Comparison of Sampling Methods for Dynamic Stochastic Programming
49 Pages Posted: 31 Dec 2014 Last revised: 3 Jan 2015
Date Written: December 29, 2014
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 the minimization of moment deviations or (Monge-Kantotrovich-)Wasserstein distance. Here we present a Wasserstein-based heuristic for discretization of a continuous state path distribution. The paper compares this heuristic to the existing methods in the literature (Monte Carlo sampling, moment matching, Latin Hypercube sampling, scenario reduction, 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 investigation of the impact of scenario generation techniques on the in- and out-of-sample stability of a simplified version of a four-period asset liability management problem employed in practice. 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 (i.e. with more early than later period branching) based on the Wasserstein distance criterion. Empirical results show that all methods outperform normal MC sampling. However when evaluated against each other all these methods perform essentially equally well, with second-order moment matching showing only marginal improvements in terms of in-sample stability and out-of-sample performance. The out-of-sample results highlight the under-estimation of portfolio risk which results from insufficient scenario samples. This discretization bias induces overly aggressive portfolio balance recommendations which can impair the performance of the model in real world applications. Thus in future research this issue needs to be carefully addressed, see e.g.
Keywords: scenario generation, sampling methods, discretization error, scenario-based approximation, stochastic programming, in-sample and out-of-sample tests
JEL Classification: C61, G11, G17
Suggested Citation: Suggested Citation