2025/09/14 by Paolo Leonetti, Leonetti, Paolo
Economics, Econometrics and Finance · #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2509.11432
Given an integer a≥ 1, a function f: ℝ→ ℝ is said to be a-subadditive if f(ax+y) ≤ af(x)+f(y) for all x,y ∈ ℝ. Of course, 1-subadditive functions (which correspond to ordinary subadditive functions) are 2-subadditive. % and 3-subadditive. Answering a question of Matkowski, we show that there exists a continuous function f satisfying f(0)=0 which is 2-subadditive but not 1-subadditive. In addition, the same example is not 3-subadditive, which shows that the sequence of families of continuous a-subadditive functions passing through the origin is not increasing with respect to a. The construction relies on a perturbation of a given subadditive function with an even Gaussian ring, which will destroy the original subadditivity while keeping the weaker property. Lastly, given a positive rational cone H⊆ (0,∞) which is not finitely generated, we prove that there exists a subadditive bijection f:H→ H such that \liminfx→ 0f(x)=0 and \limsupx→ 0f(x)=1. This is related an open question of Matkowski and Świ\k atkowski in [Proc. Amer. Math. Soc. 119 (1993), 187--197].