scholarly journals MULTIPLICATIVE APPROXIMATION OF A RANDOM PROCESS

2021 ◽  
pp. 205-214
Author(s):  
Zoya Nagolkina ◽  
Yuri Filonov
Keyword(s):  
Approximate Solution ◽  
Random Process ◽  
Stochastic Equation ◽  
Real Hilbert Space ◽  
Kolmogorov Equation ◽  
Original Equation ◽  
Infinite Dimensional ◽  
Elementary Segment ◽  
Single Solution ◽  
Stochastic Equivalence

In this paper we consider the stochastic Ito differential equation in an infinite-dimensional real Hilbert space. Using the method of multiplicative representations of Daletsky - Trotter, its approximate solution is constructed. Under classical conditions on the coefficients, there is a single to the stochastic equivalence of solutions of the stochastic equation, which is a random process. This development generates an evolutionary family of resolving operators by the formula  x(t)= S(t,  Construct the division of the segment  by the points. An equation with time-uniform coefficients is considered on each elementary segment    . There is a single solution of this equation on the elementary segment, which generates the resolving operator by the formula  The multiplicative expression  is constructed. Using the method of Dalecki-Trotter multiplicative representations, it is proved that this multiplicative expression is stochastically equivalent to the representation generated by the solution of the original equation. This means that the specified multiplicative expression is respectively a representation of the solution of the original equation. That is, the probability of one coincides with the solution of the original stochastic equation. It should be noted that this is possible under additional conditions for the coefficients of the equation. These conditions are the time continuity of the coefficients of the equation. Thus, the constructed multiplicative representation can be interpreted as an approximate solution of the original equation. This method of multiplicative approximation makes it possible to simplify the study of the corresponding random process both at the elementary segment and as a whole. It is known, that the solution of a stochastic equation by a known formula generates a solution of the inverse Kolmogorov equation in the corresponding space. This scheme of multiplicative approximation can be transferred to the solution of the parabolic equation, which is the inverse Kolmogorov equation. Thus, the method of multiplicative approximation makes it possible to simplify the study of both stochastic equations and partial differential equations.

scholarly journals About the infinite dimensional skew and obliquely reflected Ornstein–Uhlenbeck process

2015 ◽  
Vol 18 (04) ◽  
pp. 1550031
Author(s):  
Michael Röckner ◽  
Gerald Trutnau
Keyword(s):  
Real Hilbert Space ◽  
Monotonicity Condition ◽  
Normal Vector ◽  
Reflection Point ◽  
Uhlenbeck Process ◽  
Infinite Dimensional ◽  
Ornstein Uhlenbeck Process ◽  
Integration By Parts Formula ◽  
Regular Convex ◽  
Convex Subsets

Based on an integration by parts formula for closed and convex subsets [Formula: see text] of a separable real Hilbert space [Formula: see text] with respect to a Gaussian measure, we first construct and identify the infinite dimensional analogue of the obliquely reflected Ornstein–Uhlenbeck process (perturbed by a bounded drift [Formula: see text]) by means of a Skorokhod type decomposition. The variable oblique reflection at a reflection point of the boundary [Formula: see text] is uniquely described through a reflection angle and a direction in the tangent space (more precisely through an element of the orthogonal complement of the normal vector) at the reflection point. In case of normal reflection at the boundary of a regular convex set and under some monotonicity condition on [Formula: see text], we prove the existence and uniqueness of a strong solution to the corresponding SDE. Subsequently, we consider an increasing sequence [Formula: see text] of closed and convex subsets of [Formula: see text] and the skew reflection problem at the boundaries of this sequence. We present concrete examples and obtain as a special case the infinite dimensional analogue of the [Formula: see text]-skew reflected Ornstein–Uhlenbeck process.

scholarly journals Iterative methods for nonlinear quasi complementarity problems

1987 ◽  
Vol 10 (2) ◽  
pp. 339-344 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Hilbert Space ◽  
Approximate Solution ◽  
Complementarity Problems ◽  
Real Hilbert Space ◽  
Continuous Mapping ◽  
Convergence Criteria ◽  
Polar Cone ◽  
Point Mapping ◽  
Special Cases ◽  
Point To Point

In this paper, we consider and study an iterative algorithm for finding the approximate solution of the nonlinear quasi complementarity problem of findingu ϵ k(u)such thatTu ϵ k*(u)  and  (u−m(u),Tu)=0wheremis a point-to-point mapping,Tis a (nonlinear) continuous mapping from a real Hilbert spaceHinto itself andk*(u)is the polar cone of the convex conek(u)inH. We also discuss the convergence criteria and several special cases, which can be obtained from our main results.

Identification of Non-Gaussian Stochastic System

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Sung-man Park ◽  
O-shin Kwon ◽  
Jin-sung Kim ◽  
Jong-bok Lee ◽  
Hoon Heo
Keyword(s):  
Random Process ◽  
Random Noise ◽  
Stochastic Simulations ◽  
Kolmogorov Equation ◽  
Gaussian Random Process ◽  
System Parameters ◽  
Analysis Technique ◽  
Gaussian Analysis ◽  
Gaussian System ◽  
Non Gaussian

This paper proposes a method to identify non-Gaussian random noise in an unknown system through the use of a modified system identification (ID) technique in the stochastic domain, which is based on a recently developed Gaussian system ID. The non-Gaussian random process is approximated via an equivalent Gaussian approach. A modified Fokker–Planck–Kolmogorov equation based on a non-Gaussian analysis technique is adopted to utilize an effective Gaussian random process that represents an implied non-Gaussian random process. When a system under non-Gaussian random noise reveals stationary moment output, the system parameters can be extracted via symbolic computation. Monte Carlo stochastic simulations are conducted to reveal some approximate results, which are close to the actual values of the system parameters.

scholarly journals Variational Iteration Method for Solving the Generalized Degasperis-Procesi Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Qian Lijuan ◽  
Tian Lixin ◽  
Ma Kaiping
Keyword(s):  
Approximate Solution ◽  
Exact Solutions ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Initial Approximation ◽  
Variational Iteration Method ◽  
Original Equation ◽  
Variational Iteration

We introduce the variational iteration method for solving the generalized Degasperis-Procesi equation. Firstly, according to the variational iteration, the Lagrange multiplier is found after making the correction functional. Furthermore, several approximations ofun+1(x,t)which is converged tou(x,t)are obtained, and the exact solutions of Degasperis-Procesi equation will be obtained by using the traditional variational iteration method with a suitable initial approximationu0(x,t). Finally, after giving the perturbation item, the approximate solution for original equation will be expressed specifically.

Quasifree representations of Clifford algebras

1993 ◽  
Vol 113 (3) ◽  
pp. 487-497
Author(s):  
P. L. Robinson
Keyword(s):  
Hilbert Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Real Hilbert Space ◽  
Adjoint Operator ◽  
The State ◽  
Infinite Dimensional ◽  
Image Position

Let V be an infinite-dimensional real Hilbert space with associated C* Clifford algebra C[V]. To any state σ of the C* algebra C[V] there corresponds a skew-adjoint operator C of norm at most unity on V such thatwe refer to C as the covariance of the state σ.

scholarly journals ON THE KOLMOGOROV-WIENER FILTER FOR RANDOM PROCESSES WITH A POWER-LAW STRUCTURE FUNCTION BASED ON THE WALSH FUNCTIONS

2021 ◽  
pp. 39-47
Author(s):  
V. N. Gorev ◽  
A. Yu. Gusev ◽  
V. I. Korniienko ◽  
A. A. Safarov
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Numerical Calculation ◽  
Weight Function ◽  
Structure Function ◽  
Random Process ◽  
Power Law ◽  
Wiener Filter ◽  
Walsh Function ◽  
Better Than

Context. We investigate the Kolmogorov-Wiener filter weight function for the prediction of a continuous stationary random process with a power-law structure function. Objective. The aim of the work is to develop an algorithm of obtaining an approximate solution for the weight function without recourse to numerical calculation of integrals. Method. The weight function under consideration obeys the Wiener-Hopf integral equation. A search for an exact analytical solution for the corresponding integral equation meets difficulties, so an approximate solution for the weight function is sought in the framework of the Galerkin method on the basis of a truncated Walsh function series expansion. Results. An algorithm of the weight function obtaining is developed. All the integrals are calculated analytically rather than numerically. Moreover, it is shown that the accuracy of the Walsh function approximations is significantly better than the accuracy of polynomial approximations obtained in the authors’ previous papers. The Walsh function solutions are applicable in wider range of parameters than the polynomial ones. Conclusions. An algorithm of obtaining the Kolmogorov-Wiener filter weight function for the prediction of a stationary continuous random process with a power-law structure function is developed. A truncated Walsh function expansion is the basis of the developed algorithm. In opposite to the polynomial solutions investigated in the previous papers, the developed algorithm has the following advantages. First of all, all the integrals are calculated analytically, and any numerical calculation of the integrals is not needed. Secondly, the problem of the product of very small and very large numbers is absent in the framework of the developed algorithm. In our opinion, this is the reason why the accuracy of the Walsh function solutions is better than that of the polynomial solutions for many approximations and why the Walsh function solutions are applicable in a wider range of parameters than the polynomial ones. The results of the paper may be applied, for example, to practical traffic prediction in telecommunication systems with data packet transfer.

scholarly journals A gluing approach for the fractional Yamabe problem with isolated singularities

2020 ◽  
Vol 2020 (763) ◽  
pp. 25-78 ◽  
Author(s):  
Weiwei Ao ◽  
Azahara DelaTorre ◽  
María del Mar González ◽  
Juncheng Wei
Keyword(s):  
Approximate Solution ◽  
Reduction Method ◽  
Type System ◽  
Singular Solutions ◽  
Yamabe Problem ◽  
Local Problem ◽  
Infinite Dimensional ◽  
Isolated Points ◽  
Non Local ◽  
First Time

AbstractWe construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem in order to construct singular solutions. There are two main steps in the proof: to construct an approximate solution by gluing half bubble towers at each singular point, and then an infinite-dimensional Lyapunov–Schmidt reduction method, that reduces the problem to an (infinite-dimensional) Toda-type system. The main technical part is the estimate of the interactions between different bubbles in the bubble towers.

scholarly journals A spectral-based numerical method for Kolmogorov equations in Hilbert spaces

2016 ◽  
Vol 19 (03) ◽  
pp. 1650020
Author(s):  
Francisco Delgado-Vences ◽  
Franco Flandoli
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Hilbert Spaces ◽  
Stochastic Equation ◽  
Infinite System ◽  
Kolmogorov Equation ◽  
Kolmogorov Equations ◽  
Wiener Chaos ◽  
Stochastic Burgers Equation ◽  
Stochastic Diffusion Equation

We propose a numerical solution for the solution of the Fokker–Planck–Kolmogorov (FPK) equations associated with stochastic partial differential equations in Hilbert spaces. The method is based on the spectral decomposition of the Ornstein–Uhlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener–Chaos Expansion. By using this expansion we reformulate the Kolmogorov equation as an infinite system of ordinary differential equations, and by truncating it we set a linear finite system of differential equations. The solution of such system allow us to build an approximation to the solution of the Kolmogorov equations. We test the numerical method with the Kolmogorov equations associated with a stochastic diffusion equation, a Fisher–KPP stochastic equation and a stochastic Burgers equation in dimension 1.

scholarly journals Extremal solutions to a class of multivalued integral equations in Banach space

1992 ◽  
Vol 5 (3) ◽  
pp. 205-220 ◽  
Author(s):  
Sergiu Aizicovici ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Control Systems ◽  
Integral Equations ◽  
Important Application ◽  
Extremal Solutions ◽  
Original Equation ◽  
Dimensional Control ◽  
Infinite Dimensional ◽  
Integral Inclusion ◽  
Integral Solutions

We consider a nonlinear Volterra integral inclusion in a Banach space. We establish the existence of extremal integral solutions, and we show that they are dense in the solution set of the original equation. As an important application, we obtain a “bang-bang” theorem for a class of nonlinear, infinite dimensional control systems.

Sign in / Sign up

Export Citation Format

Share Document