Methods are presented for analysing data that are integrals of a diffusion process, i.e. the solution to a stochastic differential equation, observed with measurement error. First a relatively simple methodology is outlined: prediction-based estimating function. Then we present in detail a computationally more demanding method for obtaining maximum likelihood estimates. The data can be viewed as incomplete observations from a model with a tractable likelihood function. Therefore a simulated EM-algorithm is used to obtain maximum likelihood estimates of the model parameters. An essential part of the algorithm is a recent simple method for approximate simulation of diffusion bridges, which is used to simulate the full hidden data given the observations. In a simulation study the proposed method works well. Finally, a similar method for analysing discrete time observations from a stochastic volatility model is presented.
Keywords: Stochastic differential equation; EM-algorithm; Diffusion bridge; Likelihood inference
Biography: Michael Sørensen obtained his Ph.D. degree from the University of Aarhus and is now a professor of mathematical statistics at the University of Copenhagen. Over the last 30 years, he has made a large number of contributions to statistical inference for stochastic processes. Main contribitions are a book on exponential families of stochastic processes and statistical methodology for discretely sampled continuous time processes. He has also made substantial applications of statistical methods in the earth sciences and in finance. Michael Sørensen has held several offices in the Bernoulli Society, and he is an elected member of the ISI and of the Royal Danish Academy of Sciences and Letters and is an elected fellow of the IMS.