Generalized weakly corrected Milstein solutions to stochastic differential equations

TL;DR

This paper introduces generalized weakly corrected Milstein schemes for nonlinear stochastic differential equations, employing change of measures to improve solution accuracy with coarser time steps.

Contribution

It presents a novel correction method for Milstein schemes using change of measures to reduce errors in stochastic differential equation solutions.

Findings

Corrected schemes outperform classical Milstein with coarser steps

Numerical results show improved accuracy in stochastic oscillators

Method effectively accounts for measure transformation errors

Abstract

In this work, weakly corrected explicit, semi-implicit and implicit Milstein approximations are presented for the solution of nonlinear stochastic differential equations. The solution trajectories provided by the Milstein schemes are corrected by employing the \textit{change of measures}, aimed at removing the error associated with the diffusion process incurred due to the transformation between two probability measures. The change of measures invoked in the Milstein schemes ensure that the solution from the mapping is measurable with respect to the filtration generated by the error process. The proposed scheme incorporates the error between the approximated mapping and the exact representation as an innovation, that is accounted for, in the Milstein trajectories as an additive term. Numerical demonstration using a parametrically and non-parametrically excited stochastic oscillators, demonstrates the improvement in the solution accuracy for the corrected schemes with coarser time steps when compared with the classical Milstein approximation with finer time steps.