Numeraire portfolios and utility-based price systems under proportional transaction costs

In a discrete-time incomplete financial market with proportional transaction costs and with independent and bounded returns, we prove the existence of a consistent price system that can be written as the expectation of the discounted claim under the real-world probability measure P and not just under a martingale measure. In fact, the claim is then discounted by some specific dynamic portfolio called the numeraire portfolio as in the classical case of markets without transaction costs. For that specific numeraire, P will be a martingale measure. Naturally, the concept of a numeraire portfolio has here to be adapted to the concept of consistent price systems for markets with transaction costs. Moreover, again as in the classical case, the numeraire portfolio can be chosen as log-optimal portfolio. The same analysis works for power utility functions. However, then a change of measure is necessary. This paper applies methods from stochastic dynamic programming to finance.