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

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 $\alpha \in \mathbb{R}$, 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.

15 pages

References in corpus (3)

• Local asymptotic quadraticity of statistical experiments connected with a Heston model
• Asymptotic inference for a stochastic differential equation with uniformly distributed time delay
• One-parameter statistical model for linear stochastic differential equation with time delay