On the Weak Convergence of a Sequence of General Stochastic Difference Equations to a Diffusion,

1979 ◽  
Author(s):  
H. J. Kushner ◽  
Hai Huang
Keyword(s):  
Weak Convergence ◽  
Difference Equations ◽  
Stochastic Difference Equations ◽  
General Stochastic

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.

Sign in / Sign up

Export Citation Format

Share Document