Before revision:  Consider two portfolios, one is a passive buy-and-hold portfolio representing the entire market , and the other assigns a portfolio vector for each possible value of the market weights and requires trading to maintain this assignment. The evolution of stocks is taken to be any jump process in discrete time, or a path of any continuous semimartingale in continuous time. We do not make any stochastic modeling assumptions on the evolutions. We provide necessary and sufficient conditions on the assignment on portfolios to guarantee that, for all such evolutions in a diverse and sufficiently volatile market, the actively traded portfolio outperforms the buy-and-hold portfolio in the long run .
After revision:  Consider two portfolios, one is a passive buy-and-hold portfolio representing the entire market , and the other assigns a portfolio vector for each possible value of the market weight. Suppose we perform self-financing trading using this portfolio function. We study the problem of characterizing functions such that the resulting portfolio will outperform the market portfolio in the long run .