Advances in Finance and Stochastics: Essays in Honour of by Freddy Delbaen (auth.), Klaus Sandmann, Philipp J.

By Freddy Delbaen (auth.), Klaus Sandmann, Philipp J. Schönbucher (eds.)

In many parts of finance and stochastics, major advances were made on the grounds that this box of analysis was once opened by means of Black, Scholes and Merton in 1973. Advances in Finance and Stochastics encompasses a number of unique articles through a few hugely distinctive authors on study subject matters which are presently within the concentration of curiosity of either lecturers and practitioners. the subjects span possibility administration, portfolio thought and multi-asset derivatives, industry imperfections, interest-rate modelling and unique options.

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Kluwer, Dordrecht, 1997. 11. J. Diestel. Geometry of Banach space-selected topics. In Lecture Notes in Mathematics, volume 485. Springer-Verlag, Berlin Heidelberg New York, 1975. 12. N. Dunford and J. Schwartz. Linear Operators Vol I. Interscience, 1958. 13. P. Embrechts, C. Kliippelberg, and T. Mikosch. Modelling Extremal Events. Springer-Verlag, New York, 1997. Coherent Risk Measures on General Probability Spaces 37 14. A. Grothendieck. Topological Vector Spaces. Gordon and Breach, New York, 1973.

This means that the bounded functions are not dense in Lof!. This will in fact be the kernel of our example. Let us now describe the set Pu . 3, we take Pu = {h I h 2 O,Ep[h] = 1 and IIhll p ~ 2}. From de la Vallee Poussin's theorem (on uniform integrability) it follows that P u is weakly compact. Let us now take the function f = log(rl) as above and let X = 0 and Xk = - f1u>k}. It follows from Young's inequality that p( - 1) < +00. Clearly X k increases to O. Since IIXkl of! 2 8, we get the existence of an element gk such that IIgkllp = 1 and such that Ep[gk( -Xk )] 2 8/4.

The corollary now follows. 7. If p: £00 -t IR is a coherent risk measure that satisfies the Fatou property and is defined with the set P q , if the extension p: £0 -t IR U { +oo} defined as p(X) = lim sup EQ[-(X 1\ n)) n QE'P" satisfies p(X) > -00 for all X E £0, if p dominates VaR for bounded random variables, then p dominates VaR for all random variables. Proof. The proof of this proposition is obvious. o We now can give the first theorem on the relation between VaR and coherent risk measures.

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