Theoretical Statistics and Mathematics Unit, ISI Delhi

Uniqueness of solution to the Kolmogorov's forward equation: Applications to White Noise Theory of Filtering

by Abhay G. Bhatt and Rajeeva L. Karandikar

We consider a signal process $X$ taking values in a complete, separable metric space $E$. $X$ is assumed to be a Markov process charachterized via the martingale problem for an operator $A$. In the context of the finitely additive white noise theory of filtering, we show that the optimal filter $\Gamma_t(y)$ is the unique solution of the analogue of the Zakai equation for every $y$, not necessarily continuous. This is done by first proving uniqueness of solution to a (perturbed) measure valued evolution equation associated with $A$. An additional assumption of uniqueness of the local martingale problem for $A$ is imposed.

isid/ms/2009/13 [fulltext]

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