Abstract
In this work we consider the problem of numerical computation of convex risk measures, using a regularization scheme to account for undesirable fluctuations in the available historical data, combined with techniques from the Calculus of Variations.
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Notes
i.e., there exists \(r \in {{\mathbb {R}}}\) such that \(\rho (r+X)=0\), and r can be considered as the certainty equivalent of the position X.
The condition that all plausible models are absolutely continuous with respect to the reference measure P is imposed as both a technical and a qualitative condition. The technical nature will become apparent shortly while the qualitative means that the reference model takes appropriately all possible states of nature into account, though not necessarily assigning correct probabilities to them. Note that we do not need to impose the more restrictive condition of equivalence between P and Q.
We may choose \(1+\epsilon \) and not just 1 so that the entropy integral is well defined.
The positivity of u is taken care of by the inclusion of the KL-divergence term.
In general, the Sobolev space on a set \(\Omega \) is defined as \(W^{k,p}(\Omega ) := \{ f\in L^p(\Omega ) \, \Vert \, D^{\alpha }f \in L^p(\Omega ), \, \forall |\alpha |\le k \}.\)
For \(p\in [1,\infty )\) and a bounded set \(\Omega \) there exists constant \(C=C(\Omega ,p)\) such that \(\forall u\in W^{1,p}(\Omega )\): \(\Vert u\Vert _{L^p(\Omega )}\le C\Vert \nabla u\Vert _{L^p(\Omega )}\).
If F is an element of the dual space of \(W_{0}^{1,2}(I)\) and I is a bounded interval, then there exists \(g\in L^2(I)\) such that \((F,u) = \int _I g u dx,\) for all \(u\in W_{0}^{1,2}(I)\) where \(\Vert F\Vert _{(W_{0}^{1,2}(I))^{*}} = \Vert g \Vert _{L^2(I)}\). For more details see Brezis (2010).
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Acknowledgments
The authors wish to thank the four anonymous referees for their useful comments that enhanced the presentation of the results of the paper. We also wish to thank Professor G. W. Weber for his kind support and for useful discussions.
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Papayiannis, G.I., Yannacopoulos, A.N. Numerical computation of convex risk measures. Ann Oper Res 260, 417–435 (2018). https://doi.org/10.1007/s10479-016-2284-3
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DOI: https://doi.org/10.1007/s10479-016-2284-3