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Utility maximization in incomplete markets with random endowment. (English) Zbl 0993.91018

The problem of utility maximization in incomplete markets is relatively new and is the following: to find a solution to the problem of maximizing utility from terminal wealth of an agent with a random endowment process, in the general, semimartingale model for incomplete market, and to characterize it via the associated dual problem. It was solved in Itô-processes models of financial markets by I. Karatzas, J. P. Lehoczky, S. E. Shreve and G. L. Xu [SIAM J. Control Optim. 29, No. 3, 702-730 (1991; Zbl 0733.93085)] using the convex duality/martingale approach, which enabled the authors to deal with models which are not necessarily Markovian. The approach has recently been generalized to semimartingale models and under weaker conditions on the utility function by D. Kramkov and W. Schachermayer [Ann. Appl. Probab. 9, 904-950 (1999; Zbl 0967.91017)]. One of the main innovations of the latter paper, that made the approach work in this general context, was the extension of the domain of the dual problem: it is defined through a family of random variables $Y\left(T\right)$ (here $T$ denotes the terminal time) associated with nonnegative processes $Y\left(·\right)$ which are such that, for any admissible wealth process $X\left(·\right)$, the product process $X\left(·\right)Y\left(·\right)$ is a semimartingale, and not necessarily a local martingale as in the former paper.

In this paper the problem of maximizing the expected utility $E\left[U\left(X\left(T\right)\right)\right]$ of terminal wealth is solved for an agent whose income is represented as an arbitrary bounded and adapted endowment process. This is done in the general semimartingale incomplete model, under the same minimal conditions on the utility function $U$ as in the latter paper, and using a similar duality approach. The main difference is that the dual domain is even extended further – it is no longer contained in the space ${𝕃}^{1},$ but ${\left({𝕃}^{\infty }\right)}^{*}$, the dual space of ${𝕃}^{\infty }·$ In the language of control theory, the set of controls, over which the optimization is done in the dual problem, is relaxed. The solution $\stackrel{^}{Q}$ is then found in this set and the optimal terminal wealth is shown to be equal to the inverse of marginal utility evaluated at the regular part of $\stackrel{^}{Q}·$

MSC:
 91B28 Finance etc. (MSC2000) 91A24 Positional games 90C26 Nonconvex programming, global optimization 91B16 Utility theory 93E20 Optimal stochastic control (systems)