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

Eduardo Abi Jaber
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Eduardo Abi Jaber: CES

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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 $H \in (-1/2,1/2]$.

Date: 2019-12, Revised 2020-06
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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:1912.07445

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