Variance Minimization in Stochastic Systems

In portfolio selection, almost every investor would like to maximize his/her expected return while at the same time minimizing his/her risk that is often represented by a variance term. In dual control problems, the uncertainty, that can be characterized by a variance term, can be significantly reduced through active learning or probing. On the one hand, variance minimization problems are widely encountered in real-world applications. On the other hand, variance minimization is a notorious problem in optimization due to its associated properties of nonconvexity and nonseparability. The traditional optimal stochastic control theory concerns a sole objective of minimizing the expected value of a performance measure. There is a need to develop an efficient solution framework to deal with a general class of variance minimization problems. A novel solution approach is developed in this chapter to tackle variance minimization problems by exploring special features in variance minimization. Convexification and separation schemes are adopted to overcome the analytical and computational difficulties in variance minimization and to seek an analytical optimal feedback control law by a mathematically tractable setting.