Abstract: Due to the non-separability of the variance operator in sense of dynamic programming, the optimal investment policy of the multi-period mean-variance model in stochastic markets doesn't satisfy the time consistency corresponding to Bellman's optimality principle. To overcome this shortcoming, we first propose a new time consistency in stochastic markets, which is weaker than Bellman's optimality principle, and prove that the optimal investment policy satisfies the weak time consistency at any intermediate period as long as the investor's wealth is no more than a specific threshold. When the investor's wealth exceeds the given threshold, the weak time consistency no longer holds, and the investor becomes irrational since he minimizes mean and variance of the terminal wealth at the same time. To avoid this, by relaxing the self-financing constraint, we revise the optimal investment policy so that the revised policy can make the investor rational and can attain the same mean and variance of the terminal wealth as those of the optimal investment policy. It is shown that, in this process, a positive cash flow should be taken out of the market. These conclusions show that the revised policy is more efficient than the optimal investment policy. Our results extend existing conclusions for the multi-period mean-variance model in deterministic markets and the understanding about time consistency.

Key words: time inconsistency, mean-variance, stochastic markets, Bellman's optimality principle, policy revision