  We consider an expected utility maximization problem where the utility
function is not necessarily concave and the time horizon is uncertain. We
establish a necessary and sufficient condition for the optimality for general
non-concave utility function in a complete financial market. We show that the
general concavification approach of the utility function to deal with
non-concavity, while being still applicable when the time horizon is a stopping
time with respect to the financial market filtration, leads to sub-optimality
when the time horizon is independent of the financial risk, and hence can not
be directly applied. For the latter case, we suggest a recursive procedure
which is based on the dynamic programming principle. We illustrate our findings
by carrying out a multi-period numerical analysis for optimal investment
problem under a convex option compensation scheme with random time horizon. We
observe that the distribution of the non-concave portfolio in both certain and
uncertain random time horizon is right-skewed with a long right tail,
indicating that the investor expects frequent small losses and a few large
gains from the investment. While the (certain) average time horizon portfolio
at a premature stopping date is unimodal, the random time horizon portfolio is
multimodal distributed which provides the investor a certain flexibility of
switching between the local maximizers, depending on the market performance.
The multimodal structure with multiple peaks of different heights can be
explained by the concavification procedure, whereas the distribution of the
time horizon has significant impact on the amplitude between the modes.
