用多轨迹方法解决交易执行中必须清仓的难题，让模型更稳定准确。
Multi-Trajectory Physics-Informed Neural Networks for HJB Equations with Hard-Zero Terminal Inventory: Optimal Execution on Synthetic & SPY Data
- 引入多轨迹损失和反向传播清仓惩罚，强制终盘持仓为零
- 在单资产模型上误差小，终盘持仓紧贴零点，接近理论解
- 适用于高频交易场景，尤其适合风险厌恶型策略
我们研究带有硬性清仓约束的最优交易执行问题，该问题由哈密顿-雅可比-贝尔曼（HJB）方程建模。传统物理信息神经网络（PINN）常无法充分施加此约束，导致控制不稳定。为此提出多轨迹物理信息神经网络（MT-PINN），通过基于滚动预测的轨迹损失，并利用时间反向传播将终端持仓惩罚传导至训练过程，直接强制终端持仓为零。采用轻量级λ课程学习机制，在状态从风险中性简化HJB扩展到风险厌恶型HJB时保持训练稳定。在Gatheral-Schied单资产模型上，MT-PINN与已知闭式解高度吻合，沿最优路径误差小，且终盘持仓紧密聚集于零。将其应用于SPY日内数据，风险中性下匹配TWAP表现；在高风险厌恶下，暴露更低，成本更具竞争力，尤其在下跌窗口中优势明显。
原文摘要 · Abstract (English)
We study optimal trade execution with a hard-zero terminal inventory constraint, modeled via Hamilton-Jacobi-Bellman (HJB) equations. Vanilla PINNs often under-enforce this constraint and produce unstable controls. We propose a Multi-Trajectory PINN (MT-PINN) that adds a rollout-based trajectory loss and propagates a terminal penalty on terminal inventory via backpropagation-through-time, directly enforcing zero terminal inventory. A lightweight lambda-curriculum is adopted to stabilize training as the state expands from a risk-neutral reduced HJB to a risk-averse HJB. On the Gatheral-Schied single-asset model, MT-PINN aligns closely with their derived closed-form solutions and concentrates terminal inventory tightly around zero while reducing errors along optimal paths. We apply MT-PINNs on SPY intraday data, matching TWAP when risk-neutral, and achieving lower exposure and competitive costs, especially in falling windows, for higher risk-aversion.
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