Generative adversarial networks (GANs) have shown promising results when applied on partial differential equations and financial time series generation. This thesis investigates if GANs can be used to provide a strong approximation to the solution of stochastic differential equations (SDEs) of the Ito type. Standard GANs are only able to approximate processes in conditional distribution, yielding a weak approximation to the SDE.
A novel GAN architecture is proposed that enables strong approximation, called the constrained GAN. The discriminator of this GAN is informed with the random sample that corresponds to the Brownian motion increment between two time steps. This way, the constrained GAN does not only learn the conditional distribution, but the unique map from a random increment to the next asset value along the path, conditional on the previous value. The architecture was tested on geometric Brownian motion (GBM) and the Cox Ingersoll Ross (CIR) process in one dimension, where it was conditioned on a range of time steps and previous values of the asset process. The constrained GAN was shown to outperform discrete-time schemes in strong error on a discretisation with large time steps. It also outperformed the standard conditional GAN when approximating the conditional distribution. A method is proposed to extend the constrained GAN to general one-dimensional Ito SDEs, beyond the SDEs tested in this work. In future work, the constrained GAN should be conditioned on the SDE parameters as well, allowing it to learn an entire family of solutions at once. Furthermore, the architecture could be extended to higher dimensions, including systems of SDEs, such as the Heston model.