Monday, April 2, 2001


Jaya P.N. Bishwal
Bendheim Center for Finance
Princeton University

HIGHER ORDER APPROXIMATE LIKELIHOOD BASED ASYMPTOTICS
FOR DISCRETELY OBSERVED NONLINEAR DIFFUSIONS

 
Stochastic differential equations are used as models in many areas of natural sciences, engineering and especially in finance. In view of this, it becomes necessary to estimate the unknown parameters in such models based on observations of the corresponding solutions which are called diffusion processes. In practice, diffusion process though being a strong continuous Markovian semimartingale, can not be observed continuously in practice. Hence estimation in discretely observed diffusions is the current trend of investigation. We will study the asymptotic normality and local asymptotic minimaxity (in the Hajek-Le Cam sense) of approximate maximum likelihood estimators, approximate Bayes estimators and approximate maximum probability estimators of the parameter appearing nonlinearly in the drift coefficient of the model. We would use two types of approximations of the continuous likelihood: Ito type and Fisk-Stratonovich type. The approach will be through weak convergence of approximate likelihood ratio random field and local asymptotic normality for ergodic diffusions. Estimators based on the Fisk-Stratonovich type of approximation would be shown to be more efficient in the sense of having faster rate of convergence. The Bernstein-von Mises type theorems concerning the convergence of approximate posterior density to normal density will also be obtained. Asymptotics for conditional least squares estimator would be obtained as a by-product of the Ito type approximate likehood. The asymptotic framework will be long time with decreasing time lag between observations. Many open problems will be suggested.