Linear filtering with nonlinear observations

For all known finite dimensional filters, one always needs the condition that the observation terms are degree one polynomials. On the other hand, in many practical examples, e.g. tracking problems, the observation terms may be nonlinear. In this paper, we develop a new method to treat linear filtering problems with nonlinear observation terms. We first show that real time computation of Duncan-Mortensen-Zakai (DMZ) equation can be reduced to off time computation of Kolmogorov equation. An explicit algorithm of such a reduction is provided. We next show that if the drifts are linear and the observation terms are nonlinear with linear growths, then the Kolmogorov equation can be solved in real time via a system of ODEs. Consequently, the nonlinear filtering problem with linear drifts and nonlinear observations with linear growth can be solved in real time and memoryless manner.