Statistical Methods for Stochastic Differential Equations

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 SørensenIntroductionLow frequency asymptoticsMartingale estimating functionsThe likelihood functionNon-martingale estimating functionsHigh-frequency asymptoticsHigh-frequency asymptotics in a fixed time-intervalSmall-diffusion asymptoticsNon-Markovian modelsGeneral asymptotic results for estimating functionsOptimal estimating functions: General theoryThe econometrics of high frequency data, Per. A. Mykland and Lan ZhangIntroductionTime varying drift and volatilityBehavior of estimators: VarianceAsymptotic normalityMicrostructureMethods based on contiguityIrregularly spaced dataStatistics and high frequency data, Jean JacodIntroductionWhat can be estimated?Wiener plus compound Poisson processesAuxiliary limit theoremsA first LNN (Law of Large Numbers)Some other LNNsA first CLTCLT with discontinuous limitsEstimation of the integrated volatilityTesting for jumpsTesting for common jumpsThe Blumenthal–Getoor indexImportance sampling techniques for estimation of diffusion models, Omiros Papaspiliopoulos and Gareth RobertsOverview of the chapterBackgroundIS estimators based on bridge processesIS estimators based on guided processesUnbiased Monte Carlo for diffusionsAppendix: Typical problems of the projection-simulation paradigm in MC for diffusionsAppendix: Gaussian change of measureNon parametric estimation of the coefficients of ergodic diffusion processes based on high frequency data, Fabienne Comte, Valentine Genon-Catalot, and Yves RozenholcIntroductionModel and assumptionsObservations and asymptotic frameworkEstimation methodDrift estimationDiffusion coefficient estimationExamples and practical implementationBibliographical remarksAppendix. Proof of Proposition.13Ornstein–Uhlenbeck related models driven by Lévy processes, Peter J. Brockwell and Alexander LindnerIntroductionLévy processesOrnstein–Uhlenbeck related modelsSome estimation methodsParameter estimation for multiscale diffusions: an overview, Grigorios A. Pavliotis, Yvo Pokern, and Andrew M. StuartIntroductionIllustrative examplesAveraging and homogenizationSubsamplingHypoelliptic diffusionsNonparametric drift estimationConclusions 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