Abstract
This note is devoted to continuity results of the time derivative of the solution to the one-dimensional parabolic obstacle problem with variable coefficients. It applies to the smooth fit principle in numerical analysis and in financial mathematics. It relies on various tools for the study of free boundary problems: blow-up method, monotonicity formulae, Liouville’s results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Y. Achdou, An inverse problem for parabolic variational inequalities in the calibration of American options. to appear in SIAM J. Control Optim.
A. Bensoussan and J.-L. Lions, Applications des inéquations variationnelles en contrôle stochastique, Dunod, Paris, 1978. Méthodes Mathématiques de l’Informatique, No. 6.
F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), pp. 637–659.
A. Blanchet, J. Dolbeault, and R. Monneau, On the continuity of the time derivative of the solution to the one-dimensional parabolic obstacle problem with variable coefficients. In preparation.
L. Caffarelli, A. Petrosyan, and H. Shahgholian, Regularity of a free boundary in parabolic potential theory, J. Amer. Math. Soc., 17 (2004), pp. 827–869 (electronic).
P. Dupuis and H. Wang, Optimal stopping with random intervention times, Adv. in Appl. Probab., 34 (2002), pp. 141–157.
A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, J. Functional Analysis, 18 (1975), pp. 151–176.
—, Variational principles and free-boundary problems, Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1982. A Wiley-Interscience Publication.
P. Jaillet, D. Lamberton, and B. Lapeyre, Variational inequalities and the pricing of American options, Acta Appl. Math., 21 (1990), pp. 263–289.
D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, vol. 88 of Pure and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980.
D. Lamberton and B. Lapeyre, Introduction au calcul stochastique appliqué à la finance, Ellipses Édition Marketing, Paris, second ed., 1997.
R. Monneau, On the number of singularities for the obstacle problem in two dimensions, J. Geom. Anal., 13 (2003), pp. 359–389.
J.-F. Rodrigues, Obstacle problems in mathematical physics, vol. 134 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 114.
P. Van Moerbeke, An optimal stopping problem with linear reward, Acta Math., 132 (1974), pp. 111–151.
S. Villeneuve, Options américaines dans un mod`ele de Black-Scholes multidimensionnel, PhD thesis, Université De Marne la Vallée, 1999.
G.S. Weiss, Self-similar blow-up and Hausdorff dimension estimates for a class of parabolic free boundary problems, SIAM J. Math. Anal., 30 (1999), pp. 623–644 (electronic).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Blanchet, A., Dolbeault, J., Monneau, R. (2005). On the One-dimensional Parabolic Obstacle Problem with Variable Coefficients. In: Bandle, C., et al. Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 63. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7384-9_7
Download citation
DOI: https://doi.org/10.1007/3-7643-7384-9_7
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7249-1
Online ISBN: 978-3-7643-7384-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)