Euler-Maruyama method for non-Wiener processes

TL;DR

This paper generalizes the Euler-Maruyama method to simulate non-Wiener processes, particularly non-Gaussian Lévy noise, providing more physically justified models and demonstrating their equivalence to master equations.

Contribution

It introduces a generalized Euler-Maruyama scheme for non-Wiener processes, especially Lévy processes, with an example showing improved physical realism.

Findings

Non-Gaussian Lévy noise can be simulated using the generalized Euler-Maruyama method.

The method yields results superior to geometric Brownian motion in a physical example.

Additive noise results are equivalent to a master equation via Kramers-Moyal expansion.

Abstract

Descriptions of complex physical or biological systems often include stochastic contributions, and these are commonly simulated using Wiener processes. In many cases however, non-Gaussian fluctuations may originate from non-Wiener processes which remain less explored. The Euler-Maruyama method of discretising stochastic differential equations to non-Wiener processes is generalised. Non-Gaussian noise generated from a subset of L\'evy processes can be used simply and often with more physical justification, for both additive and multiplicative noise. An example of this is provided that gives superior physical results compared to using geometric Brownian motion. Finally the results of the additive noise are shown to be equivalent to a derived master equation via the Kramers-Moyal expansion.