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The theory of nonlinear filtering concerns the optimal estimation of a Markov signal in noisy observations. Such estimates necessarily depend on the model that is chosen for the signal and observations processes. This thesis studies the sensitivity of the filter to the choice of underlying model over long periods of time, within the framework of continuous time filtering with white noise type observations.
The first topic of this thesis is the asymptotic stability of the filter, which is studied using the theory of conditional diffusions. This leads to improvements on pathwise stability bounds, and to new insight into existing stability results in a fully probabilistic setting. Furthermore, I develop in detail the theory of conditional diffusions for finite-state Markov signals and clarify the duality between estimation and stochastic control in this context.
The second topic of this thesis is the sensitivity of the nonlinear filter to the model parameters of the signal and observations processes. This section concentrates on the finite state case, where the corresponding model parameters are the jump rates of the signal, the observation function, and the initial measure. The main result is that the expected difference between the filters with the true and modified model parameters is bounded uniformly on the infinite time interval, provided that the signal process satisfies a mixing property. The proof uses properties of the stochastic flow generated by the filter on the simplex, as well as the Malliavin calculus and anticipative stochastic calculus.
The third and final topic of this thesis is the asymptotic stability of quantum filters. I begin by developing quantum filtering theory using reference probability methods. The stability of the resulting filters is not easily studied using the preceding methods, as smoothing violates the nondemolition requirement. Fortunately, progress can be made by randomizing the initial state of the filter. Using this technique, I prove that the filtered estimate of the measurement observable is stable regardless of the underlying model, provided that the initial states are absolutely continuous in a suitable sense.