Multistage Optimization

Abstract

We provide a new identity for the multistage Average Value-at-Risk. The identity is based on the conditional Average Value-at-Risk at random level, which is introduced. It is of interest in situations, where the information available increases over time, so it is – among other applications – customized to multistage optimization. The identity relates to dynamic programming and is adapted to problemswhich involve the Average Value-at-Risk in its objective. We elaborate further dynamic programming equations for specific multistage optimization problems and derive a characterizing martingale property for the value function. The concept solves a particular aspect of time consistency and is adapted for situations, wheredecisions are planned and executed consecutively in subsequent instants of time. We discuss theapproach for other risk measures, which are in frequent use for decision making under uncertainty,particularly for financial decisions.