This paper introduces a new approach to generating sample paths of unknown Markovian stochastic differential equations (SDEs) using diffusion models, a class of generative AI methods commonly employed in image and video applications. Unlike the traditional Monte Carlo methods for simulating SDEs, which require explicit specifications of the drift and diffusion coefficients, ours takes a model-free, data-driven approach. Given a finite set of sample paths from an SDE, we utilize conditional diffusion models to generate new, synthetic paths of the same SDE. Numerical experiments show that our method consistently outperforms two alternative methods in terms of the Kullback--Leibler (KL) divergence between the distributions of the target SDE paths and the generated ones. Moreover, we present a theoretical error analysis deriving an explicit bound on the said KL divergence. Finally, in simulation and empirical studies, we leverage these synthetically generated sample paths to boost the performance of reinforcement learning algorithms for continuous-time mean--variance portfolio selection, hinting promising applications of our study in financial analysis and decision-making.

翻译：本文提出了一种利用扩散模型生成未知马尔可夫随机微分方程样本路径的新方法。扩散模型是一类常用于图像与视频应用的生成式人工智能方法。传统模拟随机微分方程的蒙特卡洛方法需要显式设定漂移系数与扩散系数，而我们的方法采用无模型、数据驱动的范式。给定来自某随机微分方程的有限样本路径集，我们利用条件扩散模型生成同一方程的新合成路径。数值实验表明，在目标随机微分方程路径分布与生成路径分布之间的Kullback--Leibler散度指标上，本方法始终优于两种对比方法。此外，我们通过理论误差分析推导了该KL散度的显式上界。最后，在仿真与实证研究中，我们利用这些合成样本路径提升了连续时间均值-方差投资组合选择问题中强化学习算法的性能，这预示着本研究在金融分析与决策领域具有广阔的应用前景。