Affine processes on positive semidefinite d x d matrices have jumps of finite variation in dimension d > 1

TL;DR

This paper confirms that affine processes on positive semidefinite matrices in dimensions greater than one only have jumps of finite variation, contrasting with the real line case, and explores implications for their Laplace and Fourier transforms.

Contribution

It proves the conjecture that such processes do not have infinite total variation jumps in higher dimensions, revealing a geometric phenomenon distinct from the real line case.

Findings

Affine processes in dimension d > 1 have jumps of finite variation.

The exponential affine property extends from Laplace to Fourier-Laplace transforms under certain conditions.

The result contrasts with the behavior of processes on the positive real line.

Abstract

The theory of affine processes on the space of positive semidefinite d x d matrices has been established in a joint work with Cuchiero, Filipovi\'c and Teichmann (2011). We confirm the conjecture stated therein that in dimension d greater than 1 this process class does not exhibit jumps of infinite total variation. This constitutes a geometric phenomenon which is in contrast to the situation on the positive real line (Kawazu and Watanabe, 1974). As an application we prove that the exponentially affine property of the Laplace transform carries over to the Fourier-Laplace transform if the diffusion coefficient is zero or invertible.