Numerical Methods for Random Parameter Optimal Control and the Optimal Control of Stochastic Differential Equations

This thesis considers the investigation and development of numerical methods for optimal control problems that are influenced by stochastic phenomena of various type. The first part treats tasks characterized by random parameters, while in the subsequent second part time-dependent stochastic processes are the basis of the dynamics describing the analyzed systems. In each case the investigations aim to transform the original problem into one that can be tackled by existing (direct) methods of deterministic optimal control - here we prefer Bock's direct multiple shooting approach. In the context of this transformation, in the first part approaches from stochastic programming as well as robust and probabilistic optimization are used. Regarding a specific application from mathematical economics, which considers pricing conspicuous consumption products in periods of recession, new numerical procedures are developed and analyzed with due regard to those techniques - in particular, a scenario tree approach, approximations of robust worst-case settings, and financial tools as the Value at Risk and Conditional Value at Risk. Furthermore, necessary reformulations of the resulting optimal control problems, in particular for Value at Risk and Conditional Value at Risk, as well as the discussion and interpretation of results determined depending on an uncertain recession duration, an uncertain recession strength, and control delays are in focus. The gained economic insight can be seen as an important step in the direction of a better understanding of real-world pricing strategies. In the second part of the thesis, based on the Wiener chaos expansion of a stochastic process and on Malliavin calculus, a system of coupled ordinary differential equations is developed that completely characterizes the stochastic differential equation describing the dynamics of the process. As in general this system includes infinitely many equations, a rigorous error estimation depending on the order of the chaos decomposition is proven in order to guarantee the numerical applicability. To transfer the generic procedure of the chaos expansion to stochastic optimal control problems, a method to preserve the feedback character of the occurring control process is shown. This allows the derivation of a novel direct method to solve finite-horizon stochastic optimal control problems. The appropriability and accuracy of this methodology are demonstrated by treating several problem instances numerically. Finally, the economic application of the first part is revisited under the viewpoint of dealing with a time-dependent recession strength, i.e., a stochastic process. In particular, those applications illustrate that the existing methods of deterministic optimal control can be extended to problems including stochastic differential equations

On the Duality of Optimal Control Problems with Stochastic Differential Equations

The main achievement of this work is the development of a duality theory for optimal control problems with stochastic differential equations. Incipient with the Hamilton-Jacobi-Bellman equation we established a dual problem to a given stochastic control problem and were also able to generalise the assembled theory

Numerical solution of a conspicuous consumption model with constant control delay☆

We derive optimal pricing strategies for conspicuous consumption products in periods of recession. To that end, we formulate and investigate a two-stage economic optimal control problem that takes uncertainty of the recession period length and delay effects of the pricing strategy into account