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.

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

Miture

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 processe...