The Weighted Square Integral Inequalities for the First Derivative of the Function of a Real Variable

Abstract

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.

Sensitivity of American Option Prices with Respect to the Variations of Local Volatility

“…Convexity of American option prices has been originally established by Bergman, Grundy and Wiener [2], El-Karoui, Jeanblanc-Picque and Shreve [4] and Hobson [6]. Some estimates for arbitrary finite convex functions are given in Sultan, Pecaric and Shashiashvili [7]. Continuity of the American put option prices has been proved by Rehman and Shashiashvili [13](see sections 2 and 3).…”

On approximation and energy estimates for delta 6-convex functions

“…Let be the weight function which is non-negative, twice continuously differentiable, and satisfying with . We come to the following result of Hussain, Pecaric, and Shashiashvili [ 15 ].…”

“…Using the integration by parts formula and condition ( 1.1 ), we have Now take the first and the second integrals on the right-hand side of the latter expression. Using the integration by parts formula and making use of condition ( 1.1 ), we get Proceeding in the similar way and using condition ( 1.1 ) and the definition of weight function, we obtain Now we take Using Theorem 2.1 from [ 16 ], we have Now, using (2.6) of [ 15 ], we have Substituting ( 3.12 ) and ( 3.13 ) in ( 3.11 ), we have Here, Using the above conditions, we obtain Using the integration by parts formula, we obtain …”

“…The function is said to be n-convex if , . The weighted energy estimates for the convex function and 4-convex function are derived in [ 15 ] and [ 16 ]. These estimates are important in hedging strategies in finance [ 17 ].…”

Subsolutions That Are Close in the Uniform Norm Are Close in the Sobolev Norm as Well

“…This kind of estimate with weight functions on an infinite interval [0, ∞) was subsequently applied to hedging problems of mathematical finance in S. Hussain and M. Shashiashvili [12] (see also S. Hussain, J. Peĉariè and M. Shashiashvili [11]). The natural generalization of univariate convex functions to the case of several variables are subharmonic functions that share many convenient attributes of the former functions.…”