Yihua Xu

• In this thesis, we mainly discuss the problem of parameter estimation and portfolio optimization with partial information in discrete-time. In the portfolio optimization problem, we specifically aim at maximizing the utility of terminal wealth. We focus on the logarithmic and power utility functions. We consider expert opinions as another observation in addition to stock returns to improve estimation of drift and volatility parameters at different times and for the purpose of asset optimization. In the first part, we assume that the drift term has a fixed distribution, and the volatility term is constant. We use the Kalman filter to combine the two types of observations. Moreover, we discuss how to transform this problem into a non-linear problem of Gaussian noise when the expert opinion is uniformly distributed. The generalized Kalman filter is used to estimate the parameters in this problem. In the second part, we assume that drift and volatility of asset returns are both driven by a Markov chain. We mainly use the change-of-measure technique to estimate various values required by the EM algorithm. In addition, we focus on different ways to combine the two observations, expert opinions and asset returns. First, we use the linear combination method. At the same time, we discuss how to use a logistic regression model to quantify expert opinions. Second, we consider that expert opinions follow a mixed Dirichlet distribution. Under this assumption, we use another probability measure to estimate the unnormalized filters, needed for the EM algorithm. In the third part, we assume that expert opinions follow a mixed Dirichlet distribution and focus on how we can obtain approximate optimal portfolio strategies in different observation settings. We claim the approximate strategies from the dynamic programming equations in different settings and analyze the dependence on the discretization step. Finally we compute different observation settings in a simulation study.