On the Stochastic Solution to a Cauchy Problem Associated with Nonnegative Price Processes
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.
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2012.
"Valuation equations for stochastic volatility models,"
LSE Research Online Documents on Economics
43460, London School of Economics and Political Science, LSE Library.
- Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2010. "Valuation equations for stochastic volatility models," Papers 1004.3299, arXiv.org, revised Dec 2011.
- Erhan Bayraktar & Hao Xing, 2009. "On the uniqueness of classical solutions of Cauchy problems," Papers 0908.1086, arXiv.org, revised Sep 2009.
When requesting a correction, please mention this item's handle: RePEc:arx:papers:1309.0046. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators)
If references are entirely missing, you can add them using this form.