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By N. Balakrishnan

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D(B) is an algebra that separates points in S and contains the constant functions. Definition Suppose µ is a probability measure on S. A process Zt, 0ՅtՅT, defined on some probability space (⍀, , P) and taking values in S is said to be a solution to the martingale problem for (B, µ) if: (i) (ii) for every tՅT; (iii) for all f∈D(B), is a martingale. Definition The martingale problem for (B, µ) is said to be well posed in a class of processes C if there exists a solution Z1∈C to the martingale problem for (B, µ) and if Z2∈C is also a solution to the martingale problem for (B, µ), then Z1 and Z2 have the same probability distributions.

1996). Characterization of the optimal filter: The non Markov case, Preprint. B. (1965). Markov Processes, Vols. I, II, Springer-Verlag, Berlin. N. G. (1986). Markov Processes: Characterization and Convergence, John Wiley & Sons, New York. , Kallianpur, G. and Kunita, H. (1972). Stochastic differential equations for the nonlinear filtering problem, Osaka Journal of Mathematics, 9, 19–40. W. (1976). Differential Topology, Springer-Verlag, New York. Horowitz, J. L. (1990). ), pp. 75–122, Birkhaüser, Boston.

1, . Remark As noted earlier, Mohammed (1984) has given alternative proofs of the parts (i) and (ii) of the theorem. , Theorem III. 3). 2(iii)) is not entirely correct. html. 3) with η a -measurable square integrable C-valued random variable and a, b: C→R satisfying the following Lipschitz condition: for some constant K>0. 2). 25) Suppose all the processes are defined on the probability space (Ω,F (Ft), P). First, we are going to obtain an analog of the Bayes formula due to Kallianpur and Striebel (1968), in our setup.

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