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

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  • Eduardo Abi Jaber

    (CES)

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

Suggested Citation

  • Eduardo Abi Jaber, 2019. "Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels," Papers 1912.07445, arXiv.org, revised Jun 2020.
    • Handle: RePEc:arx:papers:1912.07445
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    References listed on IDEAS

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    1. Dawson, Donald A. & Fleischmann, Klaus, 1994. "A super-Brownian motion with a single point catalyst," Stochastic Processes and their Applications, Elsevier, vol. 49(1), pages 3-40, January.
    2. Eduardo Abi Jaber, 2019. "Lifting the Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 19(12), pages 1995-2013, December.
    3. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    4. Eduardo Abi Jaber, 2019. "Lifting the Heston model," Post-Print hal-01890751, HAL.
    5. Eduardo Abi Jaber & Omar El Euch, 2019. "Multi-factor approximation of rough volatility models," Post-Print hal-01697117, HAL.
    6. Paul Jusselin & Mathieu Rosenbaum, 2018. "No-arbitrage implies power-law market impact and rough volatility," Papers 1805.07134, arXiv.org.
    7. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    8. Eduardo Abi Jaber, 2018. "Lifting the Heston model," Papers 1810.04868, arXiv.org, revised Nov 2019.
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    Cited by:

    1. Ackermann, Julia & Kruse, Thomas & Overbeck, Ludger, 2022. "Inhomogeneous affine Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 250-279.
    2. Christian Bayer & Fabian Andsem Harang & Paolo Pigato, 2020. "Log-modulated rough stochastic volatility models," Papers 2008.03204, arXiv.org, revised May 2021.

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