Are Time Consistent Valuations Information Monotone?

Multi-period risk functionals assign a risk value to discrete-time stochastic processes. While convexity and monotonicity extend in straightforward manner from the single-period case, the role of information is more problematic in the multi-period situation. In this paper, we define multi-period functionals in such a way that the development of available information over time (expressed as a filtration) enters explicitly the definition of the functional. This allows to define and study the property of information monotonicity, i.e. monotonicity w.r.t. increasing filtrations. On the other hand, time consistency of valuations is a favorable property and it is well-known that this requirement essentially leads to compositions of conditional mappings. We demonstrate that generally spoken the intersection of time consistent and information monotone valuation functionals is rather sparse, although both classes alone are quite rich. In particular, the paper gives a necessary and sufficient condition for information monotonicity of additive compositions of positively homogeneous risk/acceptability mappings. Within the class of distortion functionals only compositions of expectation or essential infima are information monotone. Furthermore, we give a sufficient condition and examples for compositions of nonhomogeneous mappings exhibiting information monotonicity.