A Quick Estimation of Fréchet Quantizers for a Dynamic Solution to Flood Risk Management Problems

Abstract

Multi-stage stochastic optimization is a well-known quantitative tool for decision-making under uncertainty. It is broadly used in financial and investment planning, inventory control, and also natural disaster risk management. Theoretical solutions of multi-stage stochastic programs can be found explicitly only in very exceptional cases due to their variational form and interdependency of uncertainty in time. Nevertheless, numerical solutions are often inaccurate, as they rely on Monte-Carlo sampling, which requires the Law of Large Numbers to hold for the approximation quality. In this article, we introduce a new approximation scheme, which computes and groups together stage-wise optimal quantizers of conditional Fréchet distributions for optimal weighting of value functions in the dynamic programming. We consider optimality of scenario quantization methods in the sense of minimal Kantorovich-Wasserstein distance at each stage of the scenario tree. By this, we bound the approximation error with convergence guarantees. We also provide global solution guarantees under convexity and monotonicity conditions on the value function. We apply the developed methods to the governmental budget allocation problem for risk management of flood events in Austria. For this, we propose an extremely efficient way to approximate optimal quantizers for conditional Fréchet distributions. Our approach allows to enhance the overall efficiency of dynamic programming via the use of different parameter estimation methods for different groups of quantizers. The groups are distinguished by a particular risk threshold and are able to differentiate between higher- and lower-impact flood events.