The Weighted Square Integral Inequalities for the First Derivative of the Function of a Real Variable
© S. Hussain et al. 2008
Received: 20 May 2008
Accepted: 30 July 2008
Published: 5 August 2008
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.
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  and subsequently applied to the hedging problems of mathematical finance by Hussain and Shashiashvili .
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  by El Karoui et al. (see also Hobson ) that the value functions of the European as well as American options are convex with respect to the underlying stock price; and the latter 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 case can be directly applied to problems of mathematical finance, especially to discrete time hedging of the European as well as American call options.
2. The Weighted Square Integral Estimates for the First Derivative of a Twice Continuously Differentiable Function
using the condition .
The latter equality together with the previous estimate (2.7) give us the required inequality (2.3).
Applying Hölder inequality to the right-hand side of estimate (2.3), we get the following.
Taking into account the latter expression in estimate (2.9), we come to the desired inequality (2.18).
Comparing the result stated in Corollary 2.4 with Theorem 2.1 from K. Shashiashvili and M. Shashiashvili , we come to the conclusion that the multiplier is twice less than obtained in the latter paper.
3. The Weighted Square Integral Estimates for the Difference of Derivatives of Two Convex Functions
In this section, we consider two arbitrary finite convex functions and on an infinite interval . It is well known that they are continuous and have finite left- and right-hand derivatives and inside the open interval
where is certain positive constant.
where and denote the left derivatives of the convex functions and respectively.
We will prove the theorem in two stages. On first stage, we verify the validity of the statement for twice continuously differentiable convex functions satisfying conditions (3.1) and (3.2) and on second stage we approximate arbitrary convex functions satisfying the same conditions by smooth ones inside the interval in an appropriate manner and afterwards we pass onto limit in the previously established estimate.
Thus let us assume at first that and are two convex functions defined on the interval which are twice continuously differentiable in the open interval and satisfy conditions (3.1) and (3.2).
then is twice continuously differentiable inside the infinite interval and at point zero, it has finite limit
Now we have to pass onto limit when and in inequality (3.16). Obviously, the left-hand side of the inequality increases and the right-hand side is bounded, when , therefore the left-hand side also converges to finite limit. Passing onto limit in inequality (3.16) using assumption (3.4) and the limit relations (3.30)–(3.36), we come to the required estimate (3.6).
Next we move to the second stage of the proof. Consider two arbitrary convex functions and defined on satisfying conditions (3.1) and (3.2). We have to construct the sequences of twice continuously differentiable (in the open interval ) convex functions and approximating, respectively, the functions and inside the interval in an appropriate manner.
For arbitrary fixed consider the restriction of functions and on the interval and let Then for
From definition (3.39), it is obvious that the functions and are infinitely differentiable, while their convexity follows from the expressions (3.41).
for and .
This shows the uniform convergence of the sequence to (and to ) on the interval .
is uniformly bounded by the constant if only
Using (3.53) let us show that for fixed the sequence converges to the left-derivative .
Finally, it remains to pass onto limit when and in the latter inequality. The left-hand side of inequality (3.65) obviously increases when and and the right-hand side is bounded by the assumption (3.4) and the limit relations (3.30)–(3.36). Therefore passing onto limit and in inequality (3.65), we arrive to the desired estimate (3.6).
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