By N. Balakrishnan
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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. Kallianpur, G.
See, for example, Mohammed (1984. 1). The main objective of this paper is to obtain a Zakai-type equation for the above filtering probl em and to show that the optimal filter is characterized as the unique solution of that equation. We do this by applying the ideas and, in some cases, extending the results of Mohammed (1984). We organize this article as follows. 2, we start with some known results on martingale problems and their connections to Markov processes. Also, we introduce the notation and definitions we will follow throughout this article.
Now let ∈>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. Case 1 sk=0. 24), we have Case 2 sk<0. MANDAL where . Clearly, φm, m=1, 2, …, are quasi-tame functions. 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.
Advances on Theoretical and Methodological Aspects of Probability and Statistics by N. Balakrishnan
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