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arXiv · 2512.12708

Multi-Trajectory Physics-Informed Neural Networks for HJB Equations with Hard-Zero Terminal Inventory: Optimal Execution on Synthetic & SPY Data

Anthime Valin
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Abstract

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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Anthime Valin. 2025-12-14. Multi-Trajectory Physics-Informed Neural Networks for HJB Equations with Hard-Zero Terminal Inventory: Optimal Execution on Synthetic & SPY Data. https://arxiv.org/abs/2512.12708

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