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 Sn of security price processes converging in distribution to S and a sequence theta^n of trading strategies converging in distribution to theta. We survey conditions under which the financial gain process f on dSn converges in distribution to f OdS. 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. Counter examples are also provided._x000B_