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Path-dependent Affine Processes

Author

Listed:
  • Boris Gunther
  • Thomas Kruse
  • Ludger Overbeck
  • Thorsten Schmidt

Abstract

We extend the classical theory of affine processes to a path-dependent setting by introducing path-dependent coefficients and provide analytic formulas for their Fourier--Laplace transform in terms of generalized Riccati-type equations. In the proposed framework, we define path-dependent affine processes through their exponential-affine Fourier--Laplace transform on the path space and establish a characterization theorem. Conversely, for path-dependent stochastic differential equations with affine path-dependent coefficients, we also provide explicit exponential-affine representations of the Fourier--Laplace functional in terms of those Riccati equations. Moreover, we derive a condition ensuring non-negativity of the path-dependent diffusion coefficient, guaranteeing well-posedness of the model. Finally, we apply these results to a path-dependent volatility model and a path-dependent extension of the Heston model, including a delayed Heston model as a special case.

Suggested Citation

  • Boris Gunther & Thomas Kruse & Ludger Overbeck & Thorsten Schmidt, 2026. "Path-dependent Affine Processes," Papers 2606.23099, arXiv.org.
    • Handle: RePEc:arx:papers:2606.23099
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    References listed on IDEAS

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    1. Christa Cuchiero & Martin Keller-Ressel & Eberhard Mayerhofer & Josef Teichmann, 2016. "Affine Processes on Symmetric Cones," Journal of Theoretical Probability, Springer, vol. 29(2), pages 359-422, June.
    2. F. Comte & L. Coutin & E. Renault, 2012. "Affine fractional stochastic volatility models," Annals of Finance, Springer, vol. 8(2), pages 337-378, May.
    3. 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.
    4. Christa Cuchiero & Damir Filipovi'c & Eberhard Mayerhofer & Josef Teichmann, 2009. "Affine processes on positive semidefinite matrices," Papers 0910.0137, arXiv.org, revised Apr 2011.
    5. Cox, John C. & Ingersoll Junior, Jonathan E. & Ross, Stephen A., 2007. "A theory of the term structure of interest rates," RAE - Revista de Administração de Empresas, FGV-EAESP Escola de Administração de Empresas de São Paulo (Brazil), vol. 47(2), April.
    6. Aurélien Alfonsi, 2015. "Affine Diffusions and Related Processes: Simulation, Theory and Applications," Post-Print hal-03127212, HAL.
    7. 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).
    8. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
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