Essays on optimal investment in mathematical finance - Zhong, Y (2012)
This thesis comprises four essays on optimal investment in mathematical finance. The first two are concerned with optimal stock buying/selling w.r.t. its global maximum (minimum). We first aim to determine an optimal selling time so as to minimize the expectation of the square error between the selling price and the global maximum. Then, we formulate four stock buying/selling problems by minimizing/maximizing the expectation of the ratio of the buying/selling price to the global maximum (minimum) price. These are optimal stopping problems that can be formulated as variational inequality problems. We solve them by a partial differential equation approach. The latter two essays are related to optimal investment with behavioral preferences. In Essay III, we consider optimal asset liquidation for an investor with an S-shaped utility. We characterize the value function in the sense of viscosity solution due to the nonsmoothness of the payoff function and show that the optimal liquidation strategies are consistent with the disposition effect. In Essay IV, we consider a mean-semi-variance portfolio selection problem with probability distortion. Using a dual argument, we change the decision variable to a quantile function. We then apply the Lagrange method to solve the problem. It turns out that the distorted mean-semi-variance problem only admits an optimal solution with some proper distortion functions.