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From Discrete‐ to Continuous‐Time Finance: Weak Convergence of the Financial Gain Process 1
Author(s) -
Duffie Darrell,
Protter Philip
Publication year - 1992
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/j.1467-9965.1992.tb00022.x
Subject(s) - convergence (economics) , weak convergence , sequence (biology) , counterexample , mathematics , discrete time and continuous time , mathematical finance , stochastic process , distribution (mathematics) , discrete time stochastic process , probability distribution , finance , economics , mathematical optimization , mathematical economics , computer science , continuous time stochastic process , discrete mathematics , statistics , mathematical analysis , computer security , biology , asset (computer security) , genetics , economic growth
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.