Summary: We consider the process , where is a process not necessarily adapted to (the filtration generated by the process and is a Brownian motion. We obtain a general representation for the likelihood ratio of the law of the process relative to a Brownian measure. This representation involves only one basic filter (expectation of conditional on the observed process . This generalizes a result of T. Kailath and M. Zakai [Ann. Math. Stat. 42, 130–140 (1971; Zbl 0226.60061)] where it is assumed that the process is adapted to .
In particular, we consider the model in which is a functional of and of a random element which is independent of the Brownian motion . For example, could be a diffusion or a Markov chain. This result can be applied to the estimation of an unknown multidimensional parameter appearing in the dynamics of the process based on continuous observation of on the time interval .
For a specific hidden diffusion financial model in which is an unobserved mean-reverting diffusion, we give an explicit form for the likelihood function of . For this model we also develop a computationally explicit E-M algorithm for the estimation of . In contrast to the likelihood ratio, the algorithm involves evaluation of a number of filtered integrals in addition to the basic filter.