Well-posedness of Filtering Equations in Weighted Sobolev Spaces with Unbounded System Coefficients

Nonlinear filtering problem is one of the core subjects in modern control theory. In this paper, we will study the well-posedness of the three fundamental evolution equations arising in continuous-time nonlinear filtering--the robust Duncan-Mortensen-Zakai (DMZ) equation, the stochastic DMZ equation, and the Kushner-Stratonovich equation--within a unified buffered weighted formulation. An exponential-type weight function and the corresponding weighted Sobolev spaces are introduced to enable a variational treatment of the filtering equations in a more general setting, in which the coefficients of the filtering system may be unbounded with polynomial growth. Under mild and easily verifiable assumptions, we first establish the well-posedness of the weak solution to the robust DMZ equation in these weighted spaces. Using the gauge (exponential) transformation and its inverse, these results are then transferred to the stochastic DMZ equation and the Kushner-Stratonovich equation, whose solutions are shown to exist and be unique in buffered weighted Sobolev spaces, yielding a unified treatment of all three filtering equations. Sufficient conditions for the well-posedness are also summarized, which illustrate the wide applicability of the proposed framework to general nonlinear filtering systems.