Likelihood inference for a discretely observed stable processde

Abstract

Parabolic and hyperbolic stochastic partial differential equations in one-dimensional space have been proposed as models for the term structure of interest rates. The solution to these equations is reviewed, and their sample path properties are studied. In the parabolic case the sample paths essentially are H\"older continuous of order $\frac{1}{2}$ in space and $\frac{1}{4}$ in time, and in the hyperbolic case the sample paths essentially are H\"older continuous of order $\frac{1}{2}$ simultaneously in time and space. Parametric likelihood inference given an observation at discrete lattice points in time and space is also considered. The associated infinite-dimensional state-space model is described, and a finite-dimensional approximation is proposed. Conditions are presented under which the resulting approximate maximum likelihood estimator is asymptotically efficient when the number of observations in time increases to infinity at a fixed time step. The asymptotic distribution of the approximative likelihood ratio test for a parabolic equation against the hyperbolic alternative is found to be a truncated chi-square. Moreover, explicit moment estimators are derived which can be used as a starting point for a numerical optimization of the likelihood function.