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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.

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