  Instead of controlling "symmetric" risks measured by central moments of
investment return or terminal wealth, more and more portfolio models have
shifted their focus to manage "asymmetric" downside risks that the investment
return is below certain threshold. Among the existing downside risk measures,
the lower-partial moments (LPM) and conditional value-at-risk (CVaR) are
probably most promising. In this paper we investigate the dynamic mean-LPM and
mean-CVaR portfolio optimization problems in continuous-time, while the current
literature has only witnessed their static versions. Our contributions are
two-fold, in 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 the class of mean-downside risk
portfolio models. The limit funding level not only enables us to solve both
dynamic mean-LPM and mean-CVaR portfolio optimization problems, but also offers
a flexibility to tame the aggressiveness of the portfolio policies generated
from such mean - downside risk models. More specifically, 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.
