  In this article, we propose a Milstein finite difference scheme for a
stochastic partial differential equation (SPDE) describing a large particle
system. We show, by means of Fourier analysis, that the discretisation on an
unbounded domain is convergent of first order in the timestep and second order
in the spatial grid size, and that the discretisation is stable with respect to
boundary data. Numerical experiments clearly indicate that the same convergence
order also holds for boundary-value problems. Multilevel path simulation,
previously used for SDEs, is shown to give substantial complexity gains
compared to a standard discretisation of the SPDE or direct simulation of the
particle system. We derive complexity bounds and illustrate the results by an
application to basket credit derivatives.
