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A simple proof of uniqueness for Kushner and Zakai equations. (English) Zbl 0732.60055

Stochastic analysis, Proc. Conf. Honor Moshe Zakai 65th Birthday, Haifa/Isr. 1991, 449-458 (1991).
[For the entire collection see Zbl 0724.00018.]
The purpose of the article is to present a simple proof of uniqueness for a generalized solution to the Kushner and Zakai equations for diffusion processes. These solutions are considered in the space of signed measures. The result holds under rather mild assumptions on the coefficients of the corresponding processes. For example, in the case of non-degenerate signal process only boundedness, continuity in t and Hölder continuity in x are assumed. In the degenerate case the coefficients are, in addition, assumed to belong to \(C_ b^{2+\alpha}(R^ d)\). Of course, uniqueness of “classical” solutions which belong to \(L_ 1\) follows from the above results.

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
93E11 Filtering in stochastic control theory

Citations:

Zbl 0724.00018