Abstract
Suppose one observes a path of a stochastic processX = (Xt)t≥0 driven by the equation
dXt=θ a(Xt)dt + dWt, t≥0, θ ≥ 0
with a(x) = x or a(x) = |x|α for some α ∈ [0,1) and given initial condition X 0. If the true but unknown parameter θ0 is positive then X is non-ergodic. It is shown that in this situation a trajectory fitting estimator for θ0 is strongly consistent and has the same limiting distribution as the maximum likelihood estimator, but converges of minor order.
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Dietz, H.M. Asymptotic Behaviour of Trajectory Fitting Estimators for Certain Non-ergodic SDE. Statistical Inference for Stochastic Processes 4, 249–258 (2001). https://doi.org/10.1023/A:1012254332474
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DOI: https://doi.org/10.1023/A:1012254332474