  We study solutions to nonlinear stochastic differential systems driven by a
multi-dimensional Wiener process. A useful algorithm for strongly simulating
such stochastic systems is the Castell--Gaines method, which is based on the
exponential Lie series. When the diffusion vector fields commute, it has been
proved that at low orders this method is more accurate in the mean-square error
than corresponding stochastic Taylor methods. However it has also been shown
that when the diffusion vector fields do not commute, this is not true for
strong order one methods. Here we prove that when there is no drift, and the
diffusion vector fields do not commute, the exponential Lie series is usurped
by the sinh-log series. In other words, the mean-square error associated with a
numerical method based on the sinh-log series, is always smaller than the
corresponding stochastic Taylor error, in fact to all orders. Our proof
utilizes the underlying Hopf algebra structure of these series, and a
two-alphabet associative algebra of shuffle and concatenation operations. We
illustrate the benefits of the proposed series in numerical studies.
