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So let be ␾ a tame function and has the representation where is C∞-bounded and s1<…0 be given. Suppose -r≤sj<0 and θ ∈ C. By uniform continuity of θ find δj>0 such that . Also suppose K is a bound for θ(t), t ∈ [-r, 0]. 24) We consider two cases.

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. Ito, K. and Nisio, M. (1964). On stationary solutions of a stochastic differential equation, J. Math. Kyoto University, 4, 1–75. Jacod, J. (1979). Calcul Stochastique et Problemes de Martingales, Lecture Notes in Mathematics 714, Springer-Verlag, Berlin.

8). 10) Then it is easy to see that the following theorem holds. See, for example, Mohammed (1998, p. 26). 4). Suppose φ ∈ . Then is a . Let us note the following properties of the operator A0. 1 Suppose A0 is defined as above. 2. MANDAL . 12) Therefore, C1 is satisfied by A0. To see that C2 holds, note that This will imply the existence of a countable set that such and hence C2 follows. That D(A0) is an algebra follows from Mohammed (1984, p. 107). It is also easy to check that D(A0) separates points in C[-r, 0] and contains the constant functions which implies that A0 satisfies C3.

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