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Holomorphic jump-diffusions

Author

Listed:
  • Cuchiero, Christa
  • Primavera, Francesca
  • Svaluto-Ferro, Sara

Abstract

We introduce a class of jump-diffusions, called holomorphic, of which the well-known classes of affine and polynomial processes are particular instances. The defining property concerns the extended generator, which is required to map a (subset of) holomorphic functions to themselves. This leads to a representation of the expectation of power series of the process’ marginals via a potentially infinite dimensional linear ODE. We apply the same procedure by considering exponentials of holomorphic functions, leading to a class of processes named affine-holomorphic for which a representation for quantities as the characteristic function of power series is provided. Relying on powerful results from complex analysis, we obtain sufficient conditions on the process’ characteristics which guarantee the holomorphic and affine-holomorphic properties and provide applications to several classes of jump-diffusions.

Suggested Citation

  • Cuchiero, Christa & Primavera, Francesca & Svaluto-Ferro, Sara, 2026. "Holomorphic jump-diffusions," Stochastic Processes and their Applications, Elsevier, vol. 191(C).
    • Handle: RePEc:eee:spapps:v:191:y:2026:i:c:s030441492500225x
    DOI: 10.1016/j.spa.2025.104781
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    References listed on IDEAS

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    1. Volker Scheidemann, 2005. "Introduction to Complex Analysis in Several Variables," Springer Books, Springer, number 978-3-7643-7491-4, August.
    2. Christa Cuchiero & Josef Teichmann, 2011. "Path properties and regularity of affine processes on general state spaces," Papers 1107.1607, arXiv.org, revised Jan 2013.
    3. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A Generative Adversarial Network Approach to Calibration of Local Stochastic Volatility Models," Risks, MDPI, vol. 8(4), pages 1-31, September.
    4. Cuchiero, Christa & Di Persio, Luca & Guida, Francesco & Svaluto-Ferro, Sara, 2024. "Measure-valued affine and polynomial diffusions," Stochastic Processes and their Applications, Elsevier, vol. 175(C).
    5. Eduardo Abi Jaber & Shaun Xiaoyuan Li & Xuyang Lin, 2025. "Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models," Post-Print hal-04567783, HAL.
    6. Eduardo Abi Jaber & Louis-Amand G'erard, 2024. "Signature volatility models: pricing and hedging with Fourier," Papers 2402.01820, arXiv.org, revised Jun 2025.
    7. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A generative adversarial network approach to calibration of local stochastic volatility models," Papers 2005.02505, arXiv.org, revised Sep 2020.
    8. Eduardo Abi Jaber & Louis-Amand Gérard, 2025. "Signature volatility models: pricing and hedging with Fourier," Post-Print hal-04435238, HAL.
    9. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
    10. Christa Cuchiero & Sara Svaluto-Ferro, 2021. "Infinite-dimensional polynomial processes," Finance and Stochastics, Springer, vol. 25(2), pages 383-426, April.
    11. Elisa Alòs & Jim Gatheral & Radoš Radoičić, 2020. "Exponentiation of conditional expectations under stochastic volatility," Quantitative Finance, Taylor & Francis Journals, vol. 20(1), pages 13-27, January.
    12. Christa Cuchiero & Sara Svaluto-Ferro & Josef Teichmann, 2023. "Signature SDEs from an affine and polynomial perspective," Papers 2302.01362, arXiv.org, revised Feb 2025.
    13. Christa Cuchiero & Martin Keller-Ressel & Josef Teichmann, 2012. "Polynomial processes and their applications to mathematical finance," Finance and Stochastics, Springer, vol. 16(4), pages 711-740, October.
    14. Peter Friz & Jim Gatheral, 2022. "Diamonds and forward variance models," Papers 2205.03741, arXiv.org.
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