

Portfolio Selection Optimization Models and Solution ApproachAbstractPortfolio optimization is a very important area for longterm 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 meanvariance 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.
Optimal Dynamic Pricing Policy for M/M/K QueueAbstractIn this report, I analyze the problem of how to maximize the longrun 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 continuoustime Markov chain problem. By using uniformization method, we can transfer it to a discreteevent 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 longrun reward. In addition, I showed that the optimal policy is a nondecreasing function of the number of customers in the system. Efficient solution approaches to solve this problem are value iteration method and policy iteration method.
Modeling the Effects of Bivalirundin in Cardiac Surgical Patients（master project）AbstractBivalirudin is infused as a “blood thinner” in patients who have, or are suspected of having, blood clots or risk of blood clotting and who have a contraindication to heparin (a common “blood thinner”). It is infused continuously, and is eliminated via the kidney and by plasma proteasemetabolism. It affects the coagulation parameters PTT an INR in a dosedependent fashion. It is used more frequently in our ICU now but residents adjusting the infusion have little experience, resulting in over or underdosing. Ideally, it would prompt the user for the desired PTT range and would recommend a bivalirudin dosage. It would take the patient’s previous response into account. We aim to create an accurate model transforming the bivalrudin infusion rate to the output PTT and INR. In the next step, I will design a dosing calculator which means the model inverted.

