Approaches to Conditional Risk

Abstract

We present and compare two different approaches to conditional risk measures. One approach draws from convex analysis in vector spaces and presents risk measures as functions on $L^p$ spaces, while the other approach utilizes module-based convex analysis where conditional risk measures are defined on $L^p$-type modules. Both approaches utilize general duality theory for vector-valued convex functions, in contrast to the current literature, in which we find ad hoc dual representations. By presenting several applications such as monotone and (sub)cash invariant hulls with corresponding examples we illustrate that module-based convex analysis is well suited to the concept of conditional risk measures. Read More: http://epubs.siam.org/doi/abs/10.1137/090773076We present and compare two different approaches to conditional risk measures. One approach draws from convex analysis in vector spaces and presents risk measures as functions on $L^p$ spaces, while the other approach utilizes module-based convex analysis where conditional risk measures are defined on $L^p$-type modules. Both approaches utilize general duality theory for vector-valued convex functions, in contrast to the current literature, in which we find ad hoc dual representations. By presenting several applications such as monotone and (sub)cash invariant hulls with corresponding examples we illustrate that module-based convex analysis is well suited to the concept of conditional risk measures.