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

TL;DR

This paper tackles dynamic portfolio choice in diffusion markets when key inputs are estimated and latent, drawn from an exogenous law q. It introduces a simulation-based two-stage solver that combines PG-DPO with a q-aggregated Pontryagin projection to produce deployable, θ-blind policies and proves a uniform BPTT-PMP correspondence along with a residual-based policy-gap bound. The projection stage stabilizes learning, enables recovery of analytic references in high-dimensional drift settings, and, in factor-driven markets, recovers intertemporal hedging demands; a PPO baseline struggles under the same deployment constraints. Practically, the approach scales to many assets and provides a principled way to incorporate parameter uncertainty without belief-state augmentation, with interactive distillation enabling fast deployment. Extensions to time-varying uncertainty and market frictions are outlined as future directions.

Abstract

We study continuous-time CRRA portfolio choice in diffusion markets with uncertain estimated coefficients. Nature draws a latent parameter from a given distribution and keeps it fixed; the investor cannot observe this parameter and must commit to a parameter-blind policy maximizing an ex-ante objective. We treat the uncertainty distribution as an inference-agnostic sampling input. We develop a simulation-only two-stage solver. Stage 1 extends Pontryagin-Guided Direct Policy Optimization (PG-DPO) by sampling parameters internally and computing gradients via backpropagation through time. Stage 2 performs an aggregated Pontryagin projection: it aggregates costates across the parameter distribution to enforce a deployable stationarity condition, yielding a structured correction amortized via interactive distillation. We prove a uniform conditional BPTT-PMP correspondence and a residual-based policy-gap bound with explicit error terms. Experiments on high-dimensional Gaussian drift and factor-driven benchmarks show that projection stabilizes learning and accurately recovers analytic references, while a model-free PPO baseline remains far from the targets.

Figures (7)

• Figure 1: Two-stage pipeline of Section \ref{['sec:pgdpo-uncertainty']}. A detailed schematic (including the coupling mechanisms and implementation notes) is provided in Appendix \ref{['app:coupling']}, Fig. \ref{['fig:sec3-pipeline-sideways']}.
• Figure 2: Decision-time Euclidean RMSE at $t=0$ versus dimension $d$ (log scale), summarized by a tail median over the late-training window. Concretely, for each condition we take the median RMSE over the last evaluation snapshots recorded during training. Rows: uncertainty magnitude $s\in\{10^{-3},10^{-2},10^{-1}\}$. Columns: aligned vs. misaligned geometry. Curves compare Stage 1 (deployable) and Stage 2 (post-hoc projection), with and without interactive distillation. Solid vs. dashed lines correspond to MC base ($100\cdot d$) vs. high ($400\cdot d$) trajectory budgets.
• Figure 3: Pathwise sanity check at $d=100$ under common random numbers. Black: analytic Gaussian constant-fraction benchmark evaluated with the remaining horizon $\tau=T-t$ (fixed $q$). Blue: Stage 1 PG--DPO. Orange: on-the-fly Stage 2 P--PGDPO teacher (computed via Monte Carlo projection at visited states). Top/bottom: aligned/misaligned.
• Figure 4: Stage 2 stationarity residual (q50). All panels report tail medians over epochs 9500--10000 (final six evaluation snapshots). Layout matches Figure \ref{['fig:crra_scaling_rmse']}: rows correspond to $s\in\{10^{-3},10^{-2},10^{-1}\}$ and columns correspond to aligned vs. misaligned uncertainty. Solid vs. dashed lines are MC base ($100\cdot d$) vs. high ($400\cdot d$). We plot the median (q50) of the estimated Hamiltonian first-order condition residual norm at the query states. Larger residual indicates the warm policy is farther from stationarity, implying a larger correction is required in the residual-form projection. Growth of this residual with $d$ (especially under misalignment) supports the mechanism that projection becomes more sensitive in high dimension due to larger correction magnitudes and amplified mixed-moment noise.
• Figure 5: Stage 2 denominator magnitude (q50). All panels report tail medians over epochs 9500--10000 (final six evaluation snapshots). Layout matches Figure \ref{['fig:crra_scaling_rmse']}: rows correspond to $s\in\{10^{-3},10^{-2},10^{-1}\}$ and columns correspond to aligned vs. misaligned uncertainty. Solid vs. dashed lines are MC base ($100\cdot d$) vs. high ($400\cdot d$). We plot a typical (q50) magnitude of the projection denominator/curvature term used in the residual-form update. Values bounded away from zero indicate that projection is not operating in a near-singular regime at typical quantiles. This helps rule out "catastrophic inversion" as the primary driver of degradation; instead, residual growth and curvature mismatch (Fig. \ref{['fig:app_stage2_kappa_med']}) provide a more consistent explanation in misaligned/high-$d$ regimes.

Theorems & Definitions (14)

• Remark 1: Latent parameter, observability, and admissible controls
• Theorem 1: $q$-aggregated first-order condition under latent $\theta$ (deployable $\theta$-blind stationarity)
• proof : Proof sketch
• Remark 2: Relation to belief-state/learning formulations
• Theorem 2: BPTT--PMP correspondence (conditional on $\theta$, including second-adjoint blocks; uniform on compacts)
• proof
• Theorem 3: Residual-based ex--ante $\theta$-blind policy-gap bound for P--PGDPO (mixed-moment, deployable, slab-wise local)
• proof
• Proposition 1: Stability of the projection map $(A,G)\mapsto -A^{-1}G$
• proof