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Global existence, regularity and a probabilistic scheme for a class of ultraparabolic Cauchy problems

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  • Christian Fries
  • Joerg Kampen

Abstract

In this paper we establish a constructive method in order to show global existence and regularity for a class of degenerate parabolic Cauchy problems which satisfy a weak Hoermander condition on a subset of the domain where the data are measurable and which have regular data on the complementary set of the domain. This result has practical incentives related to the computation of Greeks in reduced LIBOR market models, which are standard computable approximations of the HJM-description of interest rate markets. The method leads to a probabilistic scheme for the computation of the value function and its sensitivities based on Malliavin calculus. From a practical perspective the main contribution of the paper is an Monte-Carlo algorithm which includes weight corrections for paths which move in time into a region where a (weak) Hoermander condition holds.

Suggested Citation

  • Christian Fries & Joerg Kampen, 2010. "Global existence, regularity and a probabilistic scheme for a class of ultraparabolic Cauchy problems," Papers 1002.5031, arXiv.org, revised Oct 2012.
  • Handle: RePEc:arx:papers:1002.5031
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    References listed on IDEAS

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    1. Jérôme Detemple & René Garcia & Marcel Rindisbacher, 2005. "Asymptotic Properties of Monte Carlo Estimators of Derivatives," Management Science, INFORMS, vol. 51(11), pages 1657-1675, November.
    2. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
    3. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286.
    4. Nicolas Merener & Paul Glasserman, 2003. "Numerical solution of jump-diffusion LIBOR market models," Finance and Stochastics, Springer, vol. 7(1), pages 1-27.
    5. Raoul Pietersz & Antoon Pelsser & Marcel van Regenmortel, 2005. "Fast drift approximated pricing in the BGM model," Finance 0502005, EconWPA.
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