Generative Modelling of Lévy Area for High Order SDE Simulation

Abstract: It is well known that, when numerically simulating solutions to SDEs, achieving a strong convergence rate better than O(\sqrt{h}) (where h is the step size) requires the use of certain iterated integrals of Brownian motion, commonly referred to as its "L\'{e}vy 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\'{e}vyGAN, a deep-learning-based model for generating approximate samples of L\'{e}vy 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\'{e}vyGAN 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\'{e}vy area can lead to high order weak convergence and variance reduction when using multilevel Monte Carlo (MLMC).