Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels

We provide existence, uniqueness and stability results for affine stochastic Volterra equations with L1-kernels and jumps. Such equations arise as scaling limits of branching processes in population genetics and self-exciting Hawkes processes in mathematical finance. The strategy we adopt for the existence part is based on approximations using stochastic Volterra equations with L2-kernels combined with a general stability result. Most importantly, we establish weak uniqueness using a duality argument on the Fourier-Laplace transform via a deterministic Riccati-Volterra integral equation. We illustrate the applicability of our results on Hawkes processes and a class of hyper-rough Volterra Heston models with a Hurst index H ∈ (-1/2, 1/2].