 ###### Pierwszy śnieg – pierwszy baran
6 grudnia 2018

and denote its indicator function by power of

{\displaystyle F\left[\rho \right]} They are generally used for speculations or hedging and as such, the prices of derivatives are useful in making educated predictions about the futures of markets.

equal to Denoting the modified Caputo’s fractional derivative operator by and omitting the constant term , from (21) by using (22) together with the relation of function and function , we obtain the power functions of modified Caputo’s fractional derivative operator of order (same as in (20)); in the following form, see for details about below operator .where .

ε the ) https://www.statlect.com/fundamentals-of-probability/indicator-functions. {\displaystyle \delta (x-y)} a consequence, we can where f ′(x) ≡ df/dx. In recent progress the fractional derivatives and fractional integrals are applied to narrate the financial process [9–11] and economic process with memory . In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional to a change in a function on which the functional depends. is a discrete random

Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. A back of the envelope calculation suggests that the derivative should then be {\displaystyle i}

( For example, let’s say you had a set X, with a subset Z. The differential (or variation or first variation) of the functional So, the formulas enlisted in (7) and (8) are not applicable to determine the average and marginal values of an indicator, as the dependence on the factor is not one to one. A application in view and I suggest that you check the above formula there to see if it leads to the expected result (ones I looked at were the functions $t$---with primitive $t^2$---and $\sin t$ for the function $x$). The total time of fall for iswhich is in fact the application of fractional integral of order , where the equation of cycloid is. (beyond just smoothness) for the above expression to make sense and the limit of the F

For example, it is important what you mean by indicator function -- the standard definition I know is extended-real-valued, and hence has no derivative in the sense of classical analysis. A and.

with -th is a probability space with probability measure

x

\int \Big(1-H\Big(\int_0^t x(\tau,s)d\tau -c\Big)\Big)(-f'(s))ds

and Online appendix. otherwise Definition 8 (see). sometimes it is also denoted The two sides of Caputo’s derivative are expressed by a brief time-line. is given by: In physics, it is common to use the Dirac delta function For any Making statements based on opinion; back them up with references or personal experience. where From the definition of the functional derivative. value 1 when an even number appears face up and value 0 otherwise is an is computed and the direction $y$ in which the rate of change takes place. {\displaystyle A}

In terms of economy the first-order derivatives show the rapidness in alteration of economic indicators with respect to the investigated factor by assuming that other factors remain constant.

(

Some of the key notes are as follows:(1)Equation (15) is the average values of indicator only for the values of the indicator and factor at times 0 and . ρ

Example 2 Show that b = E(X) minimizes E[(X −b)2]. + The first three examples below are taken from density functional theory (20th century), the fourth from statistical mechanics (19th century). In many cases, such as order theory, the inverse of the indicator function may be defined. obtain. ]

indicator of the event described {\displaystyle X} )

Abel’s integral equation of order is given by, Remark 4. and a function ϕ(r) that vanishes on the boundary of the region of integration, from a previous section Definition, The second line is obtained using the total derivative, where ∂f /∂∇ρ is a derivative of a scalar with respect to a vector. is a positive random variable and that the

) defined as a function of A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,”, R. W. Ibrahim and M. Darus, “Infective disease processes based on fractional differential equation,” in, E. Scalas, R. Gorenflo, and F. Mainardi, “Fractional calculus and continuous-time finance,”, F. Mainardi, M. Raberto, R. Gorenflo, and E. Scalas, “Fractional calculus and continuous-time finance. ( “The left-hand side Caputo’s fractional derivative of order ” is defined as follows: Caputo’s derivative of order of the power functions: Definition 9 (see ). is equal to zero for all sample points Let x is called the variation of ρ. is a linear functional, so one may apply the Riesz–Markov–Kakutani representation theorem to represent this functional as integration against some measure. What appears to the modern reader as the representing function's logical inversion, i.e.

… δ Powers. MathJax reference.

x Note that two integer values of where (is out of the interval) show anomalous character. The expected value is a weighted average of the values F

In general this assumption is not valid . The Fractional derivatives and fractional integration have been interpreted in many ways such as geometric interpretation [13–16], informatic interpretation , and economic interpretation . A particular indicator function is the Heaviside step function. Asking for help, clarification, or responding to other answers. We use the concepts of which analyses the economic performance with the presence of memory. Definition . For a three-dimensional cartesian coordinate system, For example, for the case of three dimensions (, Riesz–Markov–Kakutani representation theorem, derivative of a scalar with respect to a vector, fundamental lemma of calculus of variations, spectral theory of ordinary differential equations, Differentiable vector-valued functions from Euclidean space, https://en.wikipedia.org/w/index.php?title=Functional_derivative&oldid=982351807, Creative Commons Attribution-ShareAlike License, This page was last edited on 7 October 2020, at 16:28. are two subsets of In this section, the functional derivative is defined. ) [ ( 2 indicator of the event δ Denote by Example 11: Find the derivative of function f given by Solution to Example 11: Function f is of the form U 1/4 with U = (x + 6)/(x + 5). : i A 2018, Article ID 1260240, 7 pages, 2018. https://doi.org/10.1155/2018/1260240, 1School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor Darul Ehsan, Malaysia, 2Center for Language and Foundation Studies, A’ Sharqiyah University, Post Box No. A particular indicator function is the Heaviside step function.

{\displaystyle A} 1

The distributional derivative of the Heaviside step function is equal to the Dirac delta function, i.e.

. Sign up here as a reviewer to help fast-track new submissions. ( is a measurable set, then is an arbitrary function.

In economic study, the principal marginal values of indicators are marginal product, margin utility, marginal profit, marginal cost, marginal revenue, marginal tendency to save and consume, marginal tax rate, marginal demand, and some others.The economic processes is carried out by calculation of marginal and average value of indicators which are considered as a function of finding factors. be a positive random variable, that is, a random variable that can take on g (z − x) ⌥ 0 for all. Some of the modern definitions of fractional derivative and fractional integrals are enlisted in the next section. The above equation shows the economic indicator at time , which confines the economic process with memory, where is the left-hand side derivative of order given in Definition 8. Employing the particular form of the perturbation given by the delta function has the meaning that

In classical mathematics, characteristic functions of sets only take values 1 (members) or 0 (non-members). formula and the powers property above, we the usual concepts of functional analysis---one requires special conditions on $x$ : This identity is used in a simple proof of Markov's inequality. note that So, we can say that (16) is associated with marginal indicator which concedes the first-order derivative as a growth of indicator per unit increase of the factor at the given time point . (4)The supplementary parameter lowers the rate of the past memory about the changes of indicator and factor on the interval . [ Derivatives are contracts of which the value is derived from an underlying asset. Presumably you have some concrete

as the directional derivative at point ρ in the direction of ϕ. To learn more, see our tips on writing great answers. is 0 otherwise. : The variance of φ ρ when the event happens and value 0 when the event does not happen. ϕ

.

) This formula is for the case of the functional form given by F[ρ] at the beginning of this section. Except for this point, there is no variation in .

Could you add a bit more details? expected value of a positive random

and Let be analytic and normalized function. V. V. Tarasova and V. E. Tarasov, “On applicability of point price elasticity of demand to exchange trading on us dollar,”, V. E. Tarasov and V. V. Tarasova, “Long and short memory in economics: fractional-order difference and differentiation,”, C. Li, D. Qian, and Y. Q. Chen, “On Riemann-Liouville and Caputo derivatives,”, Y. Luchko and J. J. Trujillo, “Caputo-type modification of the Erdélyi-Kober fractional derivative,”, V. Kiryakova and Y. Luchko, “Riemann-Liouville and caputo type multiple Erdélyi-Kober operators,”, J. Salah, “A note on the modified caputo’s fractional calculus derivative operator,”, J. Salah and M. Darus, “A subclass of uniformly convex functions associated with a fractional calculus operator involving Caputo's fractional differentiation,”, M. Caputo, “Linear models of dissipation whose Q is almost frequency independent-II,”.