Numerical Generation of Trajectories Statistically Consistent with Stochastic Differential Equations

TL;DR

This paper introduces a new numerical method for simulating stochastic differential equations with high accuracy.

Contribution

The novel method generates trajectories by matching cumulants using Gaussian variables, achieving second-order accuracy.

Findings

The method reproduces the first three cumulants of the state variable to second-order accuracy.

It outperforms the Milstein algorithm in accuracy when tested on Büttiker’s ratchet.

The approach can be extended to generate higher-order terms in the stochastic Taylor expansion.

Abstract

A weak second-order numerical method for generating trajectories based on stochastic differential equations (SDE) is developed. The proposed approach bypasses direct noise realization by updating the system’s state using independent Gaussian random variables so as to reproduce the first three cumulants of the state variable at each time step to the second order in the time-step size. The update rule for the state variable is derived based on the system’s Fokker–Planck equation in an arbitrary number of dimensions. The high accuracy of the method as compared to the standard Milstein algorithm is demonstrated on the example of Büttiker’s ratchet. While the method is second-order accurate in the time step, it can be extended to systematically generate higher-order terms of the stochastic Taylor expansion approximating the solution of the SDE.

Figures