Explicit solution of a Kolmogorov equation (Q1925029)

A rather wide class of nonlinear filtering systems with arbitrary nonlinear drift, which satisfies only some regularity assumption at infinity, is considered. A new method to solve Kolmogorov equation (i.e., Duncan-Mortensen-Zakai equation with observation \(h(x)=\text{constant}\)), which is well-known to be a fundamental equation in electrical engineering, is given. This method allows one to write down the formal solution of Kolmogorov equation explicitly in a closed form. From the truncated formal solution a convergent solution is constructed and the time interval on which that solution converges is estimated.