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

TL;DR

This paper establishes existence, uniqueness, and stability for affine stochastic Volterra equations with L1-kernels, relevant in population genetics and finance, using approximation and duality methods.

Contribution

It introduces a novel approach for weak uniqueness via a duality argument on the Fourier-Laplace transform for equations with L1-kernels.

Findings

Proved existence and stability for affine stochastic Volterra equations with L1-kernels.

Established weak uniqueness using a duality approach involving Riccati-Volterra equations.

Applied results to Hawkes processes and hyper-rough Volterra Heston models.

Abstract

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