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This paper examines an optimal investment problem in a continuous-time(essentially) complete financial market with a finite horizon. We deal with aninvestor who behaves consistently with principles of Cumulative ProspectTheory, and whose utility function on gains is bounded above. Thewell-posedness of the optimisation problem is trivial, and a necessarycondition for the existence of an optimal trading strategy is derived. Thiscondition requires that the investor's probability distortion function onlosses does not tend to 0 near 0 faster than a given rate, which is determinedby the utility function. Under additional assumptions, we show that thiscondition is indeed the borderline for attainability, in the sense that forslower convergence of the distortion function there does exist an optimalportfolio.

Continuous-Time Portfolio Optimisation for a Behavioural Investor with Bounded Utility on Gains