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Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation

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  • Aur'elien Alfonsi

Abstract

This work defines and studies convolution kernels that preserve nonnegativity. When the past dynamics of a process is integrated with a convolution kernel like in Stochastic Volterra Equations or in the jump intensity of Hawkes processes, this property allows to get the nonnegativity of the integral. We give characterizations of these kernels and show in particular that completely monotone kernels preserve nonnegativity. We then apply these results to analyze the stochastic invariance of a closed convex set by Stochastic Volterra Equations. We also get a comparison result in dimension one. Last, when the kernel is a positive linear combination of decaying exponential functions, we present a second order approximation scheme for the weak error that stays in the closed convex domain under suitable assumptions. We apply these results to the rough Heston model and give numerical illustrations.

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  • Aur'elien Alfonsi, 2023. "Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation," Papers 2302.07758, arXiv.org.
    • Handle: RePEc:arx:papers:2302.07758
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    References listed on IDEAS

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    1. Richard, Alexandre & Tan, Xiaolu & Yang, Fan, 2021. "Discrete-time simulation of Stochastic Volterra equations," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 109-138.
    2. Alfonsi Aurélien & Alexander Schied & Alla Slynko, 2012. "Order Book Resilience, Price Manipulation, and the Positive Portfolio Problem," Post-Print hal-00941333, HAL.
    3. Syoiti Ninomiya & Nicolas Victoir, 2008. "Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(2), pages 107-121.
    4. Abi Jaber, Eduardo & Bouchard, Bruno & Illand, Camille, 2019. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1726-1748.
    5. Eduardo Abi Jaber & Omar El Euch, 2019. "Multi-factor approximation of rough volatility models," Post-Print hal-01697117, HAL.
    6. E. Bacry & S. Delattre & M. Hoffmann & J. F. Muzy, 2013. "Modelling microstructure noise with mutually exciting point processes," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 65-77, January.
    7. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Modelling microstructure noise with mutually exciting point processes," Post-Print hal-01313995, HAL.
    8. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-02367200, HAL.
    9. Omar Euch & Masaaki Fukasawa & Mathieu Rosenbaum, 2018. "The microstructural foundations of leverage effect and rough volatility," Finance and Stochastics, Springer, vol. 22(2), pages 241-280, April.
    10. Wang, Zhidong, 2008. "Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1062-1071, July.
    11. Christa Cuchiero & Josef Teichmann, 2019. "Markovian lifts of positive semidefinite affine Volterra-type processes," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 407-448, December.
    12. Jim Gatheral, 2010. "No-dynamic-arbitrage and market impact," Quantitative Finance, Taylor & Francis Journals, vol. 10(7), pages 749-759.
    13. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    14. Christa Cuchiero & Josef Teichmann, 2019. "Markovian lifts of positive semidefinite affine Volterra type processes," Papers 1907.01917, arXiv.org, revised Sep 2019.
    15. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Post-Print hal-02367200, HAL.
    16. Cattiaux, Patrick & Colombani, Laetitia & Costa, Manon, 2022. "Limit theorems for Hawkes processes including inhibition," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 404-426.
    17. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    18. Aurélien Alfonsi, 2015. "Affine Diffusions and Related Processes: Simulation, Theory and Applications," Post-Print hal-03127212, HAL.
    19. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 309-348, January.
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