A nonlinear filtering problem and its applications (Q1096251)

The following nonlinear filtering model is considered: the signal \(\beta_ t\) is assumed to be a diffusion and the observation process \(\xi_ t\) is related to the signal via the relation \[ d\xi_ t=B(t,\xi_ t)B^*(t,\xi_ t)\phi (t,\beta_ t)dt+B(t,\xi_ t)dW_ t. \] It is very important that all diffusion and drift coefficients are not assumed to be of a Lipschitz-type. They obey another weaker condition which allows even discontinuity. For an integrable \({\mathcal F}_ t^{\xi,W}\)-measurable r.v. g the conditional expectation E(g\(| {\mathcal F}^{\xi}_ t)\) is explicitly written with the corresponding expression being an extension of the Bayes formula, used in \textit{G. Kallianpur} and \textit{R. L. Karandikar}'s approach [Ann. Probab. 13, 1033- 1107 (1985; Zbl 0584.60055)], known as ``white noise calculus''. The innovation problem is solved under weaker conditions and the corresponding result is applied to a problem in optimal control. The paper as a whole is very important and should be known to everyone interested in the present trends of nonlinear filtering.