Asymptotic inference for a linear stochastic differential equation with time delay

Abstract

For the stochastic differential equation

$$dX\left(t\right)=aX\left(t\right)+bX(t-1),dt+dW\left(t\right),t\ge 0,$$

the local asymptotic properties of the likelihood function are studied. They depend strongly on the true value of the parameter $\theta =(a,b{)}^{*}$. Eleven different cases are possible if $\theta$ runs through ${ℝ}^2$. Let ${\hat{\theta }}_T$ be the maximum likelihood estimator of $\theta$ based on $\left(X\right(t),t\le T)$. Applications to the asymptotic behaviour of ${\hat{\theta }}_T$ as $T\to \mathrm{\infty }$ are given.