**Basic Filtering Theory**

**UNIQUENESS FOR THE FILTERING EQUATIONS**

There is much interesting mathematics being researched at MITACS-PINTS on the basic mathematical equations of classical nonlinear filtering theory. Such work is important for determining the expected properties of the often-difficult real world problems. The first classical theoretical problem is that of uniqueness. Uniqueness of the filtering equations provides an important tool in establishing convergence and rates of convergence for computer workable approximations.

In the simplest classical setting, where the signal and observation noise are independent, Szpirglas (1978) established pathwise uniqueness for the Kushner-Stratonovich and the Duncan-Mortensen-Zakai equations, which together with the Yamada-Watanabe argument also provide distributional uniqueness. His results do not apply in the correlated or feedback case. Lucic and Heunis (2001) consider uniqueness properties of the nonlinear filter equations, both normalized and unnormalized, for a nonlinear filtering problem in which the signal is conditioned by the observation process. This situation is typically encountered in stochastic feedback controlsystems, in which the output is fed back to the input. Such feedback systems arise very commonly in the applications of nonlinear filtering to guidance and control problems in aerospace engineering.

Our particular interest is in the uniqueness in law property for the filter equations because of its applicability to the singular perturbation robustness problem discussed above. Although uniqueness in law for the nonlinear filter equations is well known in the case where the signal and observation noise are independent, it was not previously established for the case where the signal is conditioned by the observations, a basically new approach was utilized. We show uniqueness in law under rather mild restrictions for both the normalized and unnormalized filter equations. As a by-product we also get pathwise uniqueness for the unnormalized filter equation. Our main tools were the uniqueness for measure-valued evolution results obtained by Bhatt and Kanadikar as well as the Yamada-Watanabe argument. It should be mentioned that the nice uniqueness results of Kurtz and Ocone (1988) could not be used to deduce distributional uniqueness since Kurtz and Ocone's results only apply on probability spaces rich enough to contain the signal, observations and filter.

**APPLICABILITY OF THE FILTERING EQUATIONS**

The uniqueness principle established by Kurtz and Ocone (1988) says that under very general conditions, any solution to the filtering equations on a rich enough probability space must be the conditional distribution of the signal given the observations (i.e. the desired distribution). However, the conditions under which there are known solutions were far more severe. It has been known for more than thirty years that there are solutions to the basic filtering equations under the mean-square finite energy condition and right continuity. This is a much more stringent condition than those required by Kurtz and Ocone (1988) for uniqueness. Hence, there was an open problem, that we just settled in Kouritzin and Long (2002), to determine existence (i.e. whether the conditional distributions satisfy the Duncan-Mortensen-Zakai and the Kushner-Stratonovich equations) under conditions as general as those used by Kurtz and Ocone for uniqueness. Indeed, our conditions are milder that those used by Kurtz and Ocone, and demonstrate, for example, that the filtering equations continue to hold when the sensor function is linear and the signal has heavy tails like a Levy process.

**LARGE
DEVIATION PRINCIPLE FOR OPTIMAL FILTERING**

When
the signal-to-noise ratio goes to zero, Xiong (2002) derived a large deviation
principle for the optimal filter where the signal and the observation processes
take values in conuclear spaces. The approach follows from the framework
established Xiong 1996. The key is the verification of the exponential
tightness for the optimal filtering process and the exponential continuity
of the coefficients in the Zakai equation.