Strang splitting estimator for nonlinear multivariate stochastic differential equations with Pearson-type multiplicative noise

TL;DR

This paper introduces a new parameter estimator for nonlinear multivariate stochastic differential equations with Pearson-type multiplicative noise, utilizing Strang splitting to improve estimation accuracy.

Contribution

The main novelty is the development of a Strang splitting-based estimator for nonlinear Pearson-type diffusions, with proven consistency and efficiency.

Findings

Estimator outperforms Euler-Maruyama and other methods in simulations.

Introduces the Student Kramers oscillator model with proven properties.

Application to Greenland ice core data demonstrates practical utility.

Abstract

Multivariate Pearson diffusions are characterized by a linear drift and a diffusion matrix that is quadratic in the state variables. We derive closed-form expressions for the mean and covariance matrix of this class using matrix exponential integrals, and extend this framework to a broader class of nonlinear diffusions with Pearson-type multiplicative noise. The main contribution is a new parameter estimator for these nonlinear multiplicative models based on Strang splitting, which decomposes the stochastic system into a deterministic nonlinear ordinary differential equation and a multivariate Pearson diffusion. The estimator is constructed by composing their respective flows and applying a Gaussian transition approximation with exact moments from the Pearson component. We prove that the estimator is consistent and asymptotically efficient. We also introduce a new model within this class, the Student Kramers oscillator, and prove existence and uniqueness of the strong solution and of an invariant measure. We evaluate the estimator through simulation studies on this oscillator and on the multivariate Wright-Fisher diffusion from population genetics, where it outperforms the Euler-Maruyama, Gaussian approximation, and local linearization estimators. We conclude with an application to Greenland ice core data using the Student Kramers oscillator.