  A recent paper of Melbourne & Stuart, A note on diffusion limits of chaotic
skew product flows, Nonlinearity 24 (2011) 1361-1367, gives a rigorous proof of
convergence of a fast-slow deterministic system to a stochastic differential
equation with additive noise. In contrast to other approaches, the assumptions
on the fast flow are very mild.
  In this paper, we extend this result from continuous time to discrete time.
Moreover we show how to deal with one-dimensional multiplicative noise. This
raises the issue of how to interpret certain stochastic integrals; it is proved
that the integrals are of Stratonovich type for continuous time and neither
Stratonovich nor Ito for discrete time.
  We also provide a rigorous derivation of superdiffusive limit where the
stochastic differential equation is driven by a stable Levy process. In the
case of one-dimensional multiplicative noise, the stochastic integrals are of
Marcus type both in the discrete and continuous time contexts.
