  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.
