scholarly journals On Extendability of the Principle of Equivalent Utility

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 42
Author(s):  
Małgorzata Chudziak ◽  
Marek Żołdak
Keyword(s):  
Functional Equation ◽  
Utility Function ◽  
Real Number ◽  
Insurance Premium ◽  
Distortion Function ◽  
Utility Model ◽  
Continuous Solutions ◽  
The Family ◽  
Rank Dependent Utility

An insurance premium principle is a way of assigning to every risk a real number, interpreted as a premium for insuring risk. There are several methods of defining the principle. In this paper, we deal with the principle of equivalent utility under the rank-dependent utility model. The principle, generated by utility function and probability distortion function, is based on the assumption of the symmetry between the decisions of accepting and rejecting risk. It is known that the principle of equivalent utility can be uniquely extended from the family of ternary risks. However, the extension from the family of binary risks need not be unique. Therefore, the following problem arises: characterizing those principles that coincide on the family of all binary risks. We reduce the problem thus to the multiplicative Pexider functional equation on a region. Applying the form of continuous solutions of the equation, we solve the problem completely.

Projective actions, invariant sigma-curves and quadratic functional equations

1985 ◽  
Vol 98 (2) ◽  
pp. 195-212 ◽  
Author(s):  
Patrick J. McCarthy
Keyword(s):  
Functional Equation ◽  
Periodic Function ◽  
Functional Equations ◽  
Continuous Solution ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Continuous Solutions ◽  
Linear Map

AbstractThe quadratic functional equation f(f(x)) *–Tf(x) + Dx = 0 is equivalent to the requirement that the graph be invariant under a certain linear map The induced projective map is used to show that the equation admits a rich supply of continuous solutions only when L is hyperbolic (T2 > 4D), and then only when T and D satisfy certain further conditions. The general continuous solution of the equation is given explicitly in terms of either (a) an expression involving an arbitrary periodic function, function additions, inverses and composites, or(b) suitable limits of such solutions.

scholarly journals On comparison of the principles of equivalent utility and its applications

2019 ◽  
Vol 11 (2) ◽  
pp. 240-249
Author(s):  
M. Chudziak
Keyword(s):  
Decision Making ◽  
Probability Space ◽  
Random Variable ◽  
Insurance Premium ◽  
Behavioral Models ◽  
Decision Making Under Risk ◽  
Rank Dependent Utility ◽  
Positive Homogeneity ◽  
Bounded Random Variable ◽  
Negative Real Number

An insurance premium principle is a way of assigning to every risk, represented by a non-negative bounded random variable on a given probability space, a non-negative real number. Such a number is interpreted as a premium for the insuring risk. In this paper the implicitly defined principle of equivalent utility is investigated. Using the properties of the quasideviation means, we characterize a comparison in the class of principles of equivalent utility under Rank-Dependent Utility, one of the important behavioral models of decision making under risk. Then we apply this result to establish characterizations of equality and positive homogeneity of the principle. Some further applications are discussed as well.

Continuous Solutions of the Functional Equation fn(x) = f(x)

1953 ◽  
Vol 5 ◽  
pp. 101-103 ◽  
Author(s):  
G. M. Ewing ◽  
W. R. Utz
Keyword(s):  
Functional Equation ◽  
Real Axis ◽  
Continuous Solutions ◽  
The Real ◽  
Real Functions ◽  
Image Position

In this note the authors find all continuous real functions defined on the real axis and such that for an integer n > 2, and for each x,

An Analogue of the Wave Equation and Certain Related Functional Equations

1969 ◽  
Vol 12 (6) ◽  
pp. 837-846 ◽  
Author(s):  
John A. Baker
Keyword(s):  
Functional Equation ◽  
Wave Equation ◽  
Functional Equations ◽  
Geometric Interpretation ◽  
Continuous Solutions ◽  
Difference Analogue ◽  
Image Position

Consider the functional equation(1)assumed valid for all real x, y and h. Notice that (1) can be written(2)a difference analogue of the wave equation, if we interpret etc., (i. e. symmetric h differences), and that (1) has an interesting geometric interpretation. The continuous solutions of (1) were found by Sakovič [5].

scholarly journals The Generalized Pomeron Functional Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-4
Author(s):  
Yong-Guo Shi
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Linear Functional ◽  
Continuous Solution ◽  
Suitable Condition ◽  
Single Parameter ◽  
Linear Functional Equation ◽  
Continuous Solutions ◽  
Constant Coefficients

This paper investigates the linear functional equation with constant coefficients φt=κφλt+ft, where both κ>0 and 1>λ>0 are constants, f is a given continuous function on ℝ, and φ:ℝ⟶ℝ is unknown. We present all continuous solutions of this functional equation. We show that (i) if κ>1, then the equation has infinite many continuous solutions, which depends on arbitrary functions; (ii) if 0<κ<1, then the equation has a unique continuous solution; and (iii) if κ=1, then the equation has a continuous solution depending on a single parameter φ0 under a suitable condition on f.

Graphs in the plane invariant under an area preserving linear map and general continuous solutions of certain quadratic functional equations

1985 ◽  
Vol 97 (2) ◽  
pp. 261-278 ◽  
Author(s):  
P. J. McCarthy ◽  
M. Crampin ◽  
W. Stephenson
Keyword(s):  
Functional Equation ◽  
Arbitrary Function ◽  
Functional Equations ◽  
Quadratic Functional ◽  
Continuous Solutions ◽  
Linear Map ◽  
Area Preserving Maps ◽  
Image Position ◽  
Area Preserving

AbstractThe requirement that the graph of a function be invariant under a linear map is equivalent to a functional equation of f. For area preserving maps M(det (M) = 1), the functional equation is equivalent to an (easily solved) linear one, or to a quadratic one of the formfor all Here 2C = Trace (M). It is shown that (Q) admits continuous solutions ⇔ M has real eigenvalues ⇔ (Q) has linear solutions f(x) = λx ⇔ |C| ≥ 1. For |c| = 1 or C < – 1, (Q) only admits a few simple solutions. For C > 1, (Q) admits a rich supply of continuous solutions. These are parametrised by an arbitrary function, and described in the sense that a construction is given for the graphs of the functions which solve (Q).

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