Multiscale stochastic programming

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 the 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 data structure for the numerical solution of such problems. We first propose a new method for 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 framework are elaborated explicitly for various popular modeling choices, including both diffusion and jump models.