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

Eduardo Abi Jaber
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Eduardo Abi Jaber: CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique

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

Keywords: Riccati-Volterra equations; Stochastic Volterra equations; superprocesses; Affine Volterra processes; Hawkes processes; rough volatility (search for similar items in EconPapers)
Date: 2021-08
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Published in Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2021, 27 (3), pp.1583-1615. ⟨10.3150/20-BEJ1284⟩

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DOI: 10.3150/20-BEJ1284

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