Optimal strategies and values for monotone and classical mean-variance preferences coincide when asset prices are continuous

We study classical and monotone mean-variance portfolio selection when asset prices are continuous under general trading constraints. Our main result is that optimal values and strategies for both objectives coincide in this setting. The only assumptions required are that asset prices are continuous and that not investing in the risky assets is an admissible strategy. We in particular do not assume existence of an optimal strategy.