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On the Stochastic Solution to a Cauchy Problem Associated with Nonnegative Price Processes

  • Xiaoshan Chen
  • Yu-Jui Huang
  • Qingshuo Song
  • Chao Zhu
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    We consider the stochastic solution to a Cauchy problem corresponding to a nonnegative diffusion with zero drift, which represents a price process under some risk-neutral measure. When the diffusion coefficient is locally Holder continuous with some exponent in (0,1], the stochastic solution is shown to be a classical solution. A comparison theorem for the Cauchy problem is also proved, without the linear growth condition on the diffusion coefficient. Moreover, we establish the equivalence: the stochastic solution is the unique classical solution to the Cauchy problem if, and only if, a comparison theorem holds. For the case where the stochastic solution may not be smooth, we characterize it as a limit of smooth stochastic solutions associated with some approximating Cauchy problems.

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    Paper provided by in its series Papers with number 1309.0046.

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    Date of creation: Aug 2013
    Date of revision: May 2014
    Handle: RePEc:arx:papers:1309.0046
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    1. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2010. "Valuation equations for stochastic volatility models," Papers 1004.3299,, revised Dec 2011.
    2. Erhan Bayraktar & Hao Xing, 2009. "On the uniqueness of classical solutions of Cauchy problems," Papers 0908.1086,, revised Sep 2009.
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