A simple approach to numerical methods for stochastic differential equations in Lie groups

Abstract

A large number of important engineering applications require solving stochastic differential equations (SDEs) so that geometrical constraints are satisfied, e.g. the solution has to lie in a matrix Lie group. But the few papers that properly derive numerical schemes for SDEs evolving in Lie groups are in the mathematics literature and not accessible to engineers. The engineering literature is also small but plagued with problems. With this in mind we give a direct accessible derivation of numerical schemes for solving such equations. We do not rely on differential geometry or advanced random process theory. We give simulations in a simple case to show how geometric structure is preserved.

Higher strong order methods for linear Itô SDEs on matrix Lie groups

“…A simple scheme based on the Runge-Kutta-Munthe-Kaas schemes for ODEs [23] that puts the described approach into practice can be found in [18] and is presented in the following algorithm. Algorithm 3.1 Divide the time interval [0, T ] uniformly into J subintervals [t j , t j+1 ], j = 0, 1, .…”

“…Furthermore, the available literature on Lie group SDEs mainly concerns Stratonovich SDEs [1,3,8,16,21,30]. Readers interested in Itô SDEs on Lie groups will only find the geometric Euler-Maruyama scheme with strong order γ = 1 appearing in [18,19,26] and more recently the existence and convergence proof of the stochastic Magnus expansion in [12]. However, the consideration of Itô SDEs is crucial for its application in finance.…”

Higher Strong Order Methods for Itô SDEs on Matrix Lie Groups

“…A simple scheme based on the Runge-Kutta-Munthe-Kaas schemes for ODEs [20] that puts the described approach into practice can be found in [16] and is presented in the following algorithm.…”

“…Furthermore, the available literature on Lie group SDEs mainly concerns Stratonovich SDEs [3,15,1,27]. Readers interested in Itô SDEs on Lie groups will only find the geometric Euler-Maruyama scheme with strong order γ = 1 appearing in [16,17,23] and more recent the existence and convergence proof of the stochastic Magnus expansion in [11]. However, the consideration of Itô SDEs is crucial for its application in finance and due to the geometric constraints Stratonovich SDEs on matrix Lie groups cannot simply be transformed into Itô SDEs as in the traditional, non-geometric case.…”

An engineer's guide to particle filtering on matrix Lie groups

“…For a detailed derivation see our paper [34]. Special case examples of this Euler method are found in [8,35].…”