Higher Strong Order Methods for It\^o SDEs on Matrix Lie Groups

TL;DR

This paper develops higher strong order numerical methods for Itô SDEs on matrix Lie groups, extending Runge-Kutta--Munthe-Kaas schemes with convergence order 1.5, applicable in engineering and finance.

Contribution

It introduces a general procedure for higher order schemes on Lie groups, including a stochastic RKMK method with strong convergence order 1.5.

Findings

Two novel schemes with strong convergence order 1.5

Conditions for stochastic RKMK to match underlying method order

Applications demonstrated in engineering and finance

Abstract

In this paper we present a general procedure for designing higher strong order methods for It\^o stochastic differential equations on matrix Lie groups and illustrate this strategy with two novel schemes that have a strong convergence order of 1.5. Based on the Runge-Kutta--Munthe-Kaas (RKMK) method for ordinary differential equations on Lie groups, we present a stochastic version of this scheme and derive a condition such that the stochastic RKMK has the same strong convergence order as the underlying stochastic Runge-Kutta method. Further, we show how our higher order schemes can be applied in a mechanical engineering as well as in a financial mathematics setting.