Diffusion Processes and Statistical Problems

Abstract

We introduce the stochastic integral and the stochastic differential equation and present their properties. Then we give some useful formulae for the local time process and for the likelihood ratio. Finally we present an elementary theory of asymptotic inference for ergodic diffusion processes. In particular, we introduce the first definitions in the estimation and hypotheses testing problems as well as some inequalities for the risk of estimators. For several simple models of diffusion processes we describe the asymptotic behavior of the maximum likelihood, minimum distance and trajectory fitting estimators in parameter estimation problems. Then we study the asymptotic behavior of some nonparametric estimators of the invariant distribution function, density and trend coefficient. We conclude this chapter with two hypotheses testing problems.