From Discrete‐ to Continuous‐Time Finance: Weak Convergence of the Financial Gain Process 1

Source abstract

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 n of security price processes converging in distribution to S and a sequence θ n of trading strategies converging in distribution to θ . We survey conditions under which the financial gain process θ n dS 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.