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Optimal investment under behavioural criteria -- a dual approach

We consider a discrete-time, generically incomplete market model and a behavioural investor with power-like utility and distortion functions. The existence of optimal strategies in this setting has been shown in a previous paper under certain conditions on the parameters of these power functions. In the present paper we prove the existence of optimal strategies under a different set of conditions on the parameters, identical to the ones which were shown to be necessary and sufficient in the Black-Scholes model. Although there exists no natural dual problem for optimisation under behavioural criteria (due to the lack of concavity), we will rely on techniques based on the usual duality between attainable contingent claims and equivalent martingale measures.