Abstract
We show that the only dynamic risk measure which is law invariant, time consistent and relevant is the entropic one. Moreover, a real valued function c on L ∞(a, b) is normalized, strictly monotone, continuous, law invariant, time consistent and has the Fatou property if and only if it is of the form \({c(X)=u^{-1} \circ\mathbb {E}[u(X)]}\) , where \({u:(a, b) \to {\mathbb R}}\) is a strictly increasing, continuous function. The proofs rely on a discrete version of the Skorohod embedding theorem.
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We thank Freddy Delbaen, Damir Filipović, Hans Föllmer, Fabio Maccheroni, Eberhard Mayerhofer, Alexander Mürmann, Ludger Rüschendorf, Alexander Schied, Mihai Sirbu, Josef Teichmann, Nicolas Vogelpoth and especially Patrick Cheridito as well as two anonymous referees for helpful comments and suggestions. Financial support from the Vienna Science and Technology Fund (WWTF) is gratefully acknowledged. Financial support from the Austrian Science Fund (FWF) under grant P19456, from Vienna Science and Technology Fund (WWTF) under grant MA13 and by the Christian Doppler Research Association (CDG) is gratefully acknowledged.
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Kupper, M., Schachermayer, W. Representation results for law invariant time consistent functions. Math Finan Econ 2, 189–210 (2009). https://doi.org/10.1007/s11579-009-0019-9
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DOI: https://doi.org/10.1007/s11579-009-0019-9