The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations

Xiaoshan Chen, Yu-Jui Huang, Qingshuo Song and Chao Zhu

Papers from arXiv.org

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.

Date: 2013-08, Revised 2017-03
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Citations: View citations in EconPapers (1)

Published in Journal of Mathematical Analysis and Applications, Vol. 451, Issue 1 (2017), pp 448-472

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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:1309.0046

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