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Regular version of the site

Article

Asymptotic inference for a linear stochastic differential equation with time delay

Gushchin A. A., Küchler U.

For the stochastic differential equation

dX(t)=a X ( t ) + b X ( t - 1 )dt+dW(t),t≥0,

the local asymptotic properties of the likelihood function are studied. They depend strongly on the true value of the parameter ϑ=(a,b) * . Eleven different cases are possible if ϑ runs through ℝ 2 . Let ϑ ^ T be the maximum likelihood estimator of ϑ based on (X(t), t≥T). Applications to the asymptotic behaviour of ϑ ^ T as T→∞ are given.