用神经粗糙微分方程学习路径签名，提升高维金融衍生品定价与投资组合优化精度。
Rough Path Signatures: Learning Neural RDEs for Portfolio Optimization
- 结合截断对数签名与神经粗糙微分方程，构建序列到路径的深度学习架构。
- 在d=200时CVaR(0.99)降至9.80%，显著优于基线模型（12.00%-13.10%）。
- 支持尾部风险控制与曲率估计，适合高维复杂金融建模场景。
我们针对高维、路径依赖的估值与控制问题，提出一种深度反向随机微分方程（BSDE）/二阶反向随机微分方程（2BSDE）求解器，其核心为截断对数签名与神经粗糙微分方程（RDE）的耦合。该架构融合随机分析与序列到路径学习：通过面向左尾风险的CVaR-倾斜终端目标，实现风险敏感控制；可选的2BSDE分支提供曲率估计，增强控制鲁棒性。在相同计算与参数预算下，方法在亚式期权、障碍期权定价及投资组合控制任务中全面提升精度、尾部拟合度与训练稳定性。当维度d=200时，CVaR(0.99)达9.80%，显著优于强基线（12.00%-13.10%），同时获得最低哈密顿-雅可比-贝尔曼（HJB）残差（0.011）与最小的Z与伽马（Gamma）均方误差（RMSE）。消融实验验证了截断深度、局部窗口与倾斜参数的互补增益，凸显序列到路径表示与2BSDE头的协同作用。整体表明随机分析与现代深度学习间存在双向互动：前者指导表征与目标设计，后者拓展了大规模可解金融模型的范围。
原文摘要 · Abstract (English)
We tackle high-dimensional, path-dependent valuation and control and introduce a deep BSDE/2BSDE solver that couples truncated log-signatures with a neural rough differential equation (RDE) backbone. The architecture aligns stochastic analysis with sequence-to-path learning: a CVaR-tilted terminal objective targets left-tail risk, while an optional second-order (2BSDE) head supplies curvature estimates for risk-sensitive control. Under matched compute and parameter budgets, the method improves accuracy, tail fidelity, and training stability across Asian and barrier option pricing and portfolio control: at d=200 it achieves CVaR(0.99)=9.80% versus 12.00-13.10% for strong baselines, attains the lowest HJB residual (0.011), and yields the lowest RMSEs for Z and Gamma. Ablations over truncation depth, local windows, and tilt parameters confirm complementary gains from the sequence-to-path representation and the 2BSDE head. Taken together, the results highlight a bidirectional dialogue between stochastic analysis and modern deep learning: stochastic tools inform representations and objectives, while sequence-to-path models expand the class of solvable financial models at scale.
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