Comparison of Sampling Methods for Dynamic Stochastic Programming

Abstract

In solving a scenario-based dynamic (multistage) stochastic programme scenario generation plays a critical role, as it forms the input specification to the optimization process. Computational bottlenecks in this process place a limit on the number of scenarios employable in approximating the probability distribution of the paths of the underlying uncertainty. Traditional scenario generation approaches have been to find a sampling method that best approximates the path distribution in terms of some probability metrics such as minimization of moment deviations or Wasserstein distance. Here, we present a Wasserstein-based heuristic for discretization of a continuous state path probability distribution. The chapter compares this heuristic to the existing methods in the literature (Monte Carlo sampling, moment matching, Latin hypercube sampling, scenario reduction, and sequential clustering) in terms of their effectiveness in suppressing sampling error when used to generate the scenario tree of a dynamic stochastic programme. We perform an extensive computational investigation into the impact of scenario generation techniques on the in-sample and out-of-sample stability of a simplified version of a four-period asset–liability management problem employed in practice (Chapter 2, this volume). A series of out-of-sample tests are carried out to evaluate the effect of possible discretization biases. We also attempt to provide a motivation for the popular utilization of left-heavy scenario trees based on the Wasserstein distance criterion. Empirical results show that all methods outperform normal MC sampling. However, when evaluated against each other these methods essentially perform equally well, with second-order moment matching showing only marginal improvements in terms of in-sample decision stability and out-of-sample performance. The out-of-sample results highlight the problem of under-estimation of portfolio risk which results from insufficient samples. This discretization bias induces overly aggressive portfolio balance recommendations which can impair the performance of the model in real-world applications. Thus this issue needs to be carefully addressed in future research, see e.g. Dempster et al. (2010).KeywordsScenario generationSampling methodsDiscretization errorScenario-based approximationStochastic programmingIn-sample and out-of-sample tests