Abstract
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 problemis trivial, and a necessary condition for the existence of an optimal trading strategyis 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.
Highlights
The optimal investment problem is a classical one in financial mathematics, and it has been widely studied in the framework of Expected Utility Theory (EUT, for short), formulated by von Neumann and Morgenstern [21]
The third and most prominent feature of Cumulative Prospect Theory (CPT) is that the investor has a distorted perception of the actual probabilities, which is modelled by the two strictly increasing, continuous probability distortion functions w± : [0, 1] → [0, 1], with w±(0) = 0 and w±(1) = 1
We focused solely on the case where the investor’s utility on gains is bounded above and we found a necessary condition for the existence of an optimal solution
Summary
The optimal investment problem is a classical one in financial mathematics, and it has been widely studied in the framework of Expected Utility Theory (EUT, for short), formulated by von Neumann and Morgenstern [21] As some of EUT’s fundamental principles have been questioned by empirical studies, several alternative theories have emerged, amongst which the Cumulative Prospect Theory (CPT) proposed by Kahneman and Tversky [9] and Tversky and Kahneman [20] Within this framework, the utility function, which is still assumed to be strictly increasing with wealth, is no longer globally concave. The case of utilities growing slower than a power function remained open We address this problem in the present paper, in the setting of bounded above utility functions. For the sake of a simple exposition, all auxiliary results and proofs are compiled in Appendix A
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