Parameter Estimation for Partially Observed Time-Changed SDEs

2026-05-10

AI Summary

This work addresses the challenging problem of parameter estimation for time-changed stochastic differential equations (SDEs) observed partially and sampled at discrete time points. The authors propose a novel unbiased Markov chain Monte Carlo (MCMC) algorithm that integrates likelihood-based and Bayesian frameworks. By combining unbiased score estimation, stochastic approximation, and multilevel Monte Carlo techniques, the method enables, for the first time, multilevel Bayesian inference for this class of models. Theoretical analysis demonstrates that to achieve a mean squared error of 𝒪(ε²), the computational cost scales as 𝒪(ε⁻² log(ε)²). Numerical experiments and real-data analyses confirm the method’s substantial gains in both computational efficiency and estimation accuracy compared to existing approaches.

Abstract

In this paper we consider the parameter estimation problem associated to partially-observed time changed SDEs, with observations that are given at discrete times. In particular we consider both likelihood and Bayesian estimation. We develop new Markov chain Monte Carlo (MCMC) algorithms which allow an unbiased score-based stochastic approximation method to provide likelihood-type parameter estimators. We also use a variant of this MCMC algorithm to perform multilevel-based Bayesian parameter estimation. We prove that this latter method achieves a mean square error of $\mathcal{O}(ε^2)$ ($ε>0$) with a cost of $\mathcal{O}(ε^{-2}\log(ε)^2)$. Our methodologies are tested numerically on both simulated and real data.

Problem

• parameter estimation
• partially observed
• time-changed SDEs
• discrete observations

Innovation

• time-changed SDEs
• parameter estimation
• MCMC
• multilevel Monte Carlo
• stochastic approximation