We consider non-concave and non-smooth random utility functions with do- main of definition equal to the non-negative half-line. We use a dynamic pro- gramming framework together with measurable selection arguments to establish both the no-arbitrage condition characterization and the existence of an optimal portfolio in a (generically incomplete) discrete-time financial market model with finite time horizon. In contrast to the existing literature, we propose to consider a probability space which is not necessarily complete.
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