Generative Modelling of L\'evy Area for High Order SDE Simulation
Abstract
It is well understood that, when numerically simulating SDEs with general noise, achieving a strong convergence rate better than O(h) (where h is the step size) requires the use of certain iterated integrals of Brownian motion, commonly referred to as its "L\'evy areas". However, these stochastic integrals are difficult to simulate due to their non-Gaussian nature and for a d-dimensional Brownian motion with d > 2, no fast almost-exact sampling algorithm is known. In this paper, we propose L\'evyGAN, a deep-learning-based model for generating approximate samples of L\'evy area conditional on a Brownian increment. Due to our "Bridge-flipping" operation, the output samples match all joint and conditional odd moments exactly. Our generator employs a tailored GNN-inspired architecture, which enforces the correct dependency structure between the output distribution and the conditioning variable. Furthermore, we incorporate a mathematically principled characteristic-function based discriminator. Lastly, we introduce a novel training mechanism termed "Chen-training", which circumvents the need for expensive-to-generate training data-sets. This new training procedure is underpinned by our two main theoretical results. For 4-dimensional Brownian motion, we show that L\'evyGAN exhibits state-of-the-art performance across several metrics which measure both the joint and marginal distributions. We conclude with a numerical experiment on the log-Heston model, a popular SDE in mathematical finance, demonstrating that high-quality synthetic L\'evy area can lead to high order weak convergence and variance reduction when using multilevel Monte Carlo (MLMC).