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

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 by a 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 standard Wiener process. Consider a point $\vartheta \in \mathbb{R}$, where this model is unstable in the sense that it is locally asymptotically Brownian functional with 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 be nearly unstable as $T \to \infty$ if $\vartheta_T \to \vartheta$ as $T \to \infty$. For every $\alpha \in \mathbb{R}$, we prove convergence of the likelihood 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 of the maximum likelihood estimator $\hat{\alpha}_T$ of $\alpha$ based on the observations $(X^{(\vartheta+\alpha \ r_{\vartheta,T})}(t))_{t\in[-r,T]}$ 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 a parameter of a process satisfying a stochastic differential equation without time delay.