scholarly journals Upper Bounds on the Expected Value of a Convex Function Using Gradient and Conjugate Function Information

1989 ◽  
Vol 14 (4) ◽  
pp. 745-759 ◽  
Author(s):  
John Birge ◽  
Marc Teboulle
Keyword(s):  
Convex Function ◽  
Conjugate Function ◽  
Upper Bounds ◽  
Expected Value ◽  
Function Information

Some remarks on probability inequalities for sums of bounded convex random variables

1975 ◽  
Vol 12 (1) ◽  
pp. 155-158 ◽  
Author(s):  
M. Goldstein
Keyword(s):  
Convex Function ◽  
Exponential Convergence ◽  
Random Variables ◽  
Upper Bounds ◽  
Independent Random Variables ◽  
Probability Inequalities ◽  
Continuous Convex Function ◽  
Bounded Convex

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

A MEAN–VARIANCE BOUND FOR A THREE-PIECE LINEAR FUNCTION

2007 ◽  
Vol 21 (4) ◽  
pp. 611-621 ◽  
Author(s):  
Karthik Natarajan ◽  
Zhou Linyi
Keyword(s):  
Convex Function ◽  
Closed Form ◽  
Linear Function ◽  
Upper Bound ◽  
Random Variable ◽  
Expected Value ◽  
Variance Bound ◽  
The Mean ◽  
Mean Variance

In this article, we derive a tight closed-form upper bound on the expected value of a three-piece linear convex function E[max(0, X, mX − z)] given the mean μ and the variance σ2 of the random variable X. The bound is an extension of the well-known mean–variance bound for E[max(0, X)]. An application of the bound to price the strangle option in finance is provided.

scholarly journals Hermite-Hadamard, Jensen, and Fractional Integral Inequalities for Generalized P -Convex Stochastic Processes

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Fangfang Ma ◽  
Waqas Nazeer ◽  
Mamoona Ghafoor
Keyword(s):  
Probability Theory ◽  
Convex Function ◽  
Stochastic Processes ◽  
Fractional Integral ◽  
Probabilistic Models ◽  
Random Variable ◽  
Expected Value ◽  
Integral Inequalities ◽  
Stochastic Convexity ◽  
Convex Stochastic Processes

The stochastic process is one of the important branches of probability theory which deals with probabilistic models that evolve over time. It starts with probability postulates and includes a captivating arrangement of conclusions from those postulates. In probability theory, a convex function applied on the expected value of a random variable is always bounded above by the expected value of the convex function of that random variable. The purpose of this note is to introduce the class of generalized p -convex stochastic processes. Some well-known results of generalized p -convex functions such as Hermite-Hadamard, Jensen, and fractional integral inequalities are extended for generalized p -stochastic convexity.

Inheritance

1980 ◽  
Vol 12 (3) ◽  
pp. 574-590
Author(s):  
David Stirzaker
Keyword(s):  
Convex Function ◽  
Linear Approximation ◽  
Wealth Distribution ◽  
Upper Bounds ◽  
Limiting Distribution ◽  
Martingale Inequality ◽  
Distribution Of Wealth ◽  
Random Line

We consider a population of reproducing individuals who inherit, earn, consume, and bequeath wealth. A model is constructed to describe the wealth of an individual selected from the nth generation by following a random line of descent from the initial individual. It is shown that bequests are commonly a convex function of wealth. Considering a linear approximation to the bequest function enables us to obtain estimates of the limiting distribution of wealth as the number of generations increases, when earnings of parent and offspring are independent. More generally when earnings of parent and offspring are not independent we obtain upper bounds for the tail of the wealth distribution using a martingale inequality.

From FIFO to LIFO: a functional ordering of service delay via arrival discipline

1996 ◽  
Vol 33 (02) ◽  
pp. 507-512
Author(s):  
Jingwen Li
Keyword(s):  
Convex Function ◽  
Expected Value ◽  
Departure Process ◽  
Service Delay ◽  
Delay Cost ◽  
Increasing Function

Vasicek (1977) proved that among all queueing disciplines that do not change the departure process of the queue, FIFO and LIFO yield, respectively, the smallest and the largest expectation of any given convex function of the service delay. In this note we further show that, if arriving customers join the queue stochastically ‘closer' to the server(s), then the expected value of any convex function of service delay is larger. As a more interesting result, we also show that if the function under consideration is concave, then the conclusion will be exactly the opposite. This result indicates that LIFO will be the best discipline if the delay cost is an increasing function but at a diminishing rate.

scholarly journals Regularity properties and integral inequalities related to (k; h1; h2)-convexity of functions

2015 ◽  
Vol 53 (1) ◽  
pp. 19-35
Author(s):  
Gabriela Cristescu ◽  
Mihail Găianu ◽  
Awan Muhammad Uzair
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Local Minimizer ◽  
Upper Bounds ◽  
Integral Inequalities ◽  
Regularity Properties

Abstract The class of (k; h1; h2)-convex functions is introduced, together with some particular classes of corresponding generalized convex dominated functions. Few regularity properties of (k; h1; h2)-convex functions are proved by means of Bernstein-Doetsch type results. Also we find conditions in which every local minimizer of a (k; h1; h2)-convex function is global. Classes of (k; h1; h2)-convex functions, which allow integral upper bounds of Hermite-Hadamard type, are identified. Hermite-Hadamard type inequalities are also obtained in a particular class of the (k; h1; h2)- convex dominated functions.

Some remarks on probability inequalities for sums of bounded convex random variables

1975 ◽  
Vol 12 (01) ◽  
pp. 155-158
Author(s):  
M. Goldstein
Keyword(s):  
Convex Function ◽  
Exponential Convergence ◽  
Random Variables ◽  
Upper Bounds ◽  
Independent Random Variables ◽  
Probability Inequalities ◽  
Continuous Convex Function ◽  
Bounded Convex

Let X 1, X 2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi , i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi ), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

Inheritance

1980 ◽  
Vol 12 (03) ◽  
pp. 574-590
Author(s):  
David Stirzaker
Keyword(s):  
Convex Function ◽  
Linear Approximation ◽  
Wealth Distribution ◽  
Upper Bounds ◽  
Limiting Distribution ◽  
Martingale Inequality ◽  
Distribution Of Wealth ◽  
Random Line

We consider a population of reproducing individuals who inherit, earn, consume, and bequeath wealth. A model is constructed to describe the wealth of an individual selected from the nth generation by following a random line of descent from the initial individual. It is shown that bequests are commonly a convex function of wealth. Considering a linear approximation to the bequest function enables us to obtain estimates of the limiting distribution of wealth as the number of generations increases, when earnings of parent and offspring are independent. More generally when earnings of parent and offspring are not independent we obtain upper bounds for the tail of the wealth distribution using a martingale inequality.

Coefficient estimates for Libera type bi-close-to-convex functions

2021 ◽  
Vol 71 (6) ◽  
pp. 1401-1410
Author(s):  
Serap Bulut
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Function Class ◽  
Upper Bounds ◽  
Coefficient Estimates

Abstract In a very recent paper, Wang and Bulut [A note on the coefficient estimates of bi-close-to-convex functions, C. R. Acad. Sci. Paris, Ser. I 355 (2017), 876–880] determined the estimates for the general Taylor-Maclaurin coefficients of functions belonging to the bi-close-to-convex function class. In this study, we introduce the class of Libera type bi-close-to-convex functions and obtain the upper bounds for the coefficients of functions belonging to this class. Our results generalize the results in the above mentioned paper.

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