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The two-slope online leasing problem is a natural generalization of the classic online leasing problem. Due to the fact that the two-slope online leasing problem of continuous-time has already been studied, we focus on the two-slope online leasing problem of discrete time in this study. Our discussion has taken the deterministic strategy and the randomized strategy into consideration. We demonstrated an optimal deterministic strategy, which could achieve competitive factor of 2 - [1 + (s - 1)a]/s. As for the randomized strategy, we proposed the strategy of risk balanced its absolute optimality through competitive analysis. Finally, analysis and discussions are carried out for the two types of strategies, mainly based on their competitive performances. It is found that, the two-slope analysis could improve the competitive ratio of the classic online leasing problems. In addition, by taking the discreteness into consideration rather than the continuity, the effectiveness of solving online leasing problem could be much improved. [ABSTRACT FROM AUTHOR]

The maximum drawdown and the lower-semi-variance are risk indicators that investors concern about during the actual investment process. This paper combines this two indicators into analytical framework and proposes an optimal portfolio strategy under rolling economic drawdown constraints with lower-semi-variance (REDP-LSV strategy). Under the constraint of the rolling economic maximum drawdown of risky asset's prices, we have studied the dynamical allocation of the risky assets in the portfolio and derived a analytical solution by using the lower-semi-variance instead of the variance of risky asset. The investment cases of one risky asset and two risky asset are both studied in this paper. Using data from simulated experiment and real risky assets of Chinese stock market, it turns out that the REDP-LSV strategy is able to control the maximum drawdown and improve the performance of investment. [ABSTRACT FROM AUTHOR]