  One of the key ingredients to successfully apply Stein's method for
distributional approximation are solutions to the Stein equations and their
derivatives. Using Barbour's generator approach, one can solve for the
solutions to the Stein equation in terms of the semi-group of a Markov process,
which is typically a diffusion process if it is a continuous distribution. For
an arbitrary diffusion it can a difficult task to evaluate the semi-group and
its derivatives. In this paper, for polynomial test functions, instead of
calculating the semi-group of a diffusion, via a duality argument, we instead
utilise the semi-group of a much simpler Markov jump process. This approach
yields a new method for explicitly solving for the solutions of Stein equations
for diffusion processes. We present both the general idea of the approach and
examples for both univariate and multivariate distributions.
