Infinite-sample consistent estimations of parameters of the Wiener process with drift

TL;DR

This paper develops infinite-sample consistent estimators for the parameters of a Wiener process with drift, using trajectory data at fixed times, and proposes methods for estimating multiple parameters simultaneously from multiple observations.

Contribution

It introduces new infinite-sample consistent estimation methods for all parameters of the Wiener process with drift, including approaches for joint estimation from multiple trajectories.

Findings

Constructed estimators are consistent as sample size approaches infinity.

Provided methods for estimating multiple parameters simultaneously.

Demonstrated the applicability of estimators using trajectory data at fixed times.

Abstract

We consider the Wiener process with drift $$ dX_t=\mu dt +\sigma d W_t $$ with initial value problem $X_0=x_0$, where $x_0 \in R$, $ \mu \in R$ and $\sigma > 0$ are parameters. By use values $(z_k)_{k \in N}$ of corresponding trajectories at a fixed positive moment $t$, the infinite-sample consistent estimates of each unknown parameter of the Wiener process with drift are constructed under assumption that all another parameters are known. Further, we propose a certain approach for estimation of unknown parameters $x_0,\mu,\sigma$ of the Wiener process with drift by use the values $(z^{(1)}_k)_{k \in N}$ and $(z^{(2)}_k)_{k \in N}$ being the results of observations on the $2k$-th and $2k+1$-th trajectories of the Wiener process with drift at moments $t_1$ and $t_2$ , respectively.