Splitting schemes and estimators for stochastic differential equations with H\"older multiplicative noise

TL;DR

This paper introduces explicit pseudo-likelihood estimators for SDEs with H"older multiplicative noise, utilizing splitting schemes that are both convergent and preserve the state space, improving accuracy and efficiency.

Contribution

The authors develop the first explicit pseudo-likelihood estimators based on splitting schemes for this class of SDEs, with proven convergence, robustness, and improved performance.

Findings

Estimators outperform existing methods in accuracy.

Proven strong mean-square convergence and state space preservation.

Enhanced robustness and computational efficiency.

Abstract

We study parameter estimation for univariate stochastic differential equations with locally Lipschitz drift and H\"older continuous multiplicative diffusion, a class commonly arising in several applications. Existing inference methods typically rely on either the Euler-Maruyama discretisation, despite its lack of strong convergence and failure to preserve the state space, or on approximations, e.g. Gaussian approximation or truncation of Hermite's expansions, impacting on their stability and computational efficiency. We introduce the first explicit pseudo-likelihood estimators based on numerical splitting schemes that are both strong mean-square convergent and state space preserving for this class of SDEs. Our approach is based on a novel decomposition of the SDE that exploits reducibility and the Lamperti transform, leading to Lie-Trotter (LT) and Strang splitting schemes yielding explicit pseudo-likelihoods and maximum likelihood estimators based on them. We prove strong mean-square convergence, state space preservation, and improved robustness with respect to the discretisation step compared to Euler-Maruyama-based methods. We further establish consistency and asymptotic normality of the LT estimator. Because the proposed numerical scheme couples drift and diffusion parameters in the pseudo-likelihood, the asymptotic analysis requires new proof techniques. Extensive simulations demonstrate that the proposed estimators outperform existing methods in both accuracy and computational efficiency.