Multiscale stochastic optimization: modeling aspects and scenario generation

Real-world multistage stochastic optimization problems are often characterized by the fact that the decision maker may take actions only at specific points in time, even if relevant data can be observed much more frequently. In such a case there are not only multiple decision stages present but also several observation periods between consecutive decisions, where profits/costs occur contingent on the stochastic evolution of some uncertainty factors. We refer to such multistage decision problems with encapsulated multiperiod random costs, as multiscale stochastic optimization problems. In this article, we present a tailor-made modeling framework for such problems, which allows for a computational solution. We first establish new results related to the generation of scenario lattices and then incorporate the multiscale feature by leveraging the theory of stochastic bridge processes. All necessary ingredients to our proposed modeling framework are elaborated explicitly for various popular examples, including both diffusion and jump models.