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On the uniqueness of classical solutions of Cauchy problems

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  • Erhan Bayraktar
  • Hao Xing

Abstract

Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative (i.e., the volatility) is also a function of at most linear growth. In this note, we give a condition on the volatility that is necessary and sufficient for a Cauchy problem to admit a unique solution.

Suggested Citation

  • Erhan Bayraktar & Hao Xing, 2009. "On the uniqueness of classical solutions of Cauchy problems," Papers 0908.1086, arXiv.org, revised Sep 2009.
    • Handle: RePEc:arx:papers:0908.1086
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    References listed on IDEAS

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    1. Steven L. Heston & Mark Loewenstein & Gregory A. Willard, 2007. "Options and Bubbles," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 359-390.
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    Cited by:

    1. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2010. "Valuation equations for stochastic volatility models," Papers 1004.3299, arXiv.org, revised Dec 2011.
    2. Cetin, Umut, 2018. "Diffusion transformations, Black-Scholes equation and optimal stopping," LSE Research Online Documents on Economics 87261, London School of Economics and Political Science, LSE Library.
    3. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2012. "Strict local martingale deflators and valuing American call-type options," Finance and Stochastics, Springer, vol. 16(2), pages 275-291, April.
    4. Xing, Hao, 2012. "On backward stochastic differential equations and strict local martingales," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2265-2291.
    5. Qingshuo Song, 2011. "Approximating Functional of Local Martingale Under the Lack of Uniqueness of Black-Scholes PDE," Papers 1102.2285, arXiv.org, revised Sep 2012.

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