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| Format: | Preprint |
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2019
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| Online Access: | https://arxiv.org/abs/1910.07816 |
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| _version_ | 1866929686628532224 |
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| author | Benke, János Marcell Pap, Gyula |
| author_facet | Benke, János Marcell Pap, Gyula |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1910_07816 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Nearly unstable family of stochastic processes given by stochastic differential equations with time delay Benke, János Marcell Pap, Gyula Statistics Theory 62B15, 62F12 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. |
| title | Nearly unstable family of stochastic processes given by stochastic differential equations with time delay |
| topic | Statistics Theory 62B15, 62F12 |
| url | https://arxiv.org/abs/1910.07816 |