analysis (see, e. g. , [110], [132], and [155]), ecology [111], cognitive psychology (see, e. g. , [102], and [221]), neurology [109], biology [194], biomedical sciences [20], epidemi- ogy [17], political analysis and social processes [55], and many other ?elds of science and engineering. Although stochastic di?erential equations are quite popular models in the above-mentioned disciplines, there is a lot of mathem- ics behind them that is usually not trivial and for which details are not known to practitioners or experts of other ?elds. In order to make this book useful to a wider audience, we decided to keep the mathematical level of the book su?ciently low and often rely on heuristic arguments to stress the underlying ideas of the concepts introduced rather than insist on technical details. Ma- ematically oriented readers may ?nd this approach inconvenient, but detailed references are always given in the text. As the title of the book mentions, the aim of the book is twofold.

This book is unique because of its focus on the practical implementation of the simulation and estimation methods presented. The book will be useful to practitioners and students with only a minimal mathematical background because of the many R programs, and to more mathematically-educated practitioners.

Many of the methods presented in the book have not been used much in practice because the lack of an implementation in a unified framework. This book fills the gap.

With the R code included in this book, a lot of useful methods become easy to use for practitioners and students. An R package called "sde" provides functions with easy interfaces ready to be used on empirical data from real life applications. Although it contains a wide range of results, the book has an introductory character and necessarily does not cover the whole spectrum of simulation and inference for general stochastic differential equations.

The book is organized into four chapters. The first one introduces the subject and presents several classes of processes used in many fields of mathematics, computational biology, finance and the social sciences. The second chapter is devoted to simulation schemes and covers new methods not available in other publications. The third one focuses on parametric estimation techniques. In particular, it includes exact likelihood inference, approximated and pseudo-likelihood methods, estimating functions, generalized method of moments, and other techniques. The last chapter contains miscellaneous topics like nonparametric estimation, model identification and change point estimation. The reader who is not an expert in the R language will find a concise introduction to this environment focused on the subject of the book. A documentation page is available at the end of the book for each R function presented in the book.

Stefano M. Iacus is associate professor of Probability and Mathematical Statistics at the University of Milan, Department of Economics, Business and Statistics. He has a PhD in Statistics at Padua University, Italy and in Mathematics at Université du Maine, France.

He is a member of the R Core team for the development of the R statistical environment, Data Base manager for the Current Index to Statistics, and IMS Group Manager for the Institute of Mathematical Statistics. He has been associate editor of the Journal of Statistical Software.