EXPONENTIAL-FAMILIES OF STOCHASTIC-PROCESSES WITH TIME-CONTINUOUS LIKELIHOOD FUNCTIONS

The structure of exponential families of stochastic processes with a time-continuous likelihood function is investigated by means of semimartingale theory. The time-homogeneous exponential families of this kind are characterized as those for which the jump mechanism and the diffusion coefficient are the same under all probability measures in the family and the drift depends linearly on a, possibly multidimensional, parameter function. A parametrization exists for which the log-likelihood function is a quadratic form in the parameter. The derived structure of these models is utilized to show that they have nice statistical properties. Exponential families of stochastic processes that are not time-homogeneous need not be of this type. Several examples are considered.