Martingale Method for Utility Maximization
The problem of the expected utility maximization for diffusion models in complete markets has been studied by many authors; see the review article of Karatzas (1989). The same problem in incomplete markets has been extensively studied by Karatzas et al. (1991). They judiciously augmented the stocks with fictitious ones to create a complete market, such that the fictitious stocks are superfluous in the optimal portfolio for the completed market. In this case, their solution is also optimal for the original incomplete market.
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