points. The control with different initial starts was put into the program. The final result of the comparison will be the optimum of the control problem. The computing results of a particular oscillator problem are represented in graphs and tables in Section 3.6.

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Chapter 3

A FINANCIAL OSCILLATOR MODEL

1. Introduction

The financial sector is very volatile, more volatile than the business cycle instabilities of the whole economy. Modeling the oscillatory dynamics of the financial sector is well developed. This type of dynamic financial system can also be modeled as an oscillatory optimal control model with dampening where a control law is specified in the model to make the financial system stable (see Sengupta and Fanchon [80, 1997]). The objective of this type of model involves the minimization of deviations of the system variables from their desired path. Such an oscillatory financial model is developed and computed in the chapter. These type of oscillatory financial models are useful in financial decision making since general optimal control models may not provide stabilization policies and the damped oscillator models provide an effective stabilization mechanism.

In Section 2.5, a financial optimal control model, which has one state function, one control function, and the control taking two constant values sequentially, was discussed. The computational algorithms were also developed and applied to this problem (For a general discussion of this approach see also Chen and Craven [10, 2002], as most of the materials in this chapter are adapted in this paper. However, while Chen and Craven [10, 2002] provide a discussion of the approach, the present chapter shows the application of this approach to financial modeling only.) Substantial results were obtained to verify the algorithms. In this chapter a more complicated optimal control problem is introduced in Section 3.2 that can represent the oscillatory behavior of the financial sector. The financial system includes a second order differential equation. A transformation of time scale used to obtain the optimal switching times as well as a better gradient is described in Section 3.3. The transformations of