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The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations

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  • Xiaoshan Chen
  • Yu-Jui Huang
  • Qingshuo Song
  • Chao Zhu

Abstract

We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally H\"older continuous with exponent $\delta\in (0, 1]$, the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement $\delta\ge 1/2$. Uniqueness results, including a Feynman-Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale.

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  • Xiaoshan Chen & Yu-Jui Huang & Qingshuo Song & Chao Zhu, 2013. "The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations," Papers 1309.0046, arXiv.org, revised Mar 2017.
    • Handle: RePEc:arx:papers:1309.0046
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    References listed on IDEAS

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    1. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2010. "Valuation equations for stochastic volatility models," Papers 1004.3299, arXiv.org, revised Dec 2011.
    2. Hardy Hulley & Eckhard Platen, 2008. "A Visual Classification of Local Martingales," Research Paper Series 238, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
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    Cited by:

    1. Yu-Jui Huang & Li-Hsien Sun, 2023. "Partial Information in a Mean-Variance Portfolio Selection Game," Papers 2312.04045, arXiv.org, revised May 2025.

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