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).

翻译：众所周知，在数值模拟随机微分方程解的过程中，要获得优于O(√h)（h为步长）的强收敛阶，需要使用布朗运动的某些迭代积分，即所谓的“Lévy区域”。然而，由于这些随机积分具有非高斯性质，且对于维数d>2的d维布朗运动，目前尚无已知的快速几乎精确采样算法。本文提出LévyGAN——一种基于深度学习的模型，用于生成条件于布朗增量的Lévy区域的近似样本。通过我们设计的“桥翻转”操作，输出样本能够精确匹配所有联合和条件奇次矩。生成器采用定制的受图神经网络启发的架构，强制输出分布与条件变量之间具有正确的依赖结构。此外，我们引入了一个基于数学原理的特征函数判别器。最后，我们提出一种名为“陈训练”的新型训练机制，该机制避免了生成昂贵训练数据集的必要性。这一新训练过程基于我们的两个主要理论结果。对于4维布朗运动，我们证明LévyGAN在多个衡量联合分布与边缘分布的指标上展现出最先进的性能。我们以数学金融中流行的随机微分方程——对数Heston模型为例进行数值实验，结果表明高质量合成Lévy区域可在使用多级蒙特卡洛方法时实现高阶弱收敛与方差缩减。