Generative Modelling of Lévy Area for High Order SDE Simulation

TL;DR

The paper tackles the challenge of high-order SDE simulation by addressing the difficulty of sampling Lévy area for multi-dimensional Brownian motion. It introduces LévyGAN, a generative framework with Bridge-flipping, Pair-net, and Chen-training, plus a unitary characteristic-function discriminator, to approximate the Lévy area conditioned on Brownian increments without requiring real samples. The authors establish distributional and training guarantees via Chen's relation and demonstrate state-of-the-art weak-distribution performance in 4D, with practical benefits in MLMC for the log-Heston model. This approach offers a scalable, data-efficient path to higher-order SDE solvers and points to future work in adaptive solvers and broader integrals.

Abstract

It is well understood that, when numerically simulating SDEs with general noise, 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).

Figures (6)

• Figure 1: Each entry $A^{(i,j)}$ is the area between the independent Brownian motions $W^{(i)}$ and $W^{(j)}$ (diagram adapted from FosterThesis).
• Figure 2: A schematic of Lévy generation for classical conditional GAN. Throughout this article, bsz represents the training batch-size.
• Figure 3: A schematic of the Pair-net architecture when $d=3$.
• Figure 4: Probability densities estimated using samples from the generator with the first two input dimensions swapped (orange) and without swapping (blue) for Pair-net (top), and the BF generator (bottom). The blue and orange plots should overlap as best as possible.
• Figure 5: A schematic of LévyGAN. Here $bsz$ denotes the batch dimension and we recall that $nsz$ denotes the total noise dimension, namely $nsz = n \times d$.

Theorems & Definitions (45)

• Definition 1.1
• Definition 2.1: Classical conditional GAN for Lévy Area generation
• Definition 3.1: Brownian bridge
• Proposition 3.2: Distribution of Brownian bridge Lévy area foster2020bFosterThesis
• Proposition 3.3: Polynomial expansion of Lévy area FH
• Lemma 3.4
• Definition 3.5
• Theorem 3.6: Bridge-flipping
• Proposition 3.7
• Remark 3.8