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On the theory of option pricing. (English) Zbl 0554.90019
The study of speculative prices led the French mathematician L. Bachelier already in 1900 to the discovery of the mathematical theory of Brownian motion, five years before Einstein’s classic paper. Since then, several prominent economists and mathematicians, i.e. P. Samuelson, H. P. McKean, R. Merton, H. Fölmer and C. Stricker,... have made highly interesting contributions to this and closely related problems using a variety of techniques, i.e. heat equations, optimal stoppings, Itô-calculus, martingales etc. The paper under review provides, using stochastic control, drift transformation, Kunita-Watanabe representations and perturbation techniques, an axiomatic framework aimed at defining in an economic meaningful way the concept of value function for risky operations and proves completeness of the market of assets, whose prices behave like Itô processes. Further, it characterizes univocally a value function of an ”American claim”, i.e. a security issued by a company giving its owner the right to purchase or sell a share of stock at a given ”exercise” price on or before a given date.
Reviewer: M.G.L.Gomez

91B62Growth models in economics
93E20Optimal stochastic control (systems)
62L15Optimal stopping (statistics)
91B28Finance etc. (MSC2000)
60H10Stochastic ordinary differential equations