A change of measure enhanced near exact Euler Maruyama scheme for the solution to nonlinear stochastic dynamical systems

TL;DR

This paper introduces a Girsanov transformation-based method with rejection sampling to enhance the Euler-Maruyama scheme, achieving near-exact solutions for nonlinear stochastic dynamical systems.

Contribution

It develops a novel change of measure approach with rejection sampling to improve Euler-Maruyama accuracy for nonlinear stochastic systems.

Findings

Achieves near-exact approximation of system states

Demonstrates effectiveness on nonlinear test problems

Provides a more accurate alternative to traditional Euler-Maruyama

Abstract

The present study utilizes the Girsanov transformation based framework for solving a nonlinear stochastic dynamical system in an efficient way in comparison to other available approximate methods. In this approach, a rejection sampling is formulated to evaluate the Radon-Nikodym derivative arising from the change of measure due to Girsanov transformation. The rejection sampling is applied on the Euler Maruyama approximated sample paths which draw exact paths independent of the diffusion dynamics of the underlying dynamical system. The efficacy of the proposed framework is ensured using more accurate numerical as well as exact nonlinear methods. Finally, nonlinear applied test problems are considered to confirm the theoretical results. The test problems demonstrates that the proposed exact formulation of the Euler-Maruyama provides an almost exact approximation to both the displacement and velocity states of a second order non-linear dynamical system.