  We present a time change construction of affine processes with state-space
$\mathbb{R}_+^m\times \mathbb{R}^n$. These processes were systematically
studied in (Duffie, Filipovi\'c and Schachermayer, 2003) since they contain
interesting classes of processes such as L\'evy processes, continuous branching
processes with immigration, and of the Ornstein-Uhlenbeck type. The
construction is based on a (basically) continuous functional of a
multidimensional L\'evy process which implies that limit theorems for L\'evy
processes (both almost sure and in distribution) can be inherited to affine
processes. The construction can be interpreted as a multiparameter time change
scheme or as a (random) ordinary differential equation driven by discontinuous
functions. In particular, we propose approximation schemes for affine processes
based on the Euler method for solving the associated discontinuous ODEs, which
are shown to converge.
