Estimation of the parameters of the Ornstein-Uhlenbeck's stochastic process

TL;DR

This paper develops a method to consistently estimate the parameters of the Ornstein-Uhlenbeck process using observed trajectory values at a fixed time, assuming other parameters are known.

Contribution

It introduces a new estimation technique for the Ornstein-Uhlenbeck process parameters based on trajectory data at a fixed time point.

Findings

Provides explicit formulas for parameter estimates.

Demonstrates consistency of the estimators.

Applicable when all but one parameter are known.

Abstract

It is considered Ornstein-Uhlenbeck process $ x_t = x_0 e^{-\theta t} + \mu (1-e^{-\theta t}) + \sigma \int_0^t e^{-\theta (t-s)} dW_s$, where $x_0 \in R$, $\theta>0$, $ \mu \in R$ and $\sigma > 0$ are parameters. By use values $(z_k)_{k \in N}$ of corresponding trajectories at a fixed positive moment $t$, a consistent estimate of each unknown parameter of the Ornstein-Uhlenbeck's stochastic process is constructed under assumption that all another parameters are known.