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-super-n" of security price processes converging in distribution to "S" and a sequence θ-super-n of trading strategies converging in distribution to "θ". We survey conditions under which the financial gain process "θ-super-n dS-super-n" 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. Copyright 1992 Blackwell Publishers.