Generative Modeling of Lévy Area for High Order SDE Simulation

It is well known that when numerically simulating solutions to stochastic differential equations (SDEs), achieving a strong convergence rate better than O⁡(√h) (where h is the step-size) usually requires the use of certain iterated integrals of Brownian motion, commonly referred to as its “Lévy 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 LévyGAN, a deep-learning-based model for generating approximate samples of Lévy 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 graph neural network (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 four-dimensional Brownian motion, we show that LévyGAN 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 a high-quality synthetic Lévy area can lead to high order weak convergence and variance reduction when using multilevel Monte Carlo (MLMC).