Parameter Estimation for Weakly Interacting Hypoelliptic Diffusions

We study parameter estimation for interacting particle systems (IPSs) consisting of $N$ weakly interacting multivariate hypoelliptic SDEs. We propose a locally Gaussian approximation of the transition dynamics, carefully designed to address the degenerate structure of the noise (diffusion matrix), thus leading to the formation of a well-defined full likelihood. Our approach permits carrying out statistical inference for a wide class of hypoelliptic IPSs that are not covered by recent works as the latter rely on the Euler-Maruyama scheme. We analyze a contrast estimator based on the developed likelihood with $n$ high-frequency particle observations over a fixed period $[0,T]$ and show its asymptotic normality as $n, N \to \infty$ with a requirement that the step-size $Δ_n = T/n$ is such that $NΔ_n\rightarrow 0$, assuming that all particle coordinates (e.g.~position and velocity) are observed. In practical situations where only partial observations (e.g. particle positions but not velocities) are available, the proposed locally Gaussian approximation offers greater flexibility for inference, when combined with established Bayesian techniques. In particular, unlike the Euler-Maruyama-based approaches, we do not have to impose restrictive structures on the hypoelliptic IPSs. We present numerical experiments that illustrate the effectiveness of our approach, both with complete and partial particle observations.