The role of information in multi-period risk measurement

Abstract

Multi-period risk functionals assign a risk value to a discrete-time stochasticprocess $Y = (Y_1 , . . . , Y_T )$. While convexity and monotonicity properties extend ina natural way from the single-period case and several types of translation properties may be defined, the role of information becomes crucial in the multi-period situation. In this paper, we define multi-period functionals in a generic way, such that the available information (expressed as a filtration) enters explicitly the definition of the functional. This allows to study the information monotonicity property,which comes as the counterpart of value monotonicity. We discuss several ways ofconstructing concrete and computable functionals out of conditional risk mappingsand single-period risk functionals. Some of them appear as value functions of multistage stochastic programs, where the filtration appears in the non-anticipativity constraint. This approach leads in a natural way to information monotonicity. Thesubclass of polyhedral multi-period risk functionals becomes important for theiremployment in practical dynamic decision making and risk management. On the other hand, several functionals described in literature are not information-monotone, which limits their practical use.