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Let $a$ be a finite signed measure on $[-r, 0]$ with $r \in (0, \infty)$.Consider a stochastic process $(X^{(\vartheta)}(t))_{t\in[-r,\infty)}$ given bya linear stochastic delay differential equation \[ \mathrm{d}X^{(\vartheta)}(t) = \vartheta \int_{[-r,0]} X^{(\vartheta)}(t + u) \,a(\mathrm{d} u) \, \mathrm{d} t + \mathrm{d} W(t) , \qquad t \ge 0, \] where$\vartheta \in \mathbb{R}$ is a parameter and $(W(t))_{t\ge 0}$ is a standardWiener process. Consider a point $\vartheta \in \mathbb{R}$, where this modelis unstable in the sense that it is locally asymptotically Brownian functionalwith certain scalings $(r_{\vartheta,T})_{T\in(0,\infty)}$ satisfying$r_{\vartheta,T} \to 0$ as $T \to \infty$. A family$\{(X^{(\vartheta_T)}(t))_{t\in[-r,T]} : T \in (0, \infty)\}$ is said to benearly unstable as $T \to \infty$ if $\vartheta_T \to \vartheta$ as $T \to\infty$. For every $\alpha \in \mathbb{R}$, we prove convergence of thelikelihood ratio processes of the nearly unstable families$\{(X^{(\vartheta+\alpha \ r_{\vartheta,T})}(t))_{t\in[-r,T]}: T \in (0,\infty)\}$ as $T \to \infty$. As a consequence, we obtain weak convergence ofthe maximum likelihood estimator $\hat{\alpha}_T$ of $\alpha$ based on theobservations $(X^{(\vartheta+\alpha \ r_{\vartheta,T})}(t))_{t\in[-r,T]}$ as $T\to \infty$. It turns out that the limit distribution of $\hat{\alpha}_T$ as $T\to \infty$ can be represented as the maximum likelihood estimator of aparameter of a process satisfying a stochastic differential equation withouttime delay.

Nearly unstable family of stochastic processes given by stochastic differential equations with time delay