Monte Carlo Simulation of SDEs using GANs

TL;DR

This paper explores using GANs to approximate one-dimensional SDEs, proposing a supervised GAN architecture that achieves strong path-wise approximation and outperforms traditional numerical schemes on key stochastic processes.

Contribution

The paper introduces a supervised GAN framework for strong approximation of SDEs, addressing limitations of standard GANs in path-wise accuracy and demonstrating improved performance on GBM and CIR models.

Findings

Supervised GANs outperform Euler and Milstein schemes in strong error.

Standard GANs may fail to provide path-wise approximation.

Supervised GANs are more robust and accurate for large time steps.

Abstract

Generative adversarial networks (GANs) have shown promising results when applied on partial differential equations and financial time series generation. We investigate if GANs can also be used to approximate one-dimensional Ito stochastic differential equations (SDEs). We propose a scheme that approximates the path-wise conditional distribution of SDEs for large time steps. Standard GANs are only able to approximate processes in distribution, yielding a weak approximation to the SDE. A conditional GAN architecture is proposed that enables strong approximation. We inform the discriminator of this GAN with the map between the prior input to the generator and the corresponding output samples, i.e. we introduce a `supervised GAN'. We compare the input-output map obtained with the standard GAN and supervised GAN and show experimentally that the standard GAN may fail to provide a path-wise approximation. The GAN is trained on a dataset obtained with exact simulation. The architecture was tested on geometric Brownian motion (GBM) and the Cox-Ingersoll-Ross (CIR) process. The supervised GAN outperformed the Euler and Milstein schemes in strong error on a discretisation with large time steps. It also outperformed the standard conditional GAN when approximating the conditional distribution. We also demonstrate how standard GANs may give rise to non-parsimonious input-output maps that are sensitive to perturbations, which motivates the need for constraints and regularisation on GAN generators.