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

Abstract

We provide existence, uniqueness and stability results for affine stochastic Volterra equations with ${\mathit{L}}^1$-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 ${\mathit{L}}^2$-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 $\mathit{H}\in (-1/2,1/2]$.