It is well known that standard methods for SDE simulation, such as the Euler-Murayama method, are limited to a slow strong convergence rate of O(sqrt(h)) – where h denotes the step size. That said, O(h) strong convergence can be achieved if one uses both increments and iterated integrals of the underlying Brownian motion. However, these iterated integrals or “Lévy areas” are difficult to generate due to their non-Gaussian nature and for d-dimensional Brownian motion with d > 2, no exact sampling algorithm is known.
In this talk, we propose the use of a conditional Wasserstein-GAN to generate samples of Brownian Lévy area given the path increment as input. The advantage of this approach is that, once trained, GANs can achieve a good accuracy whilst being fast to evaluate and highly parallelizable. Moreover, by reflecting terms corresponding to the “Brownian bridge Lévy area”, we can ensure that the output distribution is unbiased and thus compatible with the mean-square error analysis commonly used to provide convergence guarantees in SDE numerics.
Whilst the proposed “Lévy-GAN” can be trained from a dataset of approximate Lévy areas, we show that Chen’s relation for integrals can allow GAN to learn Lévy area distributions without any data input. Finally, we demonstrate our approach in the 3-dimensional setting, where the Lévy-GAN offers faster sample generation than classical methods whilst offering a similar level of precision.
Originally from Slovenia, Andraž is a 4th year student in mathematics and computer science at the University of Oxford, specifically interested in stochastic analysis, numerical methods and machine learning.
More broadly, his interests extend to other scientific fields; he has competed in international Olympiads in mathematics, physics and astrophysics.
To become a member of the Rough Path Interest Group, register here for free.