Abstract
A discrete time model for filtering with small observation noise is considered in this paper. The observation function is assumed to be a piecewise monotone function with a finite number of intervals of monotonicity. Under a certain detectability hypothesis, a sequential quadratic variation test is proposed to detect the intervals of monotonicity of the observation function. An upper bound for the mean time of reaching a decision is given in terms of the observation noise level. Then based on the quadratic variation test, accurate approximate finite dimensional filters can be used to construct an asymptotically optimal filter as the observation noise tends to zero.
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