  Assume that we observe a stochastic process $(X(t))_{t\in[-r,T]}$, which
satisfies the linear stochastic delay differential equation \[
  \mathrm{d} X(t)
  = \vartheta \int_{[-r,0]} X(t + u) \, a(\mathrm{d} u) \, \mathrm{d} t +
\mathrm{d} W(t) ,
  \qquad t \geq 0 , \] where $a$ is a finite signed measure on $[-r, 0]$. The
local asymptotic properties of the likelihood function are studied. Local
asymptotic normality is proved in case of $v_\vartheta^* < 0$, local asymptotic
quadraticity is shown if $v_\vartheta^* = 0$, and, under some additional
conditions, local asymptotic mixed normality or periodic local asymptotic mixed
normality is valid if $v_\vartheta^* > 0$, where $v_\vartheta^*$ is an
appropriately defined quantity. As an application, the asymptotic behaviour of
the maximum likelihood estimator $\widehat{\vartheta}_T$ of $\vartheta$ based
on $(X(t))_{t\in[-r,T]}$ can be derived as $T \to \infty$.
