The aim of this thesis is to extend the current research on portfolio investment and asset pricing under uncertainty with a special focus on partial myopia and downside risk aversion. To address this goal, the thesis first proposes the use of partial myopia as an alternative approach to dynamic programming for solving a multi-period investment problem with background risks. I provide numerical examples to show that the partial myopia approach could lead to the same optimal investment decision as the dynamic programming method, even in the presence of background risks. Next, the thesis explores the drawbacks of the five existing downside risk aversion measures, and proposes a new local measure. While the proposed measure is limited by its local property, our numerical examples show that it gives the right preference ordering while the other five measures provide inconsistent signals. Finally, the thesis investigates the relationship between downside risk aversion and option pricing by analysing the elasticity of the pricing kernel. We conclude that decreasing absolute risk aversion and increasing downside risk aversion will increase the option price given the forward price. However, the two risk measures are not independent, and they interact to affect option price jointly.