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Article Dans Une Revue Bernoulli Année : 2021

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

Eduardo Abi Jaber

Résumé

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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Dates et versions

hal-02412741 , version 1 (15-12-2019)
hal-02412741 , version 2 (18-06-2020)

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Eduardo Abi Jaber. Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels. Bernoulli, 2021, 27 (3), pp.1583-1615. ⟨10.3150/20-BEJ1284⟩. ⟨hal-02412741v2⟩
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