Abstract
The behavior of affine processes, which are ubiquitous in a wide range of applications, depends crucially on the choice of state space. We study the case where the state space is compact, and prove in particular that (i) no diffusion is possible; (ii) jumps are possible and enforce a grid-like structure of the state space; (iii) jump components can feed into drift components, but not vice versa. Using our main structural theorem, we classify all bivariate affine processes with compact state space. Unlike the classical case, the characteristic function of an affine process with compact state space may vanish, even in very simple cases.
Highlights
Affine processes are ubiquitous in a wide range of applications, in particular in finance, which has motivated a rich literature developing the mathematical theory of affine processes
The following is our main result regarding the structure of affine processes on compact state spaces
We end this section by noting that any finite state Markov chain can be viewed as an affine process as follows
Summary
Affine processes are ubiquitous in a wide range of applications, in particular in finance, which has motivated a rich literature developing the mathematical theory of affine processes. The state space may be a finite discrete set, which after an affine transformation only contains points with integer coordinates. This leads to arguments with a combinatorial rather than analytical flavor. In cases where the state space is a finite set, it turns out that the characteristic function may attain the value zero, which precludes the exponential-affine structure. Further unexplained notation follows [11]
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