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Wealth-path dependent utility maximization in incomplete markets. (English) Zbl 1063.91029

The authors deal with a wealth-path dependent utility maximization problem \[ \text{maximize}\quad E\left[\int_0^T U(X_t)\, dF_t\right], \] where \(F\) is a non-decreasing right-continuous process, \(U\) is the utility function. Here, in contrast to the fixed horizon framework, the whole path of the portfolio process has to be considered. It follows that the dual variables are defined as stochastic processes as opposed to \(\mathcal F_t\)-measurable variables. A dual formulation is derived in a general incomplete semimartingale model. The existence of a solution of the original problem is proved directly. The expected duality characterization of the solution of the original problem is retrieved. The case of complete markets is also discussed. A precise description of the optimal portfolio strategy is obtained, and a sufficient condition optimality in the general case is provided. The result is then illustrated by examples.

MSC:

91G10 Portfolio theory
91B16 Utility theory
49N15 Duality theory (optimization)
49N30 Problems with incomplete information (optimization)
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