  In this paper an exact rejection algorithm for simulating paths of the
coupled Wright-Fisher diffusion is introduced. The coupled Wright-Fisher
diffusion is a family of multidimensional Wright-Fisher diffusions that have
drifts depending on each other through a coupling term and that find
applications in the study of interacting genes' networks as those encountered
in studies of antibiotic resistance. Our algorithm uses independent neutral
Wright-Fisher diffusions as candidate proposals, which can be sampled exactly
by means of existing algorithms and are only needed at a finite number of
points. Once a candidate is accepted, the remaining of the path can be
recovered by sampling from a neutral multivariate Wright-Fisher bridge, for
which we also provide an exact sampling strategy. The technique relies on a
modification of the alternating series method and extends existing algorithms
that are currently available for the one-dimensional case. Finally, the
algorithm's complexity is derived and its performance demonstrated in a
simulation study.
