From Empirical Observations to Tree Models for Stochastic Optimization: Convergence Properties

In multistage stochastic optimization we use stylized processes to model the relevant stochastic data processes. The basis for building these models is empirical observations. It is well known that the determining distance concept for multistage stochastic optimization problems is the nested distance and not the distance in distribution. In this paper we investigate the question of how to generate models out of empirical data, which approximate well the underlying stochastic processes in nested distance. We demonstrate first that the empirical measure, which is built from observed sample paths, does not converge in nested distance to the pertaining distribution if the latter has a density. On the other hand, we show that smoothing convolutions, which are appropriately adapted from classical kernel density estimation, can be employed to modify the empirical measure in order to obtain stochastic processes which converge in nested distance to the underlying process. We employ the results to estimate the conditional densities for each time stage. Finally we construct discrete tree processes from observed empirical paths, which approximate well the original stochastic process as they converge in nested distance to the underlying process.