scholarly journals Approximate formulas for the evaluation of the mathematical expectation of functionals from the solution to the linear Skorohod equation

2021 ◽  
Vol 57 (2) ◽  
pp. 198-205
Author(s):  
A. D. Egorov
Keyword(s):  
Wiener Process ◽  
Mathematical Expectation ◽  
Random Processes ◽  
Quadrature Formulas ◽  
Functions Of Bounded Variation ◽  
Random Initial Condition ◽  
Initial Condition ◽  
The Third ◽  
Nonlinear Functionals ◽  
Approximate Formulas

This paper is devoted to the construction of approximate formulas for calculating the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod stochastic differential equation with a random initial condition. To calculate the mathematical expectations of nonlinear functionals from random processes, functional analogs of quadrature formulas have been developed, based on the requirement of their accuracy for functional polynomials of a given degree. Most often, formulas are constructed that are exact for polynomials of the third degree [1–9], which are used to obtain an initial approximation and in combination with approximations of the original random process. In the latter case, they are usually also exact for polynomials of a given degree and are called compound formulas. However, in the case of processes specified in the form of compound functions from other random processes the constructed functional quadrature formulas, as a rule, have great computational complexity and cannot be used for computer implementation. This is exactly what happens in the case of functionals from the solutions of stochastic equations. In [1, 2], the approaches to solving this problem were considered for some types of Ito equations in martingales. The solution of the problem is simplified in the cases when the solution of the stochastic equation is found in explicit form: the corresponding approximations were obtained in the cases of the linear equations of Ito, Ito – Levy and Skorohod in [3–11]. In [7, 8, 11], functional quadrature formulas were constructed that are exact for the approximations of the expansions of the solutions in terms of orthonormal functional polynomials and in terms of multiple stochastic integrals. This work is devoted to the approximate calculation of the mathematical expectations of nonlinear functionals from the solution of the linear Skorokhod equation with a leading Wiener process and a random initial condition. A new approach to the construction of quadrature formulas, exact for functional polynomials of the third degree, based on the use of multiple Stieltjes integrals over functions of bounded variation in the sense of Hardy – Krause, is proposed. A composite approximate formula is also constructed, which is exact for second-order functional polynomials, converging to the exact expectation value, based on a combination of the obtained quadrature formula and an approximation of the leading Wiener process. The test examples illustrating the application of the obtained formulas are considered.

scholarly journals On composite formulas for mathematical expectation of functionals of solution of the Ito equation in Hilbert space

2019 ◽  
Vol 55 (2) ◽  
pp. 158-168
Author(s):  
A. D. Egorov
Keyword(s):  
Hilbert Space ◽  
Random Process ◽  
Wiener Process ◽  
Mathematical Expectation ◽  
Quadrature Formulas ◽  
Third Order ◽  
One Dimensional ◽  
Temporal Variables ◽  
Nonlinear Functionals ◽  
Ito Equation

This article is devoted to constructing composite approximate formulas for calculation of mathematical expectation of nonlinear functionals of solution of the linear Ito equation in Hilbert space with additive noise. As the leading process, the Wiener process taking values in Hilbert space is examined. The formulas are a sum of the approximations of the nonlinear functionals obtained by expanding the leading random process into a series of independent Gaussian random variables and correcting approximating functional quadrature formulas that ensure an approximate accuracy of compound formulas for third-order polynomials. As a test example, the application of the obtained formulas to the case of a one-dimensional wave equation with a leading Wiener process indexed by spatial and temporal variables is considered. This article continues the research begun in [1].The problem is motivated by the necessity to calculate the nonlinear functionals of solution of stochastic partial differential equations. Approximate evaluation of mathematical expectation of stochastic equations with a leading random process indexed only by the time variable is considered in [2–11]. Stochastic partial equations in various interpretations are considered [12–16]. The present article uses the approach given in [12].

An approximate formula for calculating the expectations of functionals from random processes based on using the Wiener chaos expansion

2020 ◽  
Vol 26 (4) ◽  
pp. 285-292
Author(s):  
Alexander Egorov
Keyword(s):  
Mathematical Expectation ◽  
Approximate Formula ◽  
Random Processes ◽  
New Method ◽  
Chaos Expansion ◽  
Approximation Accuracy ◽  
Wiener Chaos ◽  
Nonlinear Functionals ◽  
Approximate Formulas ◽  
Wiener Chaos Expansion

AbstractIn this work, we propose a new method for calculating the mathematical expectation of nonlinear functionals from random processes. The method is based on using Wiener chaos expansion and approximate formulas, exact for functional polynomials of given degree. Examples illustrating approximation accuracy are considered.

scholarly journals Reducing of noise structure influence on an accuracy of a desired signal extraction

2018 ◽  
Vol 15 (3) ◽  
pp. 365-370 ◽  
Author(s):  
Vladimir Marchuk
Keyword(s):  
Signal Processing ◽  
Mathematical Expectation ◽  
Random Processes ◽  
Parameters Estimation ◽  
Signal Extraction ◽  
Scientific Contribution ◽  
Noise Component ◽  
Component Structure ◽  
Noise Structure ◽  
Theoretical Results

In the paper, the issues regarding the analysis of the noise component structure are addressed and methods for reducing the error in estimating of the mathematical expectation of the noise component are proposed. The use of the proposed method of ?noise purification? makes possibility to reduce the error introduced by the noise structure when estimating the mathematical expectation and dispersion of the noise component during research. The main scientific contribution in this paper in accuracy increasing of random processes parameters estimation. These theoretical results can be applied in different spheres of data analyzing and signal processing when random processes have some structure.

scholarly journals Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Chenguang Zhou ◽  
Yongkui Zou ◽  
Shimin Chai ◽  
Fengshan Zhang
Keyword(s):  
Finite Element ◽  
Heat Equation ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Random Initial Condition ◽  
Initial Condition ◽  
Polynomial Functions ◽  
Weak Galerkin ◽  
Convergence Estimates

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

scholarly journals Natural Test for Random Numbers Generator Based on Exponential Distribution

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 920 ◽  
Author(s):  
Tanackov ◽  
Sinani ◽  
Stanković ◽  
Bogdanović ◽  
Stević ◽  
...  
Keyword(s):  
Standard Deviation ◽  
Mathematical Expectation ◽  
Random Number ◽  
Random Variable ◽  
Random Numbers ◽  
The Third ◽  
Atmospheric Noise ◽  
Constant E ◽  
Fourth Series ◽  
Natural Test

We will prove that when uniformly distributed random numbers are sorted by value, their successive differences are a exponentially distributed random variable Ex(λ). For a set of n random numbers, the parameters of mathematical expectation and standard deviation is λ =n−1. The theorem was verified on four series of 200 sets of 101 random numbers each. The first series was obtained on the basis of decimals of the constant e=2.718281…, the second on the decimals of the constant π =3.141592…, the third on a Pseudo Random Number generated from Excel function RAND, and the fourth series of True Random Number generated from atmospheric noise. The obtained results confirm the application of the derived theorem in practice.

Control and Applications of Chaos

1997 ◽  
Vol 07 (10) ◽  
pp. 2175-2197 ◽  
Author(s):  
Celso Grebogi ◽  
Ying–Cheng Lai ◽  
Scott Hayes
Keyword(s):  
Feedback Control ◽  
Power Systems ◽  
System Performance ◽  
Chaotic Systems ◽  
Electrical Power ◽  
Random Initial Condition ◽  
Initial Condition ◽  
Chaotic Orbit ◽  
Electrical Power Systems ◽  
Dissipative Dynamical Systems

This review describes a procedure for stabilizing a desirable chaotic orbit embedded in a chaotic attractor of dissipative dynamical systems by using small feedback control. The key observation is that certain chaotic orbits may correspond to a desirable system performance. By carefully selecting such an orbit, and then applying small feedback control to stabilize a trajectory from a random initial condition around the target chaotic orbit, desirable system performance can be achieved. As applications, three examples are considered: (1) synchronization of chaotic systems; (2) conversion of transient chaos into sustained chaos; and (3) controlling symbolic dynamics for communication. The first and third problems are potentially relevant to communication in engineering, and the solution of the second problem can be applied to electrical power systems to avoid catastrophic events such as the voltage collapse.

scholarly journals Improved Indonesian Language Learning Outcomes Material Intrinsic Elements of Stories through Video Media for Class IV Students

2020 ◽  
Vol 3 (3) ◽  
pp. 831
Author(s):  
Septi Dwi Harmini
Keyword(s):  
Elementary Schools ◽  
Language Learning ◽  
Learning Outcomes ◽  
Student Learning Outcomes ◽  
Visual Media ◽  
Initial Condition ◽  
The Third ◽  
Post Test ◽  
Improve Student Learning ◽  
Development Of Students

<p><em>The purpose of this study was to improve the learning outcomes of students in the Indonesian muple material of the intrinsic elements of stories in grade IV elementary schools through audio-visual media in the form of video media. The research conducted was a three-cycle Classroom Action Research (CAR). The stages of each cycle are planning, implementing, observing and reflecting. Each cycle a post test is held to determine the development of students. The initial condition before the action was carried out, the data obtained from the students who had completed was 36.8%. In the first cycle, students who completed after carrying out the post test were 52.6%. In the second cycle students who completed after carrying out the post test were 84.2%. In the third cycle students who completed after carrying out the post test were 94.7%. These results indicate that video media can improve student learning outcomes, especially Indonesian muple material, the intrinsic elements of stories at SD Negeri 1 Tirip</em></p>

Commande optimale du processus de wiener

1989 ◽  
Vol 26 (1) ◽  
pp. 50-57 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Optimal Control ◽  
Wiener Process ◽  
Mathematical Expectation ◽  
Optimal Value ◽  
The Moment ◽  
First Time

The problem of the optimal control of the Wiener process in Rn is considered. The optimal value of the control is obtained from the mathematical expectation of a quantity defined in terms of the moment and the place where the uncontrolled process hits the boundary of the continuation region for the first time. Explicit results are presented.

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