1st Edition

Statistical Methods for Stochastic Differential Equations

    508 Pages 17 B/W Illustrations
    by Chapman & Hall

    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
    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


    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