A minimum-distance estimator for diffusion processes with ergodic properties.
- Dietz, H. M.
- Kutoyants, Y.
2010 Mathematics Subject Classification
- 62M05 60J60
- minimum distance estimator, diffusion process, ergodic property, consistency, asymptotic normality
Suppose one observes one path of a stochastic process X = (Xt)t ≥ 0 which is known to solve an equation of the form
dXθt = S(θ, Xθt)dt + dWt, t ≥ 0, θ ∈ Θ ⊂ ℝd (0.1)
with a given coefficient functional S and given initial condition X0, where Θ is a non-void bounded open subset of ℝd. In order to estimate the true but unknown parameter θ0 the paper proposes the minimum distance estimator (MDE) ̂θT given by
̂θT ∈ arg inf θ ∈ Θ ∫T0 (Xt-X(θ)t)2dt, T > 0, (0.2)
X (θ)t ≔ X0 + ∫t0 S(θ,Xu) du, t ≥ 0 (0.3)
and studies its asymptotic behaviour as T → ∞. Under the main assumption that the observed process has an ergodic property and some further (less restrictive) conditions it is shown that ̂θT is strongly consistent and - in case d = 1 - asymptotically normal. In particular, the results apply to models where S(θ,x) = S(θ-x). Several examples and a comparison with likelihood estimation are added.