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 $r\in (0,\mathrm{\infty })$. Consider a stochastic process $(X^{(\vartheta )}(t){)}_{t\in [-r,\mathrm{\infty })}$ given by a linear stochastic delay differential equation $$\mathrm{d}{X}^{(\vartheta )}(t)=\vartheta {\int }_{[-r,0]}{X}^{(\vartheta )}(t+u)a(\mathrm{d}u)\mathrm{d}t+\mathrm{d}W(t),t\ge 0,$$ 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 $(r_{\vartheta ,T}{)}_{T\in (0,\mathrm{\infty })}$ satisfying $r_{\vartheta ,T}\to 0$ as $T\to \mathrm{\infty }$. A family $\{(X^{({\vartheta }_T)}(t){)}_{t\in [-r,T]}:T\in (0,\mathrm{\infty })\}$ is said to be nearly unstable as $T\to \mathrm{\infty }$ if ${\vartheta }_T\to \vartheta$ as $T \to \infty$\mathrm{.}Forevery$\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 ${\widehat{\alpha }}_T$ of $\alpha$ based on the observations $(X^{(\vartheta +\alpha r_{\vartheta ,T})}(t){)}_{t\in [-r,T]}$ as $T \to \infty$\mathrm{.}Itturnsoutthatthelimitdistributionof$\hat{\alpha}_T$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.