Breaking the Dimensional Barrier: Dynamic Portfolio Choice with Parameter Uncertainty via Pontryagin Projection

Abstract

We study continuous-time CRRA portfolio choice in diffusion markets with estimated and hence uncertain coefficients. Nature draws a latent parameter $\theta \sim q$ at time $0$ and keeps it fixed; the investor never observes $\theta$ and must commit to a single $\theta$-blind policy maximizing an ex-ante objective, treating $q$ as a decision-time input. We propose a simulation-only two-stage solver.Stage 1 (DPO) performs BPTT-based stochastic gradient ascent through an Euler simulator while sampling $\theta$ only inside the simulator. Stage 2 (Pontryagin projection) aggregates costate blocks across $\theta \sim q$ and enforces the $q$-aggregated stationarity condition within the deployable class; the resulting correction can be amortized via interactive distillation. We refer to the full Stage 1 + Stage 2 pipeline as PG-DPO.We prove a uniform conditional BPTT-PMP correspondence and a residual-based policy-gap bound with explicit discretization and Monte Carlo error terms. Experiments on high-dimensional Gaussian drift-uncertainty and factor-driven benchmarks show that projection stabilizes learning and accurately recovers analytic decision-time references, while a model-free PPO baseline remains far from the targets.