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

TL;DR

This paper develops a new mathematical representation theorem for a filtering model involving a first-passage stopping time, constructed from unobservable and observable processes, avoiding complex pinned Brownian motion measures.

Contribution

It introduces a novel representation theorem for a filtering model with first-passage stopping times, extending Nakagawa's results without using pinned Brownian motion measures.

Findings

Provides a new representation theorem for filtering models with first-passage stopping times.

Extends existing theorems to a broader class of filtrations.

Offers a different approach avoiding pinned Brownian motion measures.

Abstract

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