Asymptotic behaviour of the optimal filter of jump and slope jump processes

A piecewise constant process is considered which may contain a jump of random magnitude at a random time. The conditional density of the jump magnitude given noisy observations of the signal is derived, and the conditional density when only noisy observations of its integral can be measured. These densities are then analysed using large deviations theory and the asymptotical distributions for vanishing observation noise are characterised for times after the jump. The results suggest a surprisingly complicated behaviour of the conditional laws, which may be of some importance when assessing the performance of approximations to the optimal filter