## Logoptimal portfolios

Contrary to the previous points, let us suppose that there are a finite number of risky assets available on the market. These assets are traded continuously without any transaction costs. The investor analyses historical market data and based on this, can reset her portfolio at the end of each day. How can she maximize her wealth in the long run? If returns are independent in time, then markets are efficient in the weak sense and the time series of returns have no memory. If returns are also identically distributed (i.i.d), the optimal strategy is to set portfolio weights for example, according to the Markowitz model (see *Daróczi et al. 2013*) and to keep portfolio weights fixed over the whole time horizon. In this setting, any rearrangements would have negative effects on the portfolio value in the long run.

Now, let us suspend the assumption of longitudinal independency, hence let us allow for hidden patterns in the asset returns, therefore markets are not efficient...