Asymptotic inference for a stochastic differential equation with uniformly distributed time delay

•We have studied the local asymptotic properties of the likelihood function generated by an affine stochastic delay differential equation.•Local asymptotic normality, local asymptotic mixed normality, periodic local asymptotic mixed normality and local asymptotic quadraticity are proved depending on the value of the parameter.•Applications to the asymptotic behaviour of the maximum likelihood estimator are given based on continuous sample.

For the affine stochastic delay differential equation dX(t)=a∫−10X(t+u)dudt+dW(t),t⩾0, the local asymptotic properties of the likelihood function are studied. Local asymptotic normality is proved in case of a∈(−π22,0), local asymptotic mixed normality is shown if a∈(0,∞)a∈(0,∞), periodic local asymptotic mixed normality is valid if a∈(−∞,−π22), and only local asymptotic quadraticity holds at the points −π22 and 0. Applications to the asymptotic behaviour of the maximum likelihood estimator âT of aa based on (X(t))t∈[0,T](X(t))t∈[0,T] are given as T→∞T→∞.