GENERAL METHOD OF LYAPUNOV FUNCTIONALS CONSTRUCTION IN STABILITY INVESTIGATIONS OF NONLINEAR STOCHASTIC DIFFERENCE EQUATIONS WITH CONTINUOUS TIME

2005 ◽  
Vol 05 (02) ◽  
pp. 175-188 ◽  
Author(s):  
LEONID SHAIKHET
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Continuous Time ◽  
Type Theorem ◽  
Volterra Equations ◽  
Lyapunov Functionals ◽  
Stochastic Difference Equation ◽  
Stochastic Difference Equations ◽  
After Effect ◽  
General Method

The general method of Lyapunov functionals construction has been developed during the last decade for stability investigations of stochastic differential equations with after-effect and stochastic difference equations. After some modification of the basic Lyapunov type theorem this method was successfully used also for difference Volterra equations with continuous time. The latter often appear as useful mathematical models. Here this method is used for a stability investigation of some nonlinear stochastic difference equation with continuous time.

Linear Stochastic Difference Equations

2013 ◽  
Author(s):  
Lars Peter Hansen ◽  
Thomas J. Sargent
Keyword(s):  
Time Series ◽  
Difference Equation ◽  
Difference Equations ◽  
Martingale Difference ◽  
Stochastic Difference Equation ◽  
Stochastic Difference Equations ◽  
Order Linear ◽  
Economic Agents ◽  
First Order ◽  
Competitive Equilibria

This chapter describes the vector first-order linear stochastic difference equation. It is first used to represent information flowing to economic agents, then again to represent competitive equilibria. The vector first-order linear stochastic difference equation is associated with a tidy theory of prediction and a host of procedures for econometric application. Ease of analysis has prompted the adoption of economic specifications that cause competitive equilibria to have representations as vector first-order linear stochastic difference equations. Because it expresses next period's vector of state variables as a linear function of this period's state vector and a vector of random disturbances, a vector first-order vector stochastic difference equation is recursive. Disturbances that form a “martingale difference sequence” are basic building blocks used to construct time series. Martingale difference sequences are easy to forecast, a fact that delivers convenient recursive formulas for optimal predictions of time series.

scholarly journals Existence of almost periodic solutions to some nonautonomous higher-order stochastic difference equations

2020 ◽  
Vol 7 (1) ◽  
pp. 72-80
Author(s):  
Paul H. Bezandry
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Higher Order ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Stochastic Difference Equation ◽  
Stochastic Difference Equations

AbstractThe paper studies the existence of almost periodic solutions to some nonautonomous higher-order stochastic difference equation of the form:X\left( {t + n} \right) + \sum\limits_{r = 1}^{n - 1} {{A_r}\left( t \right)X\left( {t + r} \right) + {A_0}\left( t \right)X\left( t \right) = f\left( {t,X\left( t \right)} \right),}n ∈ 𝕑, by means of discrete dichotomy techniques.

Oscillation of Solutions to Second Order Stochastic Difference Equations

2014 ◽  
Vol 666 ◽  
pp. 282-287
Author(s):  
Jun Shan Zeng ◽  
Ya Bo Wang ◽  
Su Fang Han
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Body Part ◽  
Second Order ◽  
Stochastic Difference Equation ◽  
Stochastic Difference Equations ◽  
Oscillation Of Solutions

In this paper, we provide the criterion of existence of oscillation solutions to the following second order stochastic difference equation (please see in body part)

An invariance principle for solutions to stochastic difference equations

1981 ◽  
Vol 18 (2) ◽  
pp. 548-553
Author(s):  
Harry A. Guess
Keyword(s):  
Differential Equation ◽  
Population Genetics ◽  
Brownian Motion ◽  
Weak Convergence ◽  
Difference Equations ◽  
Continuous Time ◽  
Martingale Problem ◽  
Martingale Differences ◽  
Invariance Principles ◽  
Stochastic Difference Equations

In recent papers, McLeish and others have obtained invariance principles for weak convergence of martingales to Brownian motion. We generalize these results to prove that solutions of discrete-time stochastic difference equations defined in terms of martingale differences converge weakly to continuous-time solutions of Ito stochastic differential equations. Our proof is based on a theorem of Stroock and Varadhan which characterizes the solution of a stochastic differential equation as the unique solution of an associated martingale problem. Applications to mathematical population genetics are discussed.

ON SOLVING NONLINEAR FUNCTIONAL, FINITE DIFFERENCE, COMPOSITION, AND ITERATED EQUATIONS

Fractals ◽  
1999 ◽  
Vol 07 (03) ◽  
pp. 277-282 ◽  
Author(s):  
RAY BROWN
Keyword(s):  
Finite Difference ◽  
Difference Equation ◽  
Difference Equations ◽  
Finite Difference Equation ◽  
Euler Approximation ◽  
Nonlinear Functional ◽  
Finite Difference Equations ◽  
Wide Range ◽  
Iterated Equations ◽  
General Method

In this letter, we present a general method for solving a wide range of nonlinear functional and finite difference equations, as well as iterated equations such as the Hénon and Mandelbrot equations. The method extends to differential equations using an Euler approximation to obtain a finite difference equation.

On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations

1985 ◽  
Vol 17 (3) ◽  
pp. 666-678 ◽  
Author(s):  
K. S. Chan ◽  
H. Tong
Keyword(s):  
Lyapunov Function ◽  
Difference Equation ◽  
Difference Equations ◽  
Stochastic Difference Equations

We have shown that within the setting of a difference equation it is possible to link ergodicity with stability via the physical notion of energy in the form of a Lyapunov function.

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