We provide existence, uniqueness and stability results for affine stochastic Volterra equations with L^1-kernels  . 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  . 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  a class of hyper-rough Volterra Heston models with a Hurst index H \in (-1/2,1/2].
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].
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and jumps
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combined with a general stability result
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Fourier-Laplace
Fourier--Laplace
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Riccati-Volterra
Riccati--Volterra
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Hawkes processes and