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

TL;DR

This paper introduces a Multi-Trajectory PINN approach for solving HJB equations with hard-zero terminal inventory constraints, improving stability and accuracy in optimal trade execution models on synthetic and real market data.

Contribution

The paper proposes a novel MT-PINN method with trajectory loss and terminal penalty propagation, effectively enforcing zero terminal inventory in HJB-based optimal execution models.

Findings

MT-PINN closely matches closed-form solutions on synthetic data.

Achieves tighter terminal inventory control and lower errors.

Matches TWAP and reduces exposure on SPY data.

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.