# Sequential tracking of an unobservable two-state Markov process under Brownian noise

## TL;DR

This paper addresses an optimal control problem involving a Brownian motion with a hidden two-state Markov process, providing an explicit method to track the unobservable process based on sequential observations.

## Contribution

The paper introduces a novel explicit construction for a process that tracks an unobservable two-state Markov process under Brownian noise, advancing control strategies in stochastic systems.

## Key findings

- Explicit construction of a tracking process for unobservable Markov states
- Optimal control strategy for Brownian motion with changing drift
- Improved understanding of partially observable stochastic processes

## Abstract

We consider an optimal control problem, where a Brownian motion with drift is sequentially observed, and the sign of the drift coefficient changes at jump times of a symmetric two-state Markov process. The Markov process itself is not observable, and the problem consist in finding a {-1,1}-valued process that tracks the unobservable process as close as possible. We present an explicit construction of such a process.