Adaptive estimation for degenerate diffusion processes

Abstract

We discuss parametric estimation of a degenerate diffusion system from time-discrete observations. The first component of the degenerate diffusion system has a parameter ${\mathrm{𝜃}}_1$ in a non-degenerate diffusion coefficient and a parameter ${\mathrm{𝜃}}_2$ in the drift term. The second component has a drift term parameterized by ${\mathrm{𝜃}}_3$ and no diffusion term. Asymptotic normality is proved in two different situations for an adaptive estimator for ${\mathrm{𝜃}}_3$ with some initial estimators for $({\mathrm{𝜃}}_1,{\mathrm{𝜃}}_2)$, and an adaptive one-step estimator for $({\mathrm{𝜃}}_1,{\mathrm{𝜃}}_2,{\mathrm{𝜃}}_3)$ with some initial estimators for them. Our estimators incorporate information of the increments of both components. Thanks to this construction, the asymptotic variance of the estimators for ${\mathrm{𝜃}}_1$ is smaller than the standard one based only on the first component. The convergence of the estimators for ${\mathrm{𝜃}}_3$ 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.