Weak convergence of a sequence of stochastic difference equations to a stochastic ordinary differential equation

1987 ◽  
Vol 25 (6) ◽  
pp. 643-652 ◽  
Author(s):  
Masaru Iizuka
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Weak Convergence ◽  
Difference Equations ◽  
Stochastic Difference Equations

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.

An invariance principle for solutions to stochastic difference equations

1981 ◽  
Vol 18 (02) ◽  
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.

scholarly journals STRONGLY IRREGULAR PERIODIC SOLUTIONS OF THE FIRST-ORDER LINEAR HOMOGENEOUS DISCRETE EQUATION

2018 ◽  
Vol 62 (3) ◽  
pp. 263-267 ◽  
Author(s):  
A. K. Demenchuk
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Prime Numbers ◽  
Discrete Equation ◽  
Linear Difference Equations ◽  
Order Linear ◽  
First Order ◽  
Discrete Equations

 In 1950 J. Massera proved that a fi rst-order scalar periodic ordinary differential equation has no strongly ira proved that a first-order scalar periodic ordinary differential equation has no strongly irregular periodic solutions, that is, such solutions whose period of solution is incommensurable with the period of equation. For difference equations with discrete time, strong irregularity means that the period of the equation and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the above result of J. Massera has no complete analog.The purpose of this article is to investigate the possibility to realize Massera’s theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a first-order linear homogeneous non-stationary periodic discrete equation has no strongly irregular non-stationary periodic solutions.

scholarly journals On strongly irregular periodic solutions of the linear nonhomogeneous discrete equation of the first order

2020 ◽  
Vol 56 (1) ◽  
pp. 30-35
Author(s):  
A. K. Demenchuk
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Discrete Time ◽  
Difference Equations ◽  
Prime Numbers ◽  
Discrete Equation ◽  
Linear Difference Equations ◽  
First Order ◽  
Discrete Equations

As is proved earlier (the Massera theorem), the first-order scalar periodic ordinary differential equation does not have strongly irregular periodic solutions (solutions with a period incommensurable with the period of the equation). For difference equations with discrete time, strong irregularity means that the equation period and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the mentioned result has no complete analog.The purpose of this paper is to investigate the possibility of realizing an analog of the Massera theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a linear nonhomogeneous non-stationary periodic discrete equation of the first order does not have strongly irregular non-stationary periodic solutions.

scholarly journals The Solution of Difference Equations by Continued Fractions

1915 ◽  
Vol 34 ◽  
pp. 61-75
Author(s):  
J. A. Strang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Difference Equation ◽  
Difference Equations ◽  
Continued Fractions ◽  
Special Form ◽  
Quadratic Equation ◽  
Similar Process ◽  
Image Position ◽  
Bilinear Equation

The theorems which furnish in C.F. form the roots of a quadratic equation, and the similar process which leads to particular integrals of an ordinary differential equation of the second order, may be applied to certain types of difference equation. The types which suggest themselves for examination arethe bilinear equation, anda special form of the linear equation; the coefficients are functions of r, and s is constant.

Sign in / Sign up

Export Citation Format

Share Document