Minimizing the expected market time to reach a certain wealth level

In a financial market model, we consider variations of the problem of minimizing the expected time to upcross a certain wealth level. For exponential Lévy markets, we show the asymptotic optimality of the growth-optimal portfolio for the above problem and obtain tight bounds for the value function for any wealth level. In an Itô market, we employ the concept of market time, which is a clock that runs according to the underlying market growth. We show the optimality of the growth-optimal portfolio for minimizing the expected market time to reach any wealth level. This reveals a general definition of market time which can be useful from an investor's point of view. We utilize this last definition to extend the previous results in a general semimartingale setting.