Quasi-Monte Carlo for multivariate distributions via generative neural networks

Generative moment matching networks (GMMNs) are introduced as quasi-random number generators (QRNGs) for multivariate models with any underlying copula in order to estimate expectations with variance reduction. So far, QRNGs for multivariate distributions required a careful design, exploiting specific properties (such as conditional distributions) of the implied copula or the underlying quasi-Monte Carlo (QMC) point set, and were only tractable for a small number of models. Utilizing GMMNs allows one to construct QRNGs for a much larger variety of multivariate distributions without such restrictions. Once trained with a pseudo-random sample, these neural networks only require a multivariate standard uniform randomized QMC point set as input and are thus fast in estimating expectations of interest under dependence with variance reduction. Numerical examples are considered to demonstrate the approach, including applications inspired by risk management practice. All results are reproducible with the demo HPZ19 as part of the new R package gnn; select minimal working examples are provided in the demo GMMN_QMC of gnn