Affine Volterra processes with jumps

TL;DR

This paper extends the theory of affine Volterra processes to include jumps, addressing challenges posed by singular kernels and potential trajectory explosions, thereby broadening the modeling capabilities of such processes.

Contribution

It introduces a framework for affine stochastic Volterra equations with jumps, handling singular kernels and trajectory explosions, which was not previously developed.

Findings

Extended affine Volterra processes to include jumps.

Addressed issues of trajectory explosions due to jumps and singular kernels.

Provided theoretical foundations for rough affine processes with jumps.

Abstract

The theory of affine processes has been recently extended to the framework of stochastic Volterra equations with continuous trajectories. These so-called affine Volterra processes overcome modeling shortcomings of affine processes because they can have trajectories whose regularity is different from the regularity of the paths of Brownian motion. More specifically, singular kernels yield rough affine processes. This paper extends the theory by considering affine stochastic Volterra equations with jumps. This extension is not straightforward because the jump structure together with possible singularities of the kernel may induce explosions of the trajectories.