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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
DOI: https://doi.org/10.1090/S0002-9939-10-10306-2
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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  • 1. E. Bayraktar, K. Kardaras, and H. Xing.
    Strict local martingale deflators and American call-type options.
    Technical report, Boston University and University of Michigan, 2009.
    Available at http://arxiv.org/pdf/0908.1082.
  • 2. P. Carr, A. S. Cherny, and M. Urusov.
    On the martingale property of time-homogeneous diffusions.
    Technical report, Berlin University of Technology, 2009.
    Available at http://www.math.tu-berlin.de/˜urusov/papers/cev.pdf.
  • 3. A. Cox and D. Hobson.
    Local martingales, bubbles and option prices.
    Finance & Stochastics, 9:477-492, 2005. MR 2213778 (2006j:91146)
  • 4. F. Delbaen and H. Shirakawa.
    No arbitrage condition for positive diffusion price processes.
    Asia-Pacific Financial Markets, 9(3-4):159-168, 2002.
  • 5. E. Ekström and J. Tysk.
    Bubbles, convexity and the Black-Scholes equation.
    Annals of Applied Probability, 19 (4):1369-1384, 2009. MR 2538074
  • 6. H. J. Engelbert and W. Schmidt.
    On one-dimensional stochastic differential equations with generalized drift.
    In Stochastic differential systems (Marseille-Luminy, 1984), volume 69 of Lecture Notes in Control and Inform. Sci., pages 143-155. Springer, Berlin, 1985. MR 798317 (86m:60144)
  • 7. D. Fernholz and I. Karatzas. On optimal arbitrage. Technical report, Columbia University, 2008. Available at http://www.e-publications.org/ims/submissions/index.php/AAP/user/ submissionFile/3403?confirm=lac329b5. To appear in Annals of Applied Probability.
  • 8. A. Friedman.
    Stochastic differential equations and applications. Vol. 1.
    Probability and Mathematical Statistics, Vol. 28.
    Academic Press [Harcourt Brace Jovanovich, Publishers], New York, 1975. MR 0494490 (58:13350a)
  • 9. S. L. Heston, M. Loewenstein, and G. A. Willard.
    Options and bubbles.
    Review of Financial Studies, 20(2):359-390, 2007.
  • 10. I. Karatzas and S. E. Shreve.
    Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, second edition, 1991. MR 1121940 (92h:60127)
  • 11. I. Karatzas and S. E. Shreve.
    Methods of mathematical finance.
    Springer-Verlag, New York, 1998. MR 1640352 (2000e:91076)
  • 12. A. Mijatovic and M. Urusov.
    On the martingale property of certain local martingales.
    Technical report, Imperial College London, 2009.
    Available at http://arxiv.org/abs/0905.3701v2.

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Additional Information

Erhan Bayraktar
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48104
Email: erhan@umich.edu

Hao Xing
Affiliation: Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215
Email: haoxing@bu.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10306-2
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

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