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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract. An investor faced with a contingent claim may eliminate risk by (super-) hedging in a nancial market. As this is often quite expensive, we study partial hedges which require less capital and reduce the risk. In a previous paper we determined quantile hedges which succeed with maximal probability, given a capital constraint. Here we look for strategies which minimize the shortfall risk de ned as the expectation of the shortfall weighted by some loss function. The resulting e cient hedges allow the investor to interpolate in a systematic way b e t ween the extremes of no hedge and a perfect (super-) hedge, depending on the accepted level of shortfall risk. Terms of use: Documents in EconStor may

We study various properties of a dynamic convex risk measure for bounded random variables which describe the discounted terminal values of financial positions. In particular we characterize time-consistency by a joint supermartingale property of the risk measure and its penalty function. Moreover we discuss the limit behavior of the risk measure in terms of asymptotic safety and of asymptotic precision, a property which may be viewed as a non-linear analogue of martingale convergence. These results are illustrated by the entropic dynamic risk measure.

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Motivated by a hedging problem in mathematical nance, El Karoui and Quenez 7] and Kramkov 1 4 ] h a ve d e v eloped optional versions of the Doob-Meyer decomposition which hold simultaneously for all equivalent martingale measures. We i n vestigate the general structure of such optional decompositions, both in additive and in multiplicative form, and under constraints corresponding to di erent classes of equivalent measures. As an application, we extend results of Karatzas and Cvitani c 3] on hedging problems with constrained portfolios.

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