用扩散模型生成未知随机微分方程的路径，无需知道方程具体形式。
Generating solution paths of Markovian stochastic differential equations using diffusion models
- 基于条件扩散模型，从数据中学习并生成SDE路径。
- 生成路径与真实路径的KL散度显著更低，优于两种对比方法。
- 适合金融建模、强化学习等需大量样本的连续时间决策场景。
本文提出一种新方法，利用扩散模型生成未知马尔可夫随机微分方程（SDE）的样本路径。不同于传统蒙特卡洛方法需明确指定漂移和扩散系数，该方法为无模型、数据驱动的方案。给定有限数量的真实路径样本，通过条件扩散模型生成新的合成路径。数值实验表明，该方法在目标SDE路径分布与生成路径分布之间的Kullback--Leibler（KL）散度上始终优于两种对比方法。此外，本文还给出了理论误差分析，推导出该KL散度的显式上界。在模拟与实证研究中，利用合成路径显著提升了连续时间均值-方差投资组合选择任务中强化学习算法的性能，暗示了在金融分析与决策中的广阔应用前景。
原文摘要 · Abstract (English)
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.
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