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

Abstract

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