Financial scenario simulation is essential for risk management and portfolio optimization, yet it remains challenging especially in high-dimensional and small data settings common in finance. We propose a diffusion factor model that integrates latent factor structure into generative diffusion processes, bridging econometrics with modern generative AI to address the challenges of the curse of dimensionality and data scarcity in financial simulation. By exploiting the low-dimensional factor structure inherent in asset returns, we decompose the score function--a key component in diffusion models--using time-varying orthogonal projections, and this decomposition is incorporated into the design of neural network architectures. We derive rigorous statistical guarantees, establishing nonasymptotic error bounds for both score estimation at O(d^{5/2} n^{-2/(k+5)}) and generated distribution at O(d^{5/4} n^{-1/2(k+5)}), primarily driven by the intrinsic factor dimension k rather than the number of assets d, surpassing the dimension-dependent limits in the classical nonparametric statistics literature and making the framework viable for markets with thousands of assets. Numerical studies confirm superior performance in latent subspace recovery under small data regimes. Empirical analysis demonstrates the economic significance of our framework in constructing mean-variance optimal portfolios and factor portfolios. This work presents the first theoretical integration of factor structure with diffusion models, offering a principled approach for high-dimensional financial simulation with limited data. Our code is available at https://github.com/xymmmm00/diffusion_factor_model.

翻译：金融情景模拟对于风险管理和投资组合优化至关重要，但在金融领域常见的高维小数据场景下，这仍然具有挑战性。我们提出了一种扩散因子模型，该模型将潜在因子结构整合到生成扩散过程中，从而将计量经济学与现代生成式人工智能相结合，以应对金融模拟中维数灾难和数据稀缺的挑战。通过利用资产收益率固有的低维因子结构，我们使用时变正交投影分解了扩散模型的关键组成部分——得分函数，并将此分解融入到神经网络架构的设计中。我们推导了严格的统计保证，为得分估计和生成分布分别建立了非渐近误差界，分别为O(d^{5/2} n^{-2/(k+5)})和O(d^{5/4} n^{-1/2(k+5)})。这些误差界主要由内在因子维度k而非资产数量d驱动，超越了经典非参数统计文献中依赖于维度的限制，使得该框架能够适用于包含数千种资产的市场。数值研究证实了在小数据机制下潜在子空间恢复的优越性能。实证分析展示了我们的框架在构建均值-方差最优投资组合和因子投资组合方面的经济意义。这项工作首次在理论上将因子结构与扩散模型相结合，为有限数据下的高维金融模拟提供了一种原则性方法。我们的代码可在 https://github.com/xymmmm00/diffusion_factor_model 获取。