Journal of Financial Risk Management, 2015, 4, 22-25
Published Online March 2015 in SciRes. http://www.scirp.org/journal/jfrm
http://dx.doi.org/10.4236/jfrm.2015.41003
How to cite this paper: Kountzakis, C. E., & Konstantinides, D. G. (2015). Rearrangement Invariant, Coherent Risk Measures
on L
0
. Journal of Financial Risk Management, 4, 22-25. http://dx.doi.org/10.4236/jfrm.2015.41003
Rearrangement Invariant, Coherent Risk
Measures on L
0
Christos E. Kountzakis, Dimitrios G. Konstantinides
Department of Mathematics, University of the Aegean, Karlovassi, Greece
Email: chrkoun@aegean.gr , konstant@aegean.gr
Received 5 January 2015; accepted 2 March 2015; published 5 March 2015
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
By this paper, we give an answer to the problem of definition of coherent risk measures on rear-
rangement invariant, solid subspaces of L
0
with respect to some atom less probability space
( ) , , Ω . This problem was posed by F. Delbaen, while in this paper we proposed a solution via
ideals of L
0
and the class of the dominated variation distributions, as well.
Keywords
Rearrangement Invariance, Dominated Variation, Moment-Index
1. Introduction
In (Delbaen, 2009), the problem of defining a risk measure on a solid, rearrangement invariant subspace of
( )
0
, , L Ω -space of random variables with respect to some atomless probability space ( ) , , Ω . We recall
that a vector space E, being a vector subspace of
0
L is called rearrangement invariant if for random rariables
0
, yx L ∈ , which have the same distribution, x E ∈ implies y E ∈ . Also, the space E is solid if for andom
viariables
0
, yx L ∈ , , y x x E ≤ ∈ , implies y E ∈ . In (Delbaen, 2009), there is an extensive treatment of this
problem, related to the role of the spaces L
∞
and
1
L , compared to E, especially in (Delbaen, 2009). On the
other hand, the whole paper (Delbaen, 2002) is devoted to the difficulties of defining coherent risk measures on
subspaces of
0
L , while it is proved that if the probability space is atomless, no coherent risk measure is defined
all over
0
L (Delbaen, 2002). Of course these attempts of moving from L
∞
to appropriately defined subspaces
of
0
L , are related to the tail propertes of the random variables in actuarial science and finance and more
specifically to heavy-tailed distributed random variables. The actual problem behind these seminal article by F.
Delbaen is since we cannot define a coherent risk measure on the entire
0
L , whether subspaces of
0
L which