Near-optimal mean-variance controls under two-time-scale formulations and applications

Although the mean-variance control was initially formulated for financial portfolio management problems in which one wants to maximize the expected return and control the risk, our motivations stem from highway vehicle platoon controls that aim to maximize highway utility while ensuring zero accident. This paper develops near-optimal mean-variance controls of switching diffusion systems. To reduce the computational complexity, with motivations from earlier work on singularly perturbed Markovian systems [Sethi and Zhang, Hierarchical Decision Making in Stochastic Manufacturing Systems, Birkhäuser, Boston, MA, 1994; Yin and Zhang, Continuous-Time Markov Chains and Applications: A Singular Pertubation Approach, Springer-Verlag, New York, 1998 and Yin et al., Ann. Appl. Probab. 10 (2000), pp. 549-572], we use a two-time-scale formulation to treat the underlying system, which is represented by the use of a small parameter. As the small parameter goes to 0, we obtain a limit problem. Using the limit problem as a guide, we construct controls for the original problem, and show that the control so constructed is nearly optimal.