Parameter Inference for Degenerate Diffusion Processes

TL;DR

This paper develops a new time-discretisation method for parameter inference in highly degenerate ergodic diffusion processes, including complex models like generalized Langevin equations, supported by asymptotic analysis and simulation results.

Contribution

It introduces a tailored discretisation scheme for highly degenerate SDEs, extending inference methods to broader classes of hypo-elliptic models.

Findings

The scheme overcomes biases in high-frequency data.

Simulation studies confirm effectiveness even with partial observations.

Asymptotic results support the scheme's theoretical validity.

Abstract

We study parametric inference for ergodic diffusion processes with a degenerate diffusion matrix. Existing research focuses on a particular class of hypo-elliptic SDEs, with components split into `rough'/`smooth' and noise from rough components propagating directly onto smooth ones, but some critical model classes arising in applications have yet to be explored. We aim to cover this gap, thus analyse the highly degenerate class of SDEs, where components split into further sub-groups. Such models include e.g. the notable case of generalised Langevin equations. We propose a tailored time-discretisation scheme and provide asymptotic results supporting our scheme in the context of high-frequency, full observations. The proposed discretisation scheme is applicable in much more general data regimes and is shown to overcome biases via simulation studies also in the practical case when only a smooth component is observed. Joint consideration of our study for highly degenerate SDEs and existing research provides a general `recipe' for the development of time-discretisation schemes to be used within statistical methods for general classes of hypo-elliptic SDEs.