Bounds for moment generating functions and for extinction probabilities

1966 ◽  
Vol 3 (1) ◽  
pp. 171-178 ◽  
Author(s):  
D. Brook
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Natural Extension ◽  
Random Variable ◽  
Moment Generating Function ◽  
Multivariate Case ◽  
Extinction Probabilities ◽  
Moment Generating Functions

Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.

Bounds for moment generating functions and for extinction probabilities

1966 ◽  
Vol 3 (01) ◽  
pp. 171-178 ◽  
Author(s):  
D. Brook
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Natural Extension ◽  
Random Variable ◽  
Moment Generating Function ◽  
Multivariate Case ◽  
Extinction Probabilities ◽  
Moment Generating Functions

Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.

The Properties Related to the Moment Generating Function of the Fuzzy Variable

2014 ◽  
Vol 519-520 ◽  
pp. 863-866
Author(s):  
Sheng Ma
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Uncertainty Theory ◽  
Fuzzy Variable ◽  
Moment Generating Function ◽  
Distribution Functions ◽  
Moment Generating Functions ◽  
The Moment

In the paper, some properties related to the moment generating function of a fuzzy variable are discussed based on uncertainty theory. And we obtain the result that the convergence of moment generating functions to an moment generating function implies convergence of credibility distribution functions. Thats, the moment generating function characterizes a credibility distribution.

scholarly journals Smile Asymptotics II: Models with Known Moment Generating Functions

2008 ◽  
Vol 45 (1) ◽  
pp. 16-32 ◽  
Author(s):  
Shalom Benaim ◽  
Peter Friz
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Implied Volatility ◽  
Moment Generating Function ◽  
Tail Asymptotics ◽  
Volatility Smile ◽  
Tauberian Conditions ◽  
Black Scholes ◽  
Moment Generating Functions ◽  
Exponential Lévy Models

The tail of risk neutral returns can be related explicitly with the wing behaviour of the Black-Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we establish, under easy-to-check Tauberian conditions, tail asymptotics on logarithmic scales. Such asymptotics are enough to make the tail-wing formula (see Benaim and Friz (2008)) work and so we obtain, under generic conditions, a limiting slope when plotting the square of the implied volatility against the log strike, improving a lim sup statement obtained earlier by Lee (2004). We apply these results to time-changed exponential Lévy models and examine several popular models in more detail, both analytically and numerically.

scholarly journals Smile Asymptotics II: Models with Known Moment Generating Functions

2008 ◽  
Vol 45 (01) ◽  
pp. 16-32 ◽  
Author(s):  
Shalom Benaim ◽  
Peter Friz
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Implied Volatility ◽  
Moment Generating Function ◽  
Tail Asymptotics ◽  
Volatility Smile ◽  
Tauberian Conditions ◽  
Black Scholes ◽  
Moment Generating Functions ◽  
Exponential Lévy Models

The tail of risk neutral returns can be related explicitly with the wing behaviour of the Black-Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we establish, under easy-to-check Tauberian conditions, tail asymptotics on logarithmic scales. Such asymptotics are enough to make the tail-wing formula (see Benaim and Friz (2008)) work and so we obtain, under generic conditions, a limiting slope when plotting the square of the implied volatility against the log strike, improving a lim sup statement obtained earlier by Lee (2004). We apply these results to time-changed exponential Lévy models and examine several popular models in more detail, both analytically and numerically.

Polynomial bounds for probability generating functions

1975 ◽  
Vol 12 (3) ◽  
pp. 507-514 ◽  
Author(s):  
Henry Braun
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Probability Generating Functions ◽  
Extinction Probabilities ◽  
Polynomial Bounds

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

scholarly journals On a Class of Distributions Stable Under Random Summation

2012 ◽  
Vol 49 (02) ◽  
pp. 303-318 ◽  
Author(s):  
L. B. Klebanov ◽  
A. V. Kakosyan ◽  
S. T. Rachev ◽  
G. Temnov
Keyword(s):  
Normal Distribution ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Analytic Functions ◽  
Random Variable ◽  
Random Summation ◽  
The Stability ◽  
Rational Generating Functions ◽  
Family Of Distributions

We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

Inequalities with application in economic risk analysis

1972 ◽  
Vol 9 (2) ◽  
pp. 441-444 ◽  
Author(s):  
Robert A. Agnew
Keyword(s):  
Risk Analysis ◽  
Lower Bounds ◽  
Generating Function ◽  
Random Variable ◽  
Moment Generating Function ◽  
Upper Bounds ◽  
Economic Risk ◽  
The Moment

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

Inequalities with application in economic risk analysis

1972 ◽  
Vol 9 (02) ◽  
pp. 441-444 ◽  
Author(s):  
Robert A. Agnew
Keyword(s):  
Risk Analysis ◽  
Lower Bounds ◽  
Generating Function ◽  
Random Variable ◽  
Moment Generating Function ◽  
Upper Bounds ◽  
Economic Risk ◽  
The Moment

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

scholarly journals FPM PADA KELUARGA EKSPONENSIAL BENTUK KONONIK

2015 ◽  
Vol 3 (1) ◽  
pp. 63
Author(s):  
Mrs Entit Puspita
Keyword(s):  
Generating Function ◽  
Exponential Family ◽  
Random Variable ◽  
Moment Generating Function ◽  
The Family

We can determine the distribution of the random variable by considering its moment generating function (MGF). Unfortunately there are some distribution which have no MGF. This kind of the problem cann’t occur on an exponential family, because the comunic form of the family always can be determined its MGF.Kata kunci : Bentuk kanonik, keluarga eksponensial, fungsi pembangkitan momen.

scholarly journals Formulas derived from moment generating functions and Bernstein polynomials

2019 ◽  
Vol 13 (3) ◽  
pp. 839-848 ◽  
Author(s):  
Buket Simsek
Keyword(s):  
Partial Derivative ◽  
Generating Functions ◽  
Functional Equations ◽  
Bernstein Polynomials ◽  
Random Variable ◽  
Discrete Random Variable ◽  
Derivative Formula ◽  
Moment Generating Functions ◽  
Derivative Formulas

The purpose of this paper is to provide some identities derived by moment generating functions and characteristics functions. By using functional equations of the generating functions for the combinatorial numbers y1 (m,n,?), defined in [12, p. 8, Theorem 1], we obtain some new formulas for moments of discrete random variable that follows binomial (Newton) distribution with an application of the Bernstein polynomials. Finally, we present partial derivative formulas for moment generating functions which involve derivative formula of the Bernstein polynomials.

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