Nearly unstable family of stochastic processes given by stochastic differential equations with time delay
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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 d X^(ϑ)(t) = ϑ∫_[-r,0] X^(ϑ)(t + u) a(d u) d t + d W(t) , t > 0, where ϑ∈R is a parameter and (W(t))_t> 0 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 (r_ϑ,T)_T∈(0,∞) satisfying r_ϑ,T→ 0 as T →∞. A family {(X^(ϑ_T)(t))_t∈[-r,T] : T ∈ (0, ∞)} is said to be nearly unstable as T →∞ if ϑ_T →ϑ as T →∞. For every α∈R, we prove convergence of the likelihood ratio processes of the nearly unstable families {(X^(ϑ+α r_ϑ,T)(t))_t∈[-r,T]: T ∈ (0, ∞)} as T →∞. As a consequence, we obtain weak convergence of the maximum likelihood estimator α̂_T of α based on the observations (X^(ϑ+α r_ϑ,T)(t))_t∈[-r,T] as T →∞. It turns out that the limit distribution of α̂_T as T →∞ can be represented as the maximum likelihood estimator of a parameter of a process satisfying a stochastic differential equation without time delay.
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