Jelinčič, Andraž;
Tao, Jiajie;
Turner, William F;
Cass, Thomas;
Foster, James;
Ni, Hao;
(2025)
Generative Modeling of Lévy Area for High Order SDE Simulation.
SIAM Journal on Mathematics of Data Science (SIMODS)
, 7
(4)
pp. 1541-1567.
10.1137/23M161077X.
Preview |
Text
Ni_23m161077x.pdf Download (1MB) | Preview |
Abstract
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).
| Type: | Article |
|---|---|
| Title: | Generative Modeling of Lévy Area for High Order SDE Simulation |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1137/23M161077X |
| Publisher version: | https://doi.org/10.1137/23M161077X |
| Language: | English |
| Additional information: | This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | generative modeling, Lévy area, adversarial learning, probability theory, stochastic analysis, rough path theory, numerical approximation |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10207331 |
Archive Staff Only
 |
View Item |
