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On the uniqueness of classical solutions of Cauchy problems

Authors: Erhan Bayraktar and Hao Xing
Journal: Proc. Amer. Math. Soc. 138 (2010), 2061-2064
MSC (2010): Primary 35K65, 60G44
Published electronically: January 27, 2010
MathSciNet review: 2596042
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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 is also a function of at most linear growth. In this paper, we give a condition on the volatility that is necessary and sufficient for a Cauchy problem to admit a unique solution.

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Additional Information

Erhan Bayraktar
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48104
MR Author ID: 743030
ORCID: 0000-0002-1926-4570

Hao Xing
Affiliation: Department of Mathematics and Statistics, Boston University, Boston, Massachu- setts 02215

Keywords: Cauchy problem, a necessary and sufficient condition for uniqueness, European call-type options, strict local martingales
Received by editor(s): September 16, 2009
Published electronically: January 27, 2010
Additional Notes: This research is supported in part by the National Science Foundation under grant number DMS-0906257.
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2010 By the authors