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Strong Solutions and Quantization-Based Numerical Schemes for a Class of Non-Markovian Volatility Models

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  • Martino Grasselli
  • Gilles Pag`es

Abstract

We investigate a class of non-Markovian processes that hold particular relevance in the realm of mathematical finance. This family encompasses path-dependent volatility models, including those pioneered by [Platen and Rendek, 2018] and, more recently, by [Guyon and Lekeufack, 2023], as well as an extension of the framework proposed by [Blanc et al., 2017]. Our study unfolds in two principal phases. In the first phase, we introduce a functional quantization scheme based on an extended version of the Lamperti transformation that we propose to handle the presence of a memory term incorporated into the diffusion coefficient. For scenarios involving a Brownian integral in the diffusion term, we propose alternative numerical schemes that leverage the power of marginal recursive quantization. In the second phase, we study the problem of existence and uniqueness of a strong solution for the SDEs related to the examples that motivate our study, in order to provide a theoretical basis to correctly apply the proposed numerical schemes.

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  • Martino Grasselli & Gilles Pag`es, 2025. "Strong Solutions and Quantization-Based Numerical Schemes for a Class of Non-Markovian Volatility Models," Papers 2503.00243, arXiv.org.
    • Handle: RePEc:arx:papers:2503.00243
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    1. P. Blanc & J. Donier & J.-P. Bouchaud, 2017. "Quadratic Hawkes processes for financial prices," Quantitative Finance, Taylor & Francis Journals, vol. 17(2), pages 171-188, February.
    2. Gilles Pagès & Abass Sagna, 2015. "Recursive Marginal Quantization of the Euler Scheme of a Diffusion Process," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(5), pages 463-498, November.
    3. Gilles Zumbach, 2010. "Volatility conditional on price trends," Quantitative Finance, Taylor & Francis Journals, vol. 10(4), pages 431-442.
    4. Jim Gatheral & Paul Jusselin & Mathieu Rosenbaum, 2020. "The quadratic rough Heston model and the joint S&P 500/VIX smile calibration problem," Papers 2001.01789, arXiv.org.
    5. Enrique Sentana, 1995. "Quadratic ARCH Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 62(4), pages 639-661.
    6. 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.
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