The problem of maximizing the expected utility is well understood in the context of a complete financial market. This dissertation studies the same problem in an arbitrage-free yet incomplete market. Jin and Zhou have characterized the set of the terminal wealths that can be replicated by admissible portfolios. The problem is then transformed into a static optimization problem. It is proved that the terminal wealth is attainable for all utility functions when the market parameters are deterministic. The optimal portfolio is obtained explicitly when the utility function is logarithmic even if the market parameters follow stochastic processes. However we do not succeed in extending this result to the power utility function
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