# Affine processes with compact state space

## TL;DR

This paper investigates affine processes with compact state spaces, revealing structural constraints such as the impossibility of diffusion, the grid-like nature of jumps, and the directional influence of jumps on drift, along with a classification of bivariate cases.

## Contribution

It provides a structural theorem characterizing affine processes on compact state spaces and classifies all bivariate cases, highlighting differences from classical affine processes.

## Key findings

- No diffusion occurs in compact state space affine processes.
- Jumps create a grid-like structure in the state space.
- Characteristic functions may vanish even in simple cases.

## 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.