The framework of this paper is that of uncertainty, that is when no reference probability measure is given. To every convex regular risk measure \rho on {\cal C}_b(\Omega), we associate a canonical c_{\rho of probability measures . Furthermore{\cal   the convex risk measure admits a dual representation in terms of a weakly relatively compact set of probability measures absolutely continuous with respect to some probability measure belonging to the canonical c_{\rho . To get these results we study the topological properties of the dual of the Banach space L^1(c) associated to some capacity cand we prove a representation Theorem for convex risk measures on L^1(c). As applications, we obtain that every G-expectation \E (resp. in case of uncertain volatility every sublinear risk measure \rho), admits a representation with a numerable family of probability measures absolutely continuous with respect to some P belonging to the canonical c-class, with c(f )=  \E(|f|) , (resp. \rho(-|f|)) .
The framework of this paper is that of risk measuring under uncertainty, which is when no reference probability measure is given. To every regular convex risk measure on {\cal C}_b(\Omega), we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless non positive elements of{\cal C _b(\Omega). We prove that the convex risk measure has a dual representation with a countable set of probability measures absolutely continuous with respect to a certain probability measure in this class . To get these results we study the topological properties of the dual of the Banach space L^1(c) associated to a capacity c. As application we obtain that every G-expectation \E has a representation with a countable set of probability measures absolutely continuous with respect to a probability measure P such that P(|f| )= 0 iff \E(|f|) =0. We also apply our results to the case of uncertain volatility .
R
uncertainty, that
risk measuring under uncertainty, which
R
convex regular risk measure \rho
regular convex risk measure
R
canonical c_{\rho
unique equivalence class
R
. Furthermore
on Borel sets, characterizing the riskless non positive elements of
A
C
A
_b(\Omega). We prove that
R
admits
has
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in terms of a weakly relatively compact
with a countable
R
some probability measure belonging to the canonical c_{\rho
a certain probability measure in this class
R
some capacity cand we prove a representation Theorem for convex risk measures on L^1(c). As applications,
a capacity c. As application
R
(resp. in case of uncertain volatility every sublinear risk measure \rho), admits
has
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numerable family
countable set
R
some P belonging to the canonical c-class, with c(f
a probability measure P such that P(|f|
A
0 iff
R
, (resp. \rho(-|f|))
=0. We also apply our results to the case of uncertain volatility