Financial optimal control modeling with a cost of changing control is the main topic in this book. Normally a cost is attached to a number of switching times then added to the objective function. The cost function becomes a new objective function to be treated. The chosen cost for such an optimal control problem for optimal investment planning in the stock market has been discussed in Chapter 2. The control in the problem is approximated by a step-function. The softwares constructed in this thesis compute the optimal switching times. Basically, the time period of the problem is divided into a certain number of subintervals to solve the differential equation and calculate the objective function as an integral. For the problem in Chapter 2, a greater number N of subdivisions will lead to a better fit to the target function When a cost is attached to the switching times, the integral decreases as N increases, but the cost of switching increases with N. Hence the total cost function reaches an optimum. There are some techniques required for this research, such as the approximation of the control, the time scaled transformation for using the SCOM package nqq function, the piece-wise linear transformation for the calculation of the differential equation and integrals, non-linear transformation for the large time period, and penalty term transformations for the constraints of the problems. The computed results are also compared with the theoretical results. A certain class of optimal control problems can be put into the formula introduced in this research. The computer software packages developed in this research can then solve these problems with very little or no change.

Financing oscillator problems can form another class of financial optimal control models. These kinds of problems have a great number of applications in the real world. In these problems, the dynamic system can be described by a second-order differential equation. It is required to convert the second-order differential equation to two equivalent pair of first-order differential equations, to enable the software to be used. In order to get more accuracy in solving the differential equation, a further subdivision of the time intervals is introduced. While the control takes several discrete values, the sequences of these values may follow more than one pattern, leading to a different computed minimum. The computational algorithms are designed to handle these problems. The obtained computing results give good comparisons of them.

The modeling exercises in this book show the potential for modeling dynamic financial systems by adopting bang-bang optimal control methods. The results of this modeling can provide improved understanding about the behavior of and the decision problems in dynamic financial systems. The roles of switching times in financial strategies and transaction costs of controls are useful for financial planning as well.

The algorithms constructed in this thesis are applied to an optimal corporate financing model, which has two state functions and two control functions. The

computation experiments with two patterns of controls were tried. Effective results were obtained which also agree with the analytical solution from the original work. Another computation with the SCOM package also agreed with the computational solution in this research and the analytical solution. Further research of trying different parameters sets will be an interesting exercise. The accuracy of the computational algorithms in Chapter 3 and Chapter 4 were verified by these experiments. The STV approach may be considered satisfactory in view of its computational efficiency and time, and the plausibility of results.

In the damped oscillatory financial model, many corporate finance models are concerned with the application of optimal control. Computational approaches to the determination of the optimal financing problems were applied in a financial model which was first introduced by Davis and Elzinga [22, 1970]. The model discusses investment allocation in order to determine the optimal proportion of the sources of finance which can maximize the value of the company. In particular, this model determines the proportion of its earnings that should be retained for internal investment and what proportion should be distributed to shareholders as dividends. The model also aims to choose the smart investment program that gives the owners the most benefits.

In most of the existing optimal corporate financial structure models, the optimal proportion of various sources of funds is determined. In a linear dynamic finance model, this proportion may change depending on the bang-bang character of the time path in the model. For an improved understanding of the behavior of the dynamic path of such models, it is essential to know the optimal switching times for changing the optimal proportion of different funds. No algorithm exists in the literature which can determine the optimal switching time for corporate funds. The present book has developed such a model for optimal corporate financing and switching timing.

In this research, the computational algorithms have been improved for solving bang-bang optimal control problems. Applications of the STV algorithm to finance have been made to show the potential and methods for developing dynamic optimization methods in finance. Further research is necessary to improve the state of the art in computing bang-bang optimal control in general and financial optimal control models in particular.

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