In the first part of this thesis we develop an investment consumption model with convex transaction costs and optional stochastic returns for a finite time horizon. The model is a simplified approach for the investment in a portfolio of commodity related assets like real options or production facilities. In contrast to common models like [Awerbuch, Burger 2003] our model is a multi time step approach that optimizes the investment strategy rather then calculating a static imaginary optimal portfolio. On one hand, our numerical results are consistent with the well-known investment-consumption theory in the literature. On the other hand, this is the first in-depth numerical study of a case with convex transaction costs and optional returns. Our focus in the analyses is the form of the investment strategy and its behavior with respect to model parameters.
In the second part, an algorithm for solving continuous-time stochastic optimalcontrol problems is presented. The numerical scheme is based on the Stochastic Maximum Principle (SMP) as an alternative to the widely studied dynamic programming principle (DPP). By using the SMP, [Peng 1990] obtained a system of coupled forward-backward stochastic differential equations (FBSDE) with an external optimality condition. We extend the numerical scheme of [Delarue, Menozzi 2005] by a Newton-Raphson method to solve the FBSDE system and the optimality condition simultaneously. This is the first fully implemented algorithm for the solution of stochastic optimal control problems through the solution of the corresponding extended FBSDE system. We show that the key to its success and numerical advantage is the fact that it tracks the gradient of the value function and an adjusted Hessian backwards in time. The additional information is then exploited for the optimization.
|Supervisor:||Jäger, Prof. Dr. Willi|
|Date of thesis defense:||20 February 2013|
|Date Deposited:||27 Feb 2013 13:27|
|Faculties / Institutes:||The Faculty of Mathematics and Computer Science > Department of Applied Mathematics|
|Subjects:||500 Natural sciences and mathematics|
|Controlled Keywords:||Portfolio Selection, Rohstoffhandel, Finanzmathematik, Stochastische optimale Kontrolle, Numerische Mathematik|
|Uncontrolled Keywords:||Portfolio theory, commodity markets, real options, stochastic optimal control, forward-backward stochastic differential equations, stochastic maximum principle|