Action-Space Entropy Regularization in Bayesian Markowitz

Battle Count: 110

Directly addresses the core problem of portfolio allocation under parameter uncertainty, which is fundamental to quantitative trading. The paper clarifies that entropy regularization (common in RL-based trading) does not provide exploration benefits when observations are exogenous to the control. The closed-form Bayesian Markowitz feedback with belief-dependent gains is directly implementable. The admissibility threshold P0T < 7/2 is a practical constraint for calibration. However, the negative orthogonality result tempers expectations about combining RL-style exploration with Bayesian parameter learning in continuous-time portfolio problems.

The final formulas are closed-form and computationally simple: the optimal mean control is linear in tracking error with a belief-dependent gain, and the variance is an exponential function of the posterior mean. However, deriving and verifying these results requires deep knowledge of stochastic calculus, HJB equations, Kalman-Bucy filtering, and the exponential substitution technique. Practical implementation requires careful monitoring of the admissibility condition P0T < 7/2 and real-time computation of posterior moments.

Fully theoretical paper with complete closed-form solutions and rigorous proofs. All derivations are self-contained with explicit formulas for the value function, optimal policy, and admissibility conditions. No empirical data or code is required for verification. The numerical illustration (Figure 1) uses simple parameter choices (P0=1, T=1, τ=1, σ=0.2) that can be trivially reproduced.