Filtering With Two Sided Filtrations

摘要

The value of a diffusion at an intermediate point is observed through noisyobservations on each side. Corresponding semimartingale decompositions and recursive filtering equations are obtained. Such problems possibly arise in reconstructing images from noisy data if one wishes to estimate the signal at linear location 1.2 based on observations from 0 to t and 1-t to 1 either side. This estimation involves the two delta-fields, Yt, increasing in the positive t direction, and Z1-t increasing in the negative t direction.