scholarly journals SOLVING NOT COMPLETELY INTEGRABLE QUANTILE PFAFFIAN DIFFERENTIAL EQUATIONS

2017 ◽  
Vol 18 (3.1) ◽  
pp. 20-39
Author(s):  
L.E. Melkumova ◽  
S. Ya. Shatskikh
Keyword(s):  
Differential Equations ◽  
Integral Manifold ◽  
Probability Distributions ◽  
Maximum Dimension ◽  
Conditional Quantile ◽  
Completely Integrable ◽  
Conditional Quantiles ◽  
Integral Manifolds ◽  
Pfaffian Equations ◽  
Multidimensional Probability

The present work deals with quantile Pfaffian differential equations which are constructed using two-dimensional conditional quantiles of multidimensional probability distributions. As it was shown in [3] in case when the initial probability distributions have reproducible conditional quantiles this kind of Pfaan equations is completely integrable and the integral manifold is the conditional quantile of maximum dimension. In this paper we discuss properties of integral manifolds of maximum possible dimension for quantile Pfaffian equations which are not completely integrable. Manifolds of this type are described in terms of conditional quantiles of intermediate dimensions.

scholarly journals LINEARIZATION OF PFAFF DIFFERENTIAL EQUATIONS FOR THE CONDITIONAL QUANTILE OF MULTIVARIATE PROBABILITY DISTRIBUTIONS

2017 ◽  
Vol 21 (10) ◽  
pp. 29-39
Author(s):  
I.S. Orlova
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Probability Distributions ◽  
Nonlinear Partial Differential Equations ◽  
Linear Functions ◽  
Partial Differential ◽  
Gaussian Distributions ◽  
Conditional Quantile ◽  
Point Transformations ◽  
Constant Coefficients

The article is devoted to the task of bringing the point transformations of nonlinear partial differential equations of Pfaff for conditional quantile multi- variate probability distributions to the Pfaff differential equations with constant coefficients. Solutions of the equations of Pfaff with constant coefficients are linear functions representing the conditional quantile of multivariate Gaussian distributions.

scholarly journals On construction of a field of forces along given trajectories in the presence of random perturbations

2021 ◽  
Vol 101 (1) ◽  
pp. 98-103
Author(s):  
M.I. Tleubergenov ◽  
◽  
G.K. Vassilina ◽  
G.A. Tuzelbaeva ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Force Field ◽  
Integral Manifold ◽  
Sufficient Conditions ◽  
Second Order ◽  
Generalized Coordinates ◽  
Integral Manifolds ◽  
The Cauchy Problem ◽  
Stochastic Equivalence ◽  
The Right

In this paper, a force field is constructed along a given integral manifold in the presence of random perturbing forces. In this case, two types of integral manifolds are considered separately: 1) trajectories that depend on generalized coordinates and do not depend on generalized velocities, and 2) trajectories that depend on both generalized coordinates and generalized velocities. The construction of the force field is carried out in the class of second-order stochastic Ito differential equations. It is assumed that the functions in the right-hand sides of the equation must be continuous in time and satisfy the Lipschitz condition in generalized coordinates and generalized velocities. Also this functions satisfy the condition for linear growth in generalized coordinates and generalized velocities.These assumptions ensure the existence and uniqueness up to stochastic equivalence of the solution to the Cauchy problem of the constructed equations in the phase space, which is a strictly Markov process continuous with probability 1. To solve the two posed problems, stochastic differential equations of perturbed motion with respect to the integral manifold are constructed. Moreover, in the case when the trajectories depend on generalized coordinates and do not depend on generalized velocities, the second order equations of perturbed motion are constructed, and in the case when the trajectories depend on both generalized coordinates and generalized velocities, the first order equations of perturbed motion are constructed. And further, in both cases by Erugin’s method necessary and sufficient conditions for solving the posed problems are derived.

Integral manifolds of differential equations with rapidly rotating phase

1969 ◽  
Vol 12 (11) ◽  
pp. 1243-1252 ◽  
Author(s):  
A. S. Gurtovnik ◽  
Yu. I. Neimark
Keyword(s):  
Differential Equations ◽  
Integral Manifolds

scholarly journals QUANTILE DOUBLE AUTOREGRESSION

2021 ◽  
pp. 1-47
Author(s):  
Qianqian Zhu ◽  
Guodong Li
Keyword(s):  
Time Series ◽  
Financial Time Series ◽  
Simulation Studies ◽  
Finite Sample ◽  
Conditional Quantile ◽  
New Model ◽  
Conditional Quantiles ◽  
Financial Time ◽  
Portmanteau Tests ◽  
Strict Stationarity

Many financial time series have varying structures at different quantile levels, and also exhibit the phenomenon of conditional heteroskedasticity at the same time. However, there is presently no time series model that accommodates both of these features. This paper fills the gap by proposing a novel conditional heteroskedastic model called “quantile double autoregression”. The strict stationarity of the new model is derived, and self-weighted conditional quantile estimation is suggested. Two promising properties of the original double autoregressive model are shown to be preserved. Based on the quantile autocorrelation function and self-weighting concept, three portmanteau tests are constructed to check the adequacy of the fitted conditional quantiles. The finite sample performance of the proposed inferential tools is examined by simulation studies, and the need for use of the new model is further demonstrated by analyzing the S&P500 Index.

Averaging and integral manifolds

1970 ◽  
Vol 2 (2) ◽  
pp. 197-222 ◽  
Author(s):  
W. A. Coppel ◽  
K. J. Palmer
Keyword(s):  
Autonomous System ◽  
Method Of Characteristics ◽  
Integral Manifold ◽  
Artificial Viscosity ◽  
Original Method ◽  
Method Of Averaging ◽  
System Of Differential Equations ◽  
Viscosity Term ◽  
Integral Manifolds ◽  
The Individual

An integral manifold for a system of differential equations is a manifold such that any solution of the equations which has a point on it is entirely contained on it. The method of averaging establishes the existence of such a manifold for a system which is a perturbation of an autonomous system with a periodic orbit. The existence of the manifold is established here under more general hypotheses, namely for perturbations which are ‘integrally small’. The method differs from the original method of Bogolyubov and Mitropolskii and operates directly with the individual solutions. This is made possibly by the use of an appropriate norm, and is equivalent to solving the partial differential equation which occurs in work by Moser and Sacker by the method of characteristics rather than by the introduction of an artificial viscosity term. Moreover, detailed smoothness properties of the manifold are obtained. For periodic perturbations the integral manifold is a torus and these smoothness properties are just sufficient to permit the application of Denjoy's theorem.

Sign in / Sign up

Export Citation Format

Share Document