  This paper examines an optimal investment problem in a continuous-time
(essentially) complete financial market with a finite horizon. We deal with an
investor who behaves consistently with principles of Cumulative Prospect
Theory, and whose utility function on gains is bounded above. The
well-posedness of the optimisation problem is trivial, and a necessary
condition for the existence of an optimal trading strategy is derived. This
condition requires that the investor's probability distortion function on
losses does not tend to 0 near 0 faster than a given rate, which is determined
by the utility function. Under additional assumptions, we show that this
condition is indeed the borderline for attainability, in the sense that for
slower convergence of the distortion function there does exist an optimal
portfolio.
