Parameter Estimation with Increased Precision for Elliptic and Hypo-elliptic Diffusions

TL;DR

This work advances parametric inference for discretely observed diffusion processes by developing explicit weak second-order sampling schemes for both elliptic and hypo-elliptic SDEs and deriving a small-time density expansion to proxy the true transition density. The approach yields likelihood-based estimators with analytic guarantees in both high-frequency and low-frequency observation regimes, including improved rates for hypo-elliptic models (e.g., Δ_n = o(n^{-1/3})). Two low-frequency density schemes are proposed to support Bayesian data augmentation, with discretisation bias shown to be O(M^{-2}) for elliptic cases. Numerical experiments on neural-mass and epidemiological models illustrate substantial gains in diffusion-parameter estimation and posterior fidelity when using the new schemes versus standard methods. Overall, the methodology provides a rigorous, practical path to increased estimation precision for a broad class of diffusion models.

Abstract

This work aims at making a comprehensive contribution in the general area of parametric inference for discretely observed diffusion processes. Established approaches for likelihood-based estimation invoke a time-discretisation scheme for the approximation of the intractable transition dynamics of the Stochastic Differential Equation (SDE) model over finite time periods. The scheme is applied for a step-size that is either user-selected or determined by the data. Recent research has highlighted the critical ef-fect of the choice of numerical scheme on the behaviour of derived parameter estimates in the setting of hypo-elliptic SDEs. In brief, in our work, first, we develop two weak second order sampling schemes (to cover both hypo-elliptic and elliptic SDEs) and produce a small time expansion for the density of the schemes to form a proxy for the true intractable SDE transition density. Then, we establish a collection of analytic results for likelihood-based parameter estimates obtained via the formed proxies, thus providing a theoretical framework that showcases advantages from the use of the developed methodology for SDE calibration. We present numerical results from carrying out classical or Bayesian inference, for both elliptic and hypo-elliptic SDEs.

Figures (3)

• Figure 1: Root mean squared errors (RMSE) for estimators for $\sigma_2$ (50 replicates) in the Jansen-Rit neural mass model (\ref{['eq:jrnmm']}) under scenarios JR-1,2,3.
• Figure 2: Posterior Estimates for SIR Model. The blue (left panel) and orange (right panel) histograms and contour plots are obtained by HMC that uses the EM scheme and the weak second order scheme, respectively, both with the same discretisation step $\delta_M = 0.05$. The black histograms and contour plots superimposed in both plots show the correct quantities, as obtained from HMC that uses the EM scheme with very small $\delta_M = 10^{-3}$.
• Figure 5: Root mean squared errors (RMSEs) obtained from $20$ replications of estimates for $\sigma$ in the stochastic FitzHugh-Nagumo model (\ref{['eq:FN']}), under scenarios FN-1,2,3.

Theorems & Definitions (41)

• Remark 2.1
• Remark 2.2
• Proposition 2.1
• Remark 2.3
• Remark 2.4
• Lemma 3.1
• Remark 3.1
• Remark 3.2
• Remark 4.1
• Theorem 4.1: Consistency