Version-Independence and Nested Distributions in Multistage Stochastic Optimization

Abstract

The amount of stagewise available information is crucial in multistage stochastic optimization. But unlike data, which directly enter the profit&loss functions of a decision problem, information is invariant w.r.t. bijective transformations. The usual concept to deal with information in multistage stochastic programming is by introducing filtrations, i.e., increasing sequences of sigma-algebras, to which the decisions must be adapted. For the definition of filtrations one has to fix a certain probability space while random variables are typically given by their distributions only and all realizations of this distribution on some probability space are equivalent. We introduce here the new concept of nested distributions to describe the information structure as well as the scenario process of a stochastic optimization program in a way which is independent of specific versions of probability spaces and random variables. The setting is totally “in-distribution.” Two stochastic programs (with identical objective function and constraints) are equivalent if and only if the scenario processes have the same nested distribution. As a byproduct, we analyze the question of whether introducing extra randomness by defining randomized decisions would lead to improvement in the objective value. In the language of information this would mean that enlarging the filtration based on available information by (conditionally) independent additional random variables would have a positive effect. We show that, in general, the answer is yes, while for compound convex objectives, the answer is no. Finally, we define a distance between nested distributions, which generalizes the well-known Kantorovich distance of probability distributions and demonstrates that this distance may be used in quantifying the quality of approximation between a continuous stochastic program and a tree discretization, or between two tree discretizations.