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Martingale and duality methods for utility maximization in an incomplete market. (English) Zbl 0733.93085
Let \(X^{x,\pi}\) denote the wealth process corresponding to a portfolio \(\pi\). \(X^{x,\pi}\) is a solution of a linear Ito equation with \(X^{x,\pi}(0)=x.\) The stochastic control problem is the following: To maximize the expected utility from terminal wealth \(EU(X^{s,\pi}(T))\). This problem is an example of the utility maximization in an incomplete market containing a bound and a finite number of stocks. The prices are driven are driven by an n-dimensional Brownian motion W. The number of stocks is strictly smaller than the dimension of W. Martingale techniques and convex optimization are used.

MSC:
93E20 Optimal stochastic control
60G44 Martingales with continuous parameter
91B62 Economic growth models
49K45 Optimality conditions for problems involving randomness
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