Special Markowitz: Thermodynamic Formalism for the Joint Regularisation of Returns and Covariance

RESUMO

Special Markowitz regularises returns and covariance jointly, relative to a reference state (mu_ref, Sigma_ref). Each eigendirection of the whitened operator carries a spectral reliability potential Phi_k, derived from its estimation quality. Its exponential reliability weight exp(-Phi_k) governs both the fraction of return signal and deviation of covariance that survive regularisation. Reliable modes (low Phi_k) are preserved; unreliable modes (high Phi_k) relax to reference. The logarithmic coordinate is characterised by a multiplicative composition law; Stein loss is the unique free-energy density (within a natural class) compatible with the resulting coupling. SM pressure functional is additive across modes - defining property of Special Markowitz.