1st Edition

# Statistical Methods for Stochastic Differential Equations

508 Pages 17 B/W Illustrations
by Chapman & Hall

507 Pages
by Chapman & Hall

Also available as eBook on:

The seventh volume in the SemStat series, Statistical Methods for Stochastic Differential Equations presents current research trends and recent developments in statistical methods for stochastic differential equations. Written to be accessible to both new students and seasoned researchers, each self-contained chapter starts with introductions to the topic at hand and builds gradually towards discussing recent research.

The book covers Wiener-driven equations as well as stochastic differential equations with jumps, including continuous-time ARMA processes and COGARCH processes. It presents a spectrum of estimation methods, including nonparametric estimation as well as parametric estimation based on likelihood methods, estimating functions, and simulation techniques. Two chapters are devoted to high-frequency data. Multivariate models are also considered, including partially observed systems, asynchronous sampling, tests for simultaneous jumps, and multiscale diffusions.

Statistical Methods for Stochastic Differential Equations is useful to the theoretical statistician and the probabilist who works in or intends to work in the field, as well as to the applied statistician or financial econometrician who needs the methods to analyze biological or financial time series.

Estimating functions for diffusion-type processes, Michael Sørensen
Introduction
Low frequency asymptotics
Martingale estimating functions
The likelihood function
Non-martingale estimating functions
High-frequency asymptotics
High-frequency asymptotics in a fixed time-interval
Small-diffusion asymptotics
Non-Markovian models
General asymptotic results for estimating functions
Optimal estimating functions: General theory

The econometrics of high frequency data, Per. A. Mykland and Lan Zhang
Introduction
Time varying drift and volatility
Behavior of estimators: Variance
Asymptotic normality
Microstructure
Methods based on contiguity
Irregularly spaced data

Statistics and high frequency data, Jean Jacod
Introduction
What can be estimated?
Wiener plus compound Poisson processes
Auxiliary limit theorems
A first LNN (Law of Large Numbers)
Some other LNNs
A first CLT
CLT with discontinuous limits
Estimation of the integrated volatility
Testing for jumps
Testing for common jumps
The Blumenthal–Getoor index

Importance sampling techniques for estimation of diffusion models, Omiros Papaspiliopoulos and Gareth Roberts
Overview of the chapter
Background
IS estimators based on bridge processes
IS estimators based on guided processes
Unbiased Monte Carlo for diffusions
Appendix: Typical problems of the projection-simulation paradigm in MC for diffusions
Appendix: Gaussian change of measure

Non parametric estimation of the coefficients of ergodic diffusion processes based on high frequency data, Fabienne Comte, Valentine Genon-Catalot, and Yves Rozenholc
Introduction
Model and assumptions
Observations and asymptotic framework
Estimation method
Drift estimation
Diffusion coefficient estimation
Examples and practical implementation
Bibliographical remarks
Appendix. Proof of Proposition.13

Ornstein–Uhlenbeck related models driven by Lévy processes, Peter J. Brockwell and Alexander Lindner
Introduction
Lévy processes
Ornstein–Uhlenbeck related models
Some estimation methods

Parameter estimation for multiscale diffusions: an overview, Grigorios A. Pavliotis, Yvo Pokern, and Andrew M. Stuart
Introduction
Illustrative examples
Averaging and homogenization
Subsampling
Hypoelliptic diffusions
Nonparametric drift estimation
Conclusions and further work

### Biography

Matthieu Kessler, Department of Applied Mathematics and Statistics, University of Cartagena, Spain

Alexander Lindner, Institute of Mathematics and Statistics, TU Braunschweig, Germany

Michael Sorensen, Department of Mathematical Sciences, University of Copenhagen, Denmark

"… an excellent resource for anyone currently active in research in this area, interested in getting into research in the area, or just interested in the topic. I cannot think of another source that provides detailed yet accessible introductions of this quality and timeliness to the major issues of interest in this area. … As noted in the preface, the idea is to get young researchers ‘quickly to the forefront of knowledge and research.’ … The book succeeds in delivering on this goal. A careful reading of the chapters of this book would go a long way toward putting one in a position to begin contributing to the large and rapidly growing body of research in this important area of statistics. It would certainly be an excellent resource for teaching advanced Ph.D. courses. … This is a wonderful book for anyone interested in SDEs. I highly recommend it and am happy to have it on my bookshelf."
—Garland B. Durham, Journal of the American Statistical Association, March 2014

"The contributors are all renowned specialists in the field … the last four chapters are generally well written, informative, and cover a wide range of different aspects of statistics for SDE … the first three chapters … constitute an original and very useful contribution in a field that too often has the reputation of being technical and somehow austere. … I strongly recommend the book for anyone interested in the wide topic of statistical methods for SDE, whether she or he is a specialist or a student starting in the field."
—Marc Hoffmann, Université Paris–Dauphine Sørensen, CHANCE, 26.3

"… a good collection of useful and interesting articles … [I have] no hesitation in recommending the book."
—Tusheng Zhang, Journal of Time Series Analysis, 2013