Statistical Methods for Stochastic Differential Equations
Features
- Provides an overview of recent developments in statistical methods for stochastic differential equations
- Includes sufficient background material for each topic discussed, including high-frequency data, estimation based on simulation techniques, and diffusions with jumps
- Presents applications to finance and econometrics
- Contains mathematically rigorous contributions from leading researchers in the field
Summary
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.
Table of Contents
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
Author Bio(s)
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
Editorial Reviews
"… a good collection of useful and interesting articles … [I have] no hesitation in recommending the book."
—Tusheng Zhang, Journal of Time Series Analysis, 2013
