Approximations for the likelihood ratio for continuous multi-parameter stochastic processes

Based on finitely additive white noise theory, one may derive the likelihood ratio for random variables with values in any Hilbert space. This includes stochastic processes, defined on a one- or multi-dimensional continuous-parameter bounded domain. In certain circumstances, the likelihood ratio for continuous processes may be computed directly. In general however, one will have to approximate the likelihood ratio. In this paper approximations for the likelihood ratios for continuous-parameter processes are studied. Starting from a sequence of finite dimensional projection operators in the Hilbert space, strongly converging to identity, the authors show that the likelihood ratios for the projected processes converge to the likelihood ratio for the original process. Discretization of the stochastic process turns out to be one of the possibilities for such approximations. The discretization method is expected to give good results for signals satisfying elliptic PDEs, because discretization of these processes leads to nearest neighbor models, for which the likelihood ratio has been obtained in Luesink (1992)