Multi-stage stochastic optimization: the distance between stochastic scenario processes

Approximation techniques are challenging, important and very often irreplaceable solution methods for multi-stage stochastic optimization programs. Applications for scenario process approximation include financial and investment planning, inventory control, energy production and trading, electricity generation planning, pension fund management, supply chain management and similar fields. In multi-stage stochastic optimization problems the amount of stage-wise available information is crucial. While some authors deal with filtration distances, in this paper we consider the concepts of nested distributions and their distances which allows to keep the setup purely distributional but at the same time to introduce information and information constraints. Also we introduce the distance between stochastic process and a tree and we generalize the concept of nested distance for the case of infinite trees, i.e. for the case of two stochastic processes given by their continuous distributions. We are making a step towards to a new method for distribution quantization that is the most suitable for multi-stage stochastic optimization programs as it takes into account both the stochastic process and the stage-wise information.