# On Parameter Estimation of Hidden Ergodic Ornstein-Uhlenbeck Process

## TL;DR

This paper develops a two-step maximum likelihood estimator for unknown parameters in a partially observed Ornstein-Uhlenbeck process, demonstrating its consistency and asymptotic normality using Kalman-Bucy filtering.

## Contribution

It introduces a novel two-step MLE process for joint estimation of the unobserved process and parameters in linear stochastic differential equations.

## Key findings

- Estimator is consistent and asymptotically normal.
- Recurrent estimators are constructed using Kalman-Bucy filtering.
- Theoretical properties are established for large samples.

## Abstract

We consider the problem of parameter estimation for the partially observed linear stochastic differential equation. We assume that the unobserved Ornstein-Uhlenbeck process depends on some unknown parameter and estimate the unobserved process and the unknown parameter simultaneously. We construct the two-step MLE-process for the estimator of the parameter and describe its large sample asymptotic properties, including consistency and asymptotic normality. Using the Kalman-Bucy filtering equations we construct recurrent estimators of the state and the parameter.