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

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.