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

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  • Darrell Duffie
  • Philip Protter

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.

Suggested Citation

  • Darrell Duffie & Philip Protter, 1992. "From Discrete‐ to Continuous‐Time Finance: Weak Convergence of the Financial Gain Process1," Mathematical Finance, Wiley Blackwell, vol. 2(1), pages 1-15, January.
    • Handle: RePEc:bla:mathfi:v:2:y:1992:i:1:p:1-15
    DOI: 10.1111/j.1467-9965.1992.tb00022.x
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