Generative Modelling of Lévy Area for High Order SDE Simulation
Andraž Jelinčič Jiajie Tao William F. Turner Thomas Cass James Foster Hao Ni
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 "L\'{e}vy 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\'{e}vyGAN, a deep-learning-based model for
generating approximate samples of L\'{e}vy 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\'{e}vyGAN 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\'{e}vy area can lead to high order weak convergence
and variance reduction when using multilevel Monte Carlo (MLMC).