# Generative Modelling of L'evy Area for High Order SDE Simulation

arXiv: https://arxiv.org/abs/2308.02452 · HTML: https://mlsanity.com/paper/2308.02452/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 65 references that are in the map.
- First posted on arXiv 4 Aug 2023.

## Builds on (most cited first, 9 of 9)

- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [Sig-Wasserstein GANs for Time Series Generation](https://mlsanity.com/paper/2111.01207/) (2021)
- [Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond](https://mlsanity.com/paper/1906.07868/) (2019)
- [A Characteristic Function Approach to Deep Implicit Generative Modeling](https://mlsanity.com/paper/1909.07425/) (2019)
- [PCF-GAN: generating sequential data via the characteristic function of measures on the path space](https://mlsanity.com/paper/2305.12511/) (2023)
- [Reciprocal Adversarial Learning via Characteristic Functions](https://mlsanity.com/paper/2006.08413/) (2020)

## Built on by (most cited first, 3 of 3)

- [Single-seed generation of Brownian paths and integrals for adaptive and high order SDE solvers](https://mlsanity.com/paper/2405.06464/) (2024)
- [Sig-DEG for Distillation: Making Diffusion Models Faster and Lighter](https://mlsanity.com/paper/2508.16939/) (2025)
- [Faithful Embeddings of Irregular and Asynchronous Data for Online Log-NCDEs](https://mlsanity.com/paper/2605.30213/) (2026)

## Abstract (arXiv)

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).