Real-valued conditional convex risk measures in

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{$L^1(\mg,R)$} (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)