﻿ Qi Zhao Boston University
 QI ZHAO      Master of Science • ### Abstract

Portfolio optimization is a very important area for long-term investors. It is concerned with the problem of how to best diversify investment into different classes of assets (such as stock, bonds, real estate, and options) in order to meet liabilities and to maximize the expected profit, while minimize the unacceptable risk. Portfolio optimization problems are based on mean-variance models for returns and for risk neutral density estimation. I focus on the asset and liability management (ALM) model which involves the stochastic programming, dynamic property, and nonlinear programming model based on scenario tree. The most challenging work to solve this problem is that the for a realistic model description the size of scenario tree quickly reaches astronomical sizes. To solve this problem efficiently, there are two major methods: decomposition method and interior point method. I will analyze how to apply the Interior Point method (IPM) to solve the optimal solution. Finally, I use the Matlab to simulate the algorithm and solve a relatively small size problem modeled by some history data from the website.

• ### Abstract

In this report, I analyze the problem of how to maximize the long-run total discounted reward in a queuing system. The problem can be formulated as a Markov decision process. The control policy is increasing or decreasing the price which encourages or discourages the arrival rate of customers. We can treat the arrival process of customers as Poisson with arrival rate which is a decreasing function of the current price. The service times of the servers are independently exponentially distributed random variables with a fixed mean service rate. The total profits consist of the customer’s payment and holding cost per unit which is accumulated along the time. This is a continuous-time Markov chain problem. By using uniformization method, we can transfer it to a discrete-event Markov chain. When each event (customer arrival or service completion) happens, the manager should post a price until the next event happens. We can prove that there exists an optimal stationary policy for the infinite horizon to maximize the long-run reward. In addition, I showed that the optimal policy is a non-decreasing function of the number of customers in the system. Efficient solution approaches to solve this problem are value iteration method and policy iteration method.