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→∞.