Multi-time state mean-variance model in continuous time

Abstract: In the continuous time mean-variance model, we want to minimize the variance (risk) of the investment portfolio with a given mean at terminal time. However, the investor can stop the investment plan at any time before the terminal time. To solve this kind of problem, we consider to minimize the variances of the investment portfolio at multi-time state. The advantage of this multi-time state mean-variance model is that we can minimize the risk of the investment portfolio along the investment period. To obtain the optimal strategy of the multi-time state mean-variance model, we introduce a sequence of Riccati equations which are connected by a jump boundary condition. Based on this sequence Riccati equations, we establish the relationship between the means and variances of this multi-time state mean-variance model. Furthermore, we use an example to verify that minimizing the variances of the multi-time state can affect the average of Maximum-Drawdown of the investment portfolio.