THE EXACT DISCRETE MODEL OF A THIRD-ORDER SYSTEM OF LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS WITH OBSERVABLE STOCHASTIC TRENDS

2009 ◽  
Vol 13 (5) ◽  
pp. 656-672 ◽  
Author(s):  
Theodore Simos
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Initial Conditions ◽  
Time Lag ◽  
Estimation Procedure ◽  
Discrete Models ◽  
Order System ◽  
Third Order ◽  
Time Model ◽  
Stochastic Trends

The objective of this paper is to develop closed-form formulae for the exact discretization of a third-order system of stochastic differential equations, with fixed initial conditions, driven by observable stochastic trends and white noise innovations. The model provides a realistic alternative to first- and second-order differential equation specifications of the time lag distribution, forming the basis of a testing and estimation procedure. The exact discrete models, derived under two sampling schemes with either stock or flow variables, are put into a system error correction form that preserves the information of the underlying continuous time model regarding the order of integration and the dimension of cointegration space.

A third-order weak approximation of multidimensional Itô stochastic differential equations

2019 ◽  
Vol 25 (2) ◽  
pp. 97-120 ◽  
Author(s):  
Riu Naito ◽  
Toshihiro Yamada
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Wiener Space ◽  
Weak Approximation ◽  
Third Order ◽  
Brownian Motions ◽  
Numerical Examples ◽  
Discretization Algorithm

Abstract This paper proposes a new third-order discretization algorithm for multidimensional Itô stochastic differential equations driven by Brownian motions. The scheme is constructed by the Euler–Maruyama scheme with a stochastic weight given by polynomials of Brownian motions, which is simply implemented by a Monte Carlo method. The method of Watanabe distributions on Wiener space is effectively applied in the computation of the polynomial weight of Brownian motions. Numerical examples are shown to confirm the accuracy of the scheme.

scholarly journals Conservation Laws for a Variable Coefficient Variant Boussinesq System

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ben Muatjetjeja ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Fourth Order ◽  
Variable Coefficient ◽  
Boussinesq System ◽  
Order System ◽  
Conserved Quantities ◽  
Third Order ◽  
Partial Differential

We construct the conservation laws for a variable coefficient variant Boussinesq system, which is a third-order system of two partial differential equations. This system does not have a Lagrangian and so we transform it to a system of fourth-order, which admits a Lagrangian. Noether’s approach is then utilized to obtain the conservation laws. Lastly, the conservation laws are presented in terms of the original variables. Infinite numbers of both local and nonlocal conserved quantities are derived for the underlying system.

STOCHASTIC VERSIONS OF ANOSOV'S AND NEISTADT'S THEOREMS ON AVERAGING

2001 ◽  
Vol 01 (01) ◽  
pp. 1-21 ◽  
Author(s):  
YURI KIFER
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hamiltonian Systems ◽  
Initial Conditions ◽  
Slow Motion ◽  
Averaging Principle ◽  
Integrable Hamiltonian Systems ◽  
Constant Energy ◽  
Slow Motions ◽  
Fast And Slow Motions

In systems which combine slow and fast motions the averaging principle says that a good approximation of the slow motion can be obtained by averaging its parameters in fast variables. This setup arises, for instance, in perturbations of Hamiltonian systems where motions on constant energy manifolds are fast and across them are slow. When these perturbations are deterministic Anosov's theorem says that the averaging principle works except for a small in measure set of initial conditions while Neistadt's theorem gives error estimates in the case of perturbations of integrable Hamiltonian systems. These results are extended here to the case of fast and slow motions given by stochastic differential equations.

ON ESTIMATION OF THE LINEARIZED DRIFT FOR NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

2001 ◽  
Vol 01 (01) ◽  
pp. 23-43 ◽  
Author(s):  
R. KHASMINSKII ◽  
G. N. MILSTEIN
Keyword(s):  
Differential Equations ◽  
Equilibrium Point ◽  
Stochastic Differential Equations ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Initial Condition ◽  
Nonlinear Stochastic Differential Equations

The estimation of the linearized drift for stochastic differential equations with equilibrium points is considered. It is proved that the linearized drift matrix can be estimated efficiently if the initial condition for the system is chosen close enough to the equilibrium point. Some bounds for initial conditions ensuring the asymptotical efficiency of the estimator are found.

Periodic and Chaotic Oscillations of Laminated Composite Piezoelectric Rectangular Plate With 1:3 Internal Resonance

2007 ◽  
Author(s):  
Wei Zhang ◽  
Li-Hua Chen ◽  
Zhi-Gang Yao ◽  
Xiao-Li Yang
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Laminated Composite ◽  
Rectangular Plates ◽  
Third Order ◽  
Chaotic Motions ◽  
Transverse Loads ◽  
Averaged Equations ◽  
Laminate Theory

The chaotic dynamics of parametrically excited, simply supported laminated composite piezoelectric rectangular plates are analyzed, The plates are forced by transverse loads. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded in them. Firstly, based on von Karman-type equations and third-order shear deformation laminate theory of Reddy, the nonlinear equations of motions of the laminated composite piezoelectric rectangular plates are derived. Here, we consider the piezoelectric parametric loads and in-plane parametric loads acting in both x-direction and y-direction. Then, the Galerkin’s approach is applied to convert partial differential equations to the ordinary differential equations. The method of multiple scales is used to obtain the averaged equations. Finally, based on the averaged equations, periodic and chaotic motions of the plates are found by using numerical simulation. The numerical results show the existence of periodic and chaotic motions in averaged equations. The chaotic responses are sensitive to initial conditions especially to forcing loads and the parametric excitation.

scholarly journals Conservation Laws for a Generalized Coupled Korteweg-de Vries System

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Daniel Mpho Nkwanazana ◽  
Ben Muatjetjeja ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Fourth Order ◽  
Nonlinear Partial Differential Equations ◽  
Order System ◽  
Conserved Quantities ◽  
Third Order ◽  
The Third ◽  
New System

We construct conservation laws for a generalized coupled KdV system, which is a third-order system of nonlinear partial differential equations. We employ Noether's approach to derive the conservation laws. Since the system does not have a Lagrangian, we make use of the transformationu=Ux,v=Vxand convert the system to a fourth-order system inU,V. This new system has a Lagrangian, and so the Noether approach can now be used to obtain conservation laws. Finally, the conservation laws are expressed in theu,vvariables, and they constitute the conservation laws for the third-order generalized coupled KdV system. Some local and infinitely many nonlocal conserved quantities are found.

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