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

Darrell Duffie and Philip Protter

Mathematical Finance, 1992, vol. 2, issue 1, 1-15

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 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.

Date: 1992
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