The stochastic solution to a Cauchy problem for degenerate parabolic equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:451

Article

Chen, Xiaoshan¹  Huang, Yu-Jui²  Song, Qingshuo³  Zhu, Chao⁴ 

[1] South China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R China

[2] Univ Colorado, Dept Appl Math, Boulder, CO 80303 USA

[3] City Univ Hong Kong, Dept Math, 83 Tat Chee Ave, Kowloon Tong, Hong Kong, Peoples R China

[4] Univ Wisconsin Milwaukee, Dept Math Sci, Milwaukee, WI 53201 USA

关键词: Local martingales;    Stochastic solutions;    Degenerate Cauchy problems;    Feynman Kac formula;    Necessary and sufficient condition for uniqueness;    Comparison principle;   

DOI  :  10.1016/j.jmaa.2017.02.021

来源: Elsevier

【 摘 要 】

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 Holder continuous with exponent delta is an element of (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 >= 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. (C) 2017 Elsevier Inc. All rights reserved.

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