Distributional Portfolio Optimization (DPO): A Unified Framework for Distributions over Weights, Returns, and Parameters

Classical portfolio optimization treats expected returns, covariances, and allocations as deterministic. Modern practice replaces at least one by a distribution: a posterior over parameters, a law of future returns, a stochastic allocation policy, or a distributional-robustness set. We call distributional portfolio optimization (DPO) the unified framework in which weights, returns, and parameters are all modeled as probability measures, organized around the joint coupling Gamma_theta(dw,dr) and its marginal triple (W,R,P). The contribution is synthetic and structural: we organize Bayesian, robust, chance-constrained, stochastic-allocation, and distributional reinforcement-learning portfolio methods through this coupling and prove boundary results connecting them, including a portfolio specialization of Wasserstein-CVaR duality, a static no-randomization theorem, a Bayesian credible-radius calibration of Wasserstein DRO, a Gaussian-isotropic second-order conservatism bound, a conditional two-sided rate W_1 = Theta(n^{-(1+alpha)/2}) governed by the local boundary Holder exponent alpha in [0,1], and a risk-shifted distributional Bellman contraction. A controlled experiment shows that across factor models at K in {10,25,50}, the credible-radius rule lands within 3-7 bp of the oracle out-of-sample tail risk and beats a 24-month validation-tuned radius while spending no validation data. On a K=25 DJIA backtest, equal-weight, no-view Black-Litterman, and Ledoit-Wolf shrinkage attain higher Sharpe than every distributional method; the operational claim is therefore confined to calibration-without-validation and turnover, not raw-return dominance.