Norm constrained empirical portfolio optimization with stochastic dominance: Robust optimization non-asymptotics

Abstract This note provides an initial theoretical justification for how <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {\ell }_{p} -norm regularizations can control the non-asymptotic probability of false dominance (FD) classification in empirically optimal portfolios that satisfy empirical stochastic dominance constraints under an independent and identically distributed setting. The analysis employs a dual characterization of the norm-constrained problem as one of distributionally robust optimization, which enables the application of concentration inequalities involving the Wasserstein distance from the empirical distribution. This approach yields explicit upper bounds for the non-asymptotic FD probability, offering insights into the minimal sample size requirements necessary for maintaining this probability below a pre-specified significance level. The results provide a theoretical framework that outlines directions for future extensions to more general settings involving temporally dependent financial time series.