## 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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