Pontryagin-Guided Deep Learning for Large-Scale Constrained Dynamic Portfolio Choice

We present a Pontryagin-Guided Direct Policy Optimization (PG-DPO) method for constrained dynamic portfolio choice - incorporating consumption and multi-asset investment - that scales to thousands of risky assets. By combining neural-network controls with Pontryagin's Maximum Principle (PMP), it circumvents the curse of dimensionality that renders dynamic programming (DP) grids intractable beyond a handful of assets. Unlike value-based PDE or BSDE approaches, PG-DPO enforces PMP conditions at each gradient step, naturally accommodating no-short-selling or borrowing constraints and optional consumption bounds. A "one-shot" variant rapidly computes Pontryagin-optimal controls after a brief warm-up, leading to substantially higher accuracy than naive baselines. On modern GPUs, near-optimal solutions often emerge within just one or two minutes of training. Numerical experiments confirm that, for up to 1,000 assets, PG-DPO accurately recovers the known closed-form solution in the unconstrained case and remains tractable under constraints -- far exceeding the longstanding DP-based limit of around seven assets.