Abstract
For stationary ergodic diffusions satisfying nonlinear homogeneous Itô stochastic differential equations, this paper obtains the Berry–Esseen bounds on the rates of convergence to normality of the distributions of the quasi maximum likelihood estimators based on stochastic Taylor approximation, under some regularity conditions, when the diffusion is observed at equally spaced dense time points over a long time interval, the high-frequency regime. It shows that the higher-order stochastic Taylor approximation-based estimators perform better than the basic Euler approximation in the sense of having smaller asymptotic variance.
Highlights
Parameter estimation in diffusion processes based on discrete observations is the recent trend of investigation in financial econometrics and mathematical biology since the data available in finance and biology are high-frequency discrete, though the model is continuous
Where {Wt, t ≥ 0} is a one-dimensional standard Wiener process, θ ∈ Θ, Θ is a compact subset of R, f is a known real valued function defined on Θ × R, the unknown parameter θ is to be estimated on the basis of observation of the process { Xt, t ≥ 0}
The Itô approximate maximum likelihood estimate (IAMLE) based on Ln,T is defined as θn,T := arg max Ln,T (θ )
Summary
Introduction and PreliminariesParameter estimation in diffusion processes based on discrete observations is the recent trend of investigation in financial econometrics and mathematical biology since the data available in finance and biology are high-frequency discrete, though the model is continuous. Maximum Likelihood Estimators for the Discretely Observed Diffusions. This estimator was first studied by Dorogovcev (1976) [3], who obtained its weak consistency under some regularity conditions as T → ∞ and Tn → 0. A contrast for the estimation of θ is derived from the above log-likelihood by substituting { Zti , 0 ≤ i ≤ n} with { Xti , 0 ≤ i ≤ n}.
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