LévyGAN

Jelincic Andraz, University of Bath

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 ``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 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 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 high-quality synthetic Lévy area can lead to high order weak convergence and variance reduction when using multilevel Monte Carlo (MLMC).

Area: CS42 - Rough paths and data science (Christian Bayer, Paul Hager and Sebastian Riedel)

Keywords: Generative modelling, Lévy area, adversarial learning, probability theory, stochastic analysis, rough path theory, numerical approximation.

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