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- W2980323153 abstract "Let $a$ be a finite signed measure on $[-r, 0]$ with $r in (0, infty)$. Consider a stochastic process $(X^{(vartheta)}(t))_{tin[-r,infty)}$ 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 $(W(t))_{tge 0}$ 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 $(r_{vartheta,T})_{Tin(0,infty)}$ satisfying $r_{vartheta,T} to 0$ as $T to infty$. A family ${(X^{(vartheta_T)}(t))_{tin[-r,T]} : T in (0, infty)}$ 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 ${(X^{(vartheta+alpha r_{vartheta,T})}(t))_{tin[-r,T]}: T in (0, infty)}$ 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 $(X^{(vartheta+alpha r_{vartheta,T})}(t))_{tin[-r,T]}$ 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." @default.
- W2980323153 created "2019-10-25" @default.
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- W2980323153 date "2019-10-17" @default.
- W2980323153 modified "2023-09-27" @default.
- W2980323153 title "Nearly unstable family of stochastic processes given by stochastic differential equations with time delay" @default.
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