A representation theorem on a filtering model with first-passage-type stopping time

We present a representation theorem for a filtering model with first-passage-type stopping time. The model is constructed from two unobservable processes and one observable process that is under the influence of two unobservable processes.A filter is constructed using Brownian motion in the observable process and a first-passage-type stopping time in an unobservable process.Though our theorems are similar to those of Nakagawa\cite{Nakagawa}, we do not use pinned Brownian motion measure, which is difficult to deal with. In addition, we describe a representation theorem for another filtration that was not discussed by Nakagawa