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We study the backward Kolmogorov equation on the space of probabilitymeasures associated to the Kushner-Stratonovich equation of nonlinearfiltering. We prove existence and uniqueness in the viscosity sense and, inparticular, we provide a comparison theorem. In the context of stochasticfiltering it is natural to consider measure-valued processes that satisfystochastic differential equations. In the literature, a classical way toaddress this problem is by assuming that these measure-valued processes admit adensity. Our approach is different and we work directly with measures. Thus,the backward Kolmogorov equation we study is a second-order partialdifferential equation of parabolic type on the space of probability measureswith compact support. In the literature only few results are available onviscosity solutions for this kind of problems and in particular the uniquenessis a very challenging issue. Here we find a viscosity solution and then weprove that it is unique via comparison theorem.

Kolmogorov equations on the space of probability measures associated to the nonlinear filtering equation: the viscosity approach