From Discrete- to Continuous-Time Finance: Weak Convergence of the Financial Gain Process1

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 θⁿ of trading strategies converging in distribution to θ. We survey conditions under which the financial gain process θⁿ dSⁿ 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.