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

David Reinhardt · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.14029 · cited by 0 works (OpenAlex) · open-access PDF

French

Markowitz spécial (SM) régularise les rendements et la covariance conjointement, par rapport à un état de référence (mu_ref, Sigma_ref). Chaque eigendirection de l'opérateur relatif blanchi porte un potentiel spectral signé Phi_k, avec le facteur de persistance psi_k = exp(-Phi_k) > 0. Les potentiels positifs atténuent les déviations empiriques de la géométrie de référence, le potentiel zéro les conserve et les potentiels négatifs les amplifient. Le facteur de persistance psi_k gouverne à la fois le signal de retour et la déviation de covariance: la déviation régularisée de la référence est psi_k fois la déviation empirique. La coordinate

Machine translation (Meta M2M100); the English original follows.

Abstract

Special Markowitz (SM) regularises returns and covariance jointly, relative to a reference state (mu_ref, Sigma_ref). Each eigendirection of the whitened relative operator carries a signed spectral potential Phi_k, with persistence factor psi_k = exp(-Phi_k) > 0. Positive potentials attenuate empirical deviations from the reference geometry, zero potential preserves them, and negative potentials amplify them. The persistence factor psi_k governs both the return signal and the covariance deviation: the regularised deviation from the reference is psi_k times the empirical deviation. The logarithmic potential coordinate is characterised by a multiplicative composition law on the multiplicative group of positive real numbers; the Stein loss is characterised as the unique free-energy density (within a natural class) compatible with the resulting coupling. The SM pressure functional is additive across modes the defining property of Special Markowitz.

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Topics: Multiplicative function, Covariance, Mathematics, Covariance operator, Logarithm, Applied mathematics

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