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

Abstract: We study continuous-time portfolio choice in diffusion markets with parameter $θ\in Θ$ and uncertainty law $q(dθ)$. Nature draws latent $θ\sim q$ at time 0; the investor cannot observe it and must deploy a single $θ$-blind feedback policy maximizing an ex-ante CRRA objective averaged over diffusion noise and $θ$. Our methods access $q$ only by sampling and assume no parametric form. We extend Pontryagin-Guided Direct Policy Optimization (PG-DPO) by sampling $θ$ inside the simulator and computing discrete-time gradients via backpropagation through time (BPTT), and we propose projected PG-DPO (P-PGDPO) that projects costate estimates to satisfy the $q$-aggregated Pontryagin first-order condition, yielding a deployable rule. We prove a BPTT-PMP correspondence uniform on compacts and a residual-based $θ$-blind policy-gap bound under local stability with explicit discretization/Monte Carlo errors; experiments show projection-driven stability and accurate decision-time benchmark recovery in high dimensions.