From Discrete to Continuous Time Finance : Weak Convergence of the Financial Gain Process

Conditions, suitable for applications in finance, are given for the weak convergence (or convergence in probability) of stochastic integrals. For example, consider a sequence S of security price processes converging in distribution to S and a sequence θ of trading strategies converging in distribution to θ. We survey conditions under which the financial gain process ∫ θ dS converges in distribution to ∫ θ dS. Examples include convergence from discrete to continuous time settings, and in particular, generalizations of the convergence of binomial option replication models to the Black-Scholes model. Counterexamples are also provided. Duffie is with the Graduate School of Business, Stanford University, Stanford, CA 94305, and acknowledges the financial support of Batterymarch Financial Management. Protter is with the Departments of Mathematics and Statistics, Purdue University, West Lafeyette, IN 47907, and is supported in part by NSF grant #DMS8805595. We thank Jaime San Martin, Jean Jacod, Tom Kurtz, Ernst Eberlein, and an unusually attentive referee for helpful comments.