从单对金融轨迹中恢复随机微分方程，实现预测与数据生成
Data-driven Feynman-Kac Discovery with Applications to Prediction and Data Generation
- 基于风险中性测度的随机SINDy方法，无需遍历性假设
- 仅需一对股价与期权路径即可恢复后向随机微分方程
- 可生成符合概率规律的新合成路径，适用于金融建模
本文提出一种全新的数据驱动框架，用于发现费曼-卡茨公式背后的概率规律。我们首次在风险中性概率测度下构建了随机SINDy方法，仅需一对股票与期权轨迹，即可恢复后向随机微分方程（BSDE）。与以往需遍历性假设的方法不同，该框架摆脱了对平稳性的要求，从而可在有限金融时间序列数据下实现BSDE识别。利用该算法，不仅能进行前瞻性预测，还可生成与底层概率规律一致的新合成数据路径。
原文摘要 · Abstract (English)
In this paper, we propose a novel data-driven framework for discovering probabilistic laws underlying the Feynman-Kac formula. Specifically, we introduce the first stochastic SINDy method formulated under the risk-neutral probability measure to recover the backward stochastic differential equation (BSDE) from a single pair of stock and option trajectories. Unlike existing approaches to identifying stochastic differential equations-which typically require ergodicity-our framework leverages the risk-neutral measure, thereby eliminating the ergodicity assumption and enabling BSDE recovery from limited financial time series data. Using this algorithm, we are able not only to make forward-looking predictions but also to generate new synthetic data paths consistent with the underlying probabilistic law.
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