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3) and , then (πtX, 0ՅtՅT) is a solution to the martingale problem corresponding to A0. This will be one of the main tools in dealing with the nonlinear filtering problem with delay equations in the next section. 3) and . 7). 2, pp. 8) for θ ∈C with the understanding that when k=1, . Let the space of quasi-tame functions be denoted by . 8). 9) Define an operator A0 on Cb with as follows. 8). 10) Then it is easy to see that the following theorem holds. See, for example, Mohammed (1998, p. 26). 4).

59) Indeed, if P(. 37) satisfied by measure-valued Zakai equation for ␷t(. , Yt). 63). 4). 2). 25) holds. 2 corresponding to A0. 36) holds. 35), is the unnormalized conditional distribution of πtX given . 64). 65) also. Since X and the observation noise W are independent, so are πtX and W. 3, A0 satisfies the conditions C1–C5. MANDAL of Bhatt, Kallianpur and Karadikar (1995) on the modified signal process πtX completes the proof. 64) one can, in principle, calculate the unnormalized conditional expectation of given and hence can obtain .

36) holds. 35), is the unnormalized conditional distribution of πtX given . 64). 65) also. Since X and the observation noise W are independent, so are πtX and W. 3, A0 satisfies the conditions C1–C5. MANDAL of Bhatt, Kallianpur and Karadikar (1995) on the modified signal process πtX completes the proof. 64) one can, in principle, calculate the unnormalized conditional expectation of given and hence can obtain . 66). 2 Suppose conditional expectation (given -bounded. Let Et denote the ). 67) PROOF Let quasitame function be a C∞-bounded function.

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