The Weighted Square Integral Inequalities for the First Derivative of the Function of a Real Variable
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Research Article The Weighted Square Integral Inequalities for the First Derivative of the Function of a Real Variable ˇ c, ´ 1, 2 and M. Shashiashvili1, 3 S. Hussain,1 J. Pecari 1
Abdus-Salam School of Mathematical Sciences, GC University, 35-C-2 Gulberg-III, Lahore 54660, Pakistan 2 Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia 3 Department of Probability Theory and Mathematical Statistics Chair, Tbilisi State University, 2 University Street, Tblisi 0143, Georgia Correspondence should be addressed to S. Hussain, [email protected] Received 20 May 2008; Accepted 30 July 2008 Recommended by Wing-Sum Cheung We generalize the square integral estimate for the derivative of the convex function by Shashiashvili 2005 to the case of the family of the weight functions, satisfying certain conditions. This kind of generalization is especially valuable in the problems of mathematical finance for construction of the discrete time hedging strategies. Copyright q 2008 S. Hussain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction The role of mathematical inequalities within the mathematical branches as well as in its enormous applications should not be underestimated. The appearance of the new mathematical inequality often puts on firm foundation for the heuristic algorithms and procedures used in applied sciences. The present paper considers new type of weighted square integral inequalities for the first derivative of the convex function, the particular case which has been originally established by K. Shashiashvili and M. Shashiashvili 1 and subsequently applied to the hedging problems of mathematical finance by Hussain and Shashiashvili 2. The convexity property of the value functions of the various problems in finance leads to deep and unexpected results of great practical importance for the traders and practitioners dealing with the real-world financial markets. For example, it is shown in the article 3 by El Karoui et al. see also Hobson 4 that the value functions of the European as well as American options are convex with respect to the underlying stock price; and the latter
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Journal of Inequalities and Applications
property gives us the following remarkable robustness result. Even if the writer of the option uses incorrect mathematical model to describe the dynamics of stock prices, he is able to carry out his liabilities if only the incorrectly chosen volatility dominates the true volatility function. The present paper is organized as follows. In Section 2 we prove the weighted square integral estimates for the first derivative of a function that is assumed to be twice continuously differentiable. Afterwards in Section 3 we consider more general case of the arbitrary convex functions which are not supposed even one time continuously differentiable. We emphasize the fact that the latter c
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