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

Anthime Valin

Keywords: Optimal execution, Optimal liquidation, Stochastic control, Hamilton-Jacobi-Bellman (HJB) equations, Physics-informed neural networks (PINNs), Multi-trajectory training, Hard-zero terminal inventory constraint, Market impact, Algorithmic trading

TL;DR: We propose a multi-trajectory PINN to solve optimal execution HJB with a hard-zero terminal inventory constraint, improving zero-inventory enforcement with reduced error along optimal paths, matching risk-neutral TWAP and adapting with risk aversion.

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 $X_T$ via backpropagation-through-time, directly enforcing $X_T=0$. 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.

Submission Number: 70