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An invariance principle for solutions to stochastic difference equations

Published online by Cambridge University Press:  14 July 2016

Harry A. Guess
Harry A. Guess*
Affiliation:
National Institutes of Health
*
∗Postal address: Department of Pediatrics, University of North Carolina, School of Medicine, Chapel Hill, NC 27514, U.S.A.
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Abstract

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.

Keywords

STOCHASTIC DIFFERENCE EQUATIONSSTOCHASTIC DIFFERENTIAL EQUATIONSWEAK CONVERGENCEMARTINGALESPOPULATION GENETICS
Type
Short Communications
Information
Journal of Applied Probability , Volume 18 , Issue 2 , June 1981 , pp. 548 - 553
Copyright
Copyright © Applied Probability Trust 

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