Nearly unstable family of stochastic processes given by stochastic differential equations with time delay

Abstract Let a be finite signed measure on [ − r , 0 ] with ∈ ( ∞ ) . Consider stochastic process X ϑ t given by linear delay differential equation d = ∫ + u W R where is parameter and standard Wiener process. point this model unstable in the sense that it locally asymptotically Brownian functional certain scalings T satisfying → as A family { : } said to nearly if For every α we prove convergence of likelihood ratio processes families As consequence, obtain weak maximum estimator based observations It turns out limit distribution can represented without time delay.