Adaptive and non-adaptive estimation for degenerate diffusion processes

We discuss parametric estimation of a degenerate diffusion system from time-discrete observations. The first component of the degenerate diffusion system has a parameter $θ_1$ in a non-degenerate diffusion coefficient and a parameter $θ_2$ in the drift term. The second component has a drift term parameterized by $θ_3$ and no diffusion term. Asymptotic normality is proved in three different situations for an adaptive estimator for $θ_3$ with some initial estimators for ($θ_1$ , $θ_2$), an adaptive one-step estimator for ($θ_1$ , $θ_2$ , $θ_3$) with some initial estimators for them, and a joint quasi-maximum likelihood estimator for ($θ_1$ , $θ_2$ , $θ_3$) without any initial estimator. Our estimators incorporate information of the increments of both components. Thanks to this construction, the asymptotic variance of the estimators for $θ_1$ is smaller than the standard one based only on the first component. The convergence of the estimators for $θ_3$ is much faster than the other parameters. The resulting asymptotic variance is smaller than that of an estimator only using the increments of the second component.

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