  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.

Numerical Analysis;Numerical Analysis