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Optimization problems in the theory of continuous trading. (English) Zbl 0701.90008

The paper treats comprehensively applications of stochastic analysis to economics and finance. It narrows the gap in literature after the book by A. Malliaris [“Stochastic methods in economics and finance” (1982; Zbl 0479.90003)], written nearly 10 years ago. Here the following questions are considered:

a) A general treatment of the pricing of contingent claims such as options, which can be exercised only at maturity (European) and which can be exercised any time before or at maturity (American); b) The resolution of consumption/investment problems for a small investor with quite general utility functions; c) The associated study of equilibrium models.

The mathematical background includes two fundamental stochastic results: the Girsanov change of probability measure and the representation of Brownian martingales as stochastic integrals.

The text is organized in the following sections: 1) Introduction and summary. 2) The financial market model. 3) A small investor. 4) Admissible strategies. 5) The pricing of European options. 6) The pricing of Amerian options. 7) Utility functions. 8) Maximization of utility from consumption. 9) Maximization of utility from investment. 10) Maximization of utility from both consumption and terminal wealth. 11) The case of constant coefficients. 12) An equilibrium model.

At the end of the paper there are notes, where detailed comments and references are given. Finally, it should be pointed out that the aim of this invited expository article is completely achieved.

Reviewer: S.Gaidov
MSC:
91B28Finance etc. (MSC2000)
93E20Optimal stochastic control (systems)
91B16Utility theory
60G44Martingales with continuous parameter
91B62Growth models in economics
60G40Stopping times; optimal stopping problems; gambling theory
91B50General equilibrium theory in economics
60H20Stochastic integral equations
60J65Brownian motion
91B24Price theory and market structure