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

Erick, 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 -