  We address parameter estimation in second-order stochastic differential
equations (SDEs), prevalent in physics, biology, and ecology. Second-order SDE
is converted to a first-order system by introducing an auxiliary velocity
variable raising two main challenges. First, the system is hypoelliptic since
the noise affects only the velocity, making the Euler-Maruyama estimator
ill-conditioned. To overcome that, we propose an estimator based on the Strang
splitting scheme. Second, since the velocity is rarely observed we adjust the
estimator for partial observations. We present four estimators for complete and
partial observations, using full likelihood or only velocity marginal
likelihood. These estimators are intuitive, easy to implement, and
computationally fast, and we prove their consistency and asymptotic normality.
Our analysis demonstrates that using full likelihood with complete observations
reduces the asymptotic variance of the diffusion estimator. With partial
observations, the asymptotic variance increases due to information loss but
remains unaffected by the likelihood choice. However, a numerical study on the
Kramers oscillator reveals that using marginal likelihood for partial
observations yields less biased estimators. We apply our approach to
paleoclimate data from the Greenland ice core and fit it to the Kramers
oscillator model, capturing transitions between metastable states reflecting
observed climatic conditions during glacial eras.
