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

TL;DR

This work extends near-unit-root phenomena to linear stochastic differential equations with time delay by analyzing nearly unstable families where $dX^{(\theta)}(t)=\theta\int_{[-r,0]}X^{(\theta)}(t+u)a(du)\,dt+dW(t)$. Under the boundary instability condition $v_{\theta}^*=0$ and an appropriate scaling $r_{\theta,T}=T^{-m_{\theta}^*-1}$, the authors prove convergence of the likelihood ratio processes to those of a non-delayed multidimensional OU model, enabling the limit distribution of the MLE for the perturbation parameter $\alpha$ to be represented as the MLE of a non-delayed process. Consequently, the nearly unstable MLE $\widehat{\alpha}_T$ converges in distribution to $\widehat{\alpha}$, the MLE in the corresponding non-delayed model. This provides a rigorous generalization of near-unit-root inference to SDDEs with delays, with potential applications in econometrics and engineering where delays are intrinsic.

Abstract

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 $α\in \mathbb{R}$, we prove convergence of the likelihood ratio processes of the nearly unstable families $\{(X^{(\vartheta+α\ r_{\vartheta,T})}(t))_{t\in[-r,T]}: T \in (0, \infty)\}$ as $T \to \infty$. As a consequence, we obtain weak convergence of the maximum likelihood estimator $\hatα_T$ of $α$ based on the observations $(X^{(\vartheta+α\ r_{\vartheta,T})}(t))_{t\in[-r,T]}$ as $T \to \infty$. It turns out that the limit distribution of $\hatα_T$ 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.