We investigate in this paper dynamic mean-downside risk portfolio optimization problems in continuous-time, where the downside risk measures can be either the lower-partial moments (LPM) or the conditional value-at-risk (CVaR). Our contributions are twofold, both building up tractable formulations and deriving corresponding analytical solutions. By imposing a limit funding level on the terminal wealth, we conquer the ill-posedness exhibited in a class of mean-downside risk portfolio models. For a general market setting, we prove the existence and uniqueness of the Lagrangian multiplies, which is a key step in applying the martingale approach, and establish a theoretical foundation for developing efficient numerical solution approaches. Moreover, for situations where the opportunity set of the market setting is deterministic, we derive analytical portfolio policies for both dynamic mean-LPM and mean-CVaR formulations.