# Real-valued conditional convex risk measures in Lp(ℱ, R)

ESAIM: Proceedings (2011)

- Volume: 31, page 101-115
- ISSN: 1270-900X

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topErick, Treviño-Aguilar. Emilia Caballero, Ma., et al, eds. "Real-valued conditional convex risk measures in Lp(ℱ, R)." ESAIM: Proceedings 31 (2011): 101-115. <http://eudml.org/doc/251201>.

@article{Erick2011,

abstract = {The numerical representation of convex risk measures beyond essentially bounded financial
positions is an important topic which has been the theme of recent literature. In other
direction, it has been discussed the assessment of essentially bounded risks taking
explicitly new information into account, i.e., conditional convex risk measures. In this
paper we combine these two lines of research. We discuss the numerical representation of
conditional convex risk measures which are defined in a space
Lp(ℱ, R), for
p ≥ 1, and take values in \hbox\{$L^1(\mg,R)$\}L1(𝒢, R)
(in this sense, real-valued). We show how to characterize such a class of
real-valued conditional convex risk measures. In the first result of the paper, we see
that real-valued conditional convex risk measures always admit a numerical representation
in terms of a nice class of “locally equivalent”probability measures
\hbox\{$\mq^\{q,\infty\}_\{e,loc\}$\}. To
this end, we use the recent extended Namioka-Klee Theorem, due to Biagini and Frittelli.
The second result of the paper says that a conditional convex risk measure defined in a
space Lp(ℱ, R) is real-valued
if and only if the corresponding minimal penalty function satisfies a coerciveness
property, as introduced by Cheridito and Li in the non-conditional case. This
characterization, together with an invariance property will allow us to characterize
conditional convex risk measures defined in a space
L∞(ℱ, R) which can be extended to a space
Lp(ℱ, R), and at the same
time continue to be real-valued. In particular we see that the measures of risk, AVaR and
Shortfall, assign real values even if we extend their natural domain
L∞(ℱ, R) to a space
Lp(ℱ, R).},

author = {Erick, Treviño-Aguilar},

editor = {Emilia Caballero, Ma., Chaumont, Loïc, Hernández-Hernández, Daniel, Rivero, Víctor},

journal = {ESAIM: Proceedings},

keywords = {convex risk measure; average value at risk; -spaces},

language = {eng},

month = {3},

pages = {101-115},

publisher = {EDP Sciences},

title = {Real-valued conditional convex risk measures in Lp(ℱ, R)},

url = {http://eudml.org/doc/251201},

volume = {31},

year = {2011},

}

TY - JOUR

AU - Erick, Treviño-Aguilar

AU - Emilia Caballero, Ma.

AU - Chaumont, Loïc

AU - Hernández-Hernández, Daniel

AU - Rivero, Víctor

TI - Real-valued conditional convex risk measures in Lp(ℱ, R)

JO - ESAIM: Proceedings

DA - 2011/3//

PB - EDP Sciences

VL - 31

SP - 101

EP - 115

AB - The numerical representation of convex risk measures beyond essentially bounded financial
positions is an important topic which has been the theme of recent literature. In other
direction, it has been discussed the assessment of essentially bounded risks taking
explicitly new information into account, i.e., conditional convex risk measures. In this
paper we combine these two lines of research. We discuss the numerical representation of
conditional convex risk measures which are defined in a space
Lp(ℱ, R), for
p ≥ 1, and take values in \hbox{$L^1(\mg,R)$}L1(𝒢, R)
(in this sense, real-valued). We show how to characterize such a class of
real-valued conditional convex risk measures. In the first result of the paper, we see
that real-valued conditional convex risk measures always admit a numerical representation
in terms of a nice class of “locally equivalent”probability measures
\hbox{$\mq^{q,\infty}_{e,loc}$}. To
this end, we use the recent extended Namioka-Klee Theorem, due to Biagini and Frittelli.
The second result of the paper says that a conditional convex risk measure defined in a
space Lp(ℱ, R) is real-valued
if and only if the corresponding minimal penalty function satisfies a coerciveness
property, as introduced by Cheridito and Li in the non-conditional case. This
characterization, together with an invariance property will allow us to characterize
conditional convex risk measures defined in a space
L∞(ℱ, R) which can be extended to a space
Lp(ℱ, R), and at the same
time continue to be real-valued. In particular we see that the measures of risk, AVaR and
Shortfall, assign real values even if we extend their natural domain
L∞(ℱ, R) to a space
Lp(ℱ, R).

LA - eng

KW - convex risk measure; average value at risk; -spaces

UR - http://eudml.org/doc/251201

ER -

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