An invariance principle for solutions to 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.