  In this article, we consider diffusion approximations for a general class of
stochastic recursions. Such recursions arise as models for population growth,
genetics, financial securities, multiplicative time series, numerical schemes
and MCMC algorithms. We make no particular probabilistic assumptions on the
type of noise appearing in these recursions. Thus, our technique is well suited
to recursions where the noise sequence is not a semi-martingale, even though
the limiting noise may be. Our main theorem assumes a weak limit theorem on the
noise process appearing in the random recursions and lifts it to diffusion
approximation for the recursion itself. To achieve this, we approximate the
recursion (pathwise) by the solution to a stochastic equation driven by
piecewise smooth paths; this can be thought of as a pathwise version of
backward error analysis for SDEs. We then identify the limit of this stochastic
equation, and hence the original recursion, using tools from rough path theory.
We provide several examples of diffusion approximations, both new and old, to
illustrate this technique.
