Mirror Descent Algorithms for Risk Budgeting Portfolios

This paper introduces and examines numerical approximation schemes for computing risk budgeting portfolios associated to positive homogeneous and subadditive risk measures. We employ mirror descent algorithms to determine the optimal risk budgeting weights in both deterministic and stochastic settings, establishing convergence along with an explicit nonasymptotic quantitative rate for the averaged algorithm. A comprehensive numerical analysis follows, illustrating our theoretical findings across various risk measures—including standard deviation, expected shortfall, deviation measures, and variantiles—and comparing the performance with that of the standard stochastic gradient descent method recently proposed in the literature. Funding: N. Frikha’s work was supported by the Institut Europlace de Finance.