Statistical Methods for Stochastic Differential Equations: 1st Edition (Hardback) book cover

Statistical Methods for Stochastic Differential Equations

1st Edition

By Mathieu Kessler, Alexander Lindner, Michael Sorensen

Chapman and Hall/CRC

507 pages | 17 B/W Illus.

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pub: 2012-05-17
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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.


"… 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

Table of Contents

Estimating functions for diffusion-type processes, Michael Sørensen


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


Time varying drift and volatility

Behavior of estimators: Variance

Asymptotic normality


Methods based on contiguity

Irregularly spaced data

Statistics and high frequency data, Jean Jacod


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


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


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


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


Illustrative examples

Averaging and homogenization


Hypoelliptic diffusions

Nonparametric drift estimation

Conclusions and further work

About the Authors

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

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
MATHEMATICS / Probability & Statistics / General