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Publication Date:
February 2012
ISSN:
1572-9176
DOI:
10.1515/gmj-2012-0007

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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Shervashidze, T.

Editorial Board Member: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, J. / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

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Stability estimates for the multidimensional elliptic obstacle problem

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1Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore, 54600, Pakistan

2A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 2 University Str., Tbilisi 0186, Georgia

Citation Information: Georgian Mathematical Journal. Volume 19, Issue 1, Pages 1–18, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: 10.1515/gmj-2012-0007, February 2012

Publication History:
Received:
2011-04-02
Published Online:
2012-02-29

Abstract.

A Hölder type stability estimate in the second order Sobolev space of functions is established for the solution of the elliptic obstacle problem with respect to variations of the coefficients of the corresponding differential operator. In modern finance, this estimate provides results on the sensitivity of a perpetual American option price with respect to variations of the volatilities in the underlying assets prices. To meet the objective of this paper, the stability result with respect to external force functions, which was proved by J.-F. Rodrigues under the nondegeneracy condition, is generalized to the case of arbitrary functions belonging to the space , .

Keywords.: Unilateral elliptic obstacle problem; stability estimates with respect to the coefficients of a differential operator; sensitivity of perpetual American option prices with respect to volatilities

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