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

Affine processes on $\mathbb{R}_+^n \times \mathbb{R}^n$ and multiparameter time changes