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is shown in Figure 2.3 because of the increased switching times.




Figure 2.3. Plot of n=6, forcing function ut= 1,0,1.0,1,0

When a very close fit between state xm and the given fitting function
is shown in Figure 2.4. The result proves a very good convergence of the
algorithms.




Figure 2.4. Plot of n=8, forcing function ut= 1,0,1,0,1,0,1,0

In Figure 2.5, although the approximation between xm and is still get-
ting better when the difference between the result of and
is not as big as between and The decreasing of the results of the
objective function slows down when is very large.
The STV Approach to Financial Optimal Control Models 33




Figure 2.5. Plot of n=10, forcing function ut= 1,0,1,0,1,0,1,0,1,0


The results of the objective function according to different numbers of the
time intervals are put in Table 2.1. The decreasing of the results of the
objective function follows the increases number of the time intervals When
is small, the result of the objective function will decrease very fast, however
this decrease will slow down when becomes big. A good illustration is shown
in Figure 2.6.




In Figure 2.6, the results of the objective function against the different num-
ber of the time intervals are shown. From the graph, we will find out that the
descent of the results of the objective function slows down when the number of
time intervals increases. The conclusion can be made that more jumps of the
control in the time period [0, 1] give a better association between the state
and fitting function (i.e. makes the financial system more stable along
34 OPTIMAL CONTROL MODELS IN FINANCE

its desired path) and leads the financial system to reach the best fit when
becomes infinity.




Figure 2.6. Plot of the values of the objective function to the number of the switching times


The optimal control problem in (2.11)-(2.15) is computed and the results are
shown in the above graphs. The number of time intervals is
set consequently. A better fit comes into being Figures 2.1, 2.2, 2.3, 2.4 and 2.5.
In Figure 2.6, it is shown that the result of the objective function J decreases
while the number of time intervals increases. Same results are also shown in
Table 2.1. Now a proper cost K is set and attached with the number of the time
intervals to the objective function J. The objective function (2.11) becomes:




Use the algorithms to solve this new optimal control problem with the same
constraints as in (2.12)-(2.15). Three different values of cost K are chosen for
The results of adding K * n to the objective function
are shown in Table 2.2, corresponding to
The STV Approach to Financial Optimal Control Models 35

From the results in Table 2.2, a conclusion can be made that only when
cost K takes a certain value, it will lead the cost function to a minimum in-
finite switching times. When K = 0.02, the results of the cost function are
increasing from to and the increases become faster. When
K = 0.0002, the results are decreasing, but this decrease begins to slow down
when increases. It is conjectured that when the number of the switching
times is getting very big, this decreases will stop at a certain value of
Meaningful results from K = 0.002 are given, and a minimum is obtained at
The following Figure 2.7 shows the results of the cost function against
the number of the time intervals while the cost K = 0.002 is attached to the
objective function. The bottom of the line in the figure is the minimum point




Figure 2.7. Plot of the cost function to the cost of switching control




This is an effective example of computational algorithms. All the results
confirm the accuracy of computational algorithms in Section 2.6. A hand cal-
culation of the part of the differential equation to exam the algorithms also
gives a proof of computational accuracy. Some other experiments with differ-
ent fitting functions will be discussed later in Chapter 5.
36 OPTIMAL CONTROL MODELS IN FINANCE

8. Financial Implications and Conclusion
The crucial aspect of the computational experiments undertaken here is that
switching times and and the cost of switching times have significant implica-
tions for optimal investment planning. As the graphical results showed in
Section 2.7, different numbers of switching times lead to different results for
the objective function of the financial model. While the number of switching
times increases, the value of the objective function decreases. That means a
better fit is obtained. When a cost of changing control at each switching time
is added to the original objective, the perspective changes. In the present case,
when the number of switching times increases, the value of the term, which
includes cost and the number of the switching, also increases. This increase
slows down the decreasing of the original objective function. When the number
of switching times increase to a certain value, the cost function stops decreas-
ing and instead increases. The intermediate point between the decreasing and
increasing is the optimal point.
The value of the cost of switching significantly influence the values of the
switching times and optimal control. Comparing with the result of the objective
function, if the cost is too big, the cost functional will increase all the time. It
can be explained that it costs too much to change a control at each time interval.
The financial system can never reach an optimal solution. But if the cost is too
small, it could not affect the cost function at all. Then the control will jump
infinitely in the time period to search an optimal solution, which is difficult
to realize in real computation. Therefore, the value of the cost of switching
time affects the optimal number of switching time. In terms of the value of the
optimal control, it is found that several jumps in the strategy of investment in
the stock market, shown by 1,0 values of investment in the stock market are
optimal. A higher cost for switching reduces the optimal value of switching
times compared to a situation when there is no cost for control switching.
An optimal investment strategy, therefore, should always be made on the
consideration of the cost of control switching to determine how often the in-
vestment strategy can be changed between 1 and 0.
In the next chapter, a financial oscillator model for optimal aggregative in-
vestment planning will be presented. Since the oscillator problem has a state
function, which is a second-order differential equation, the computational al-
gorithms require transforming the second-order differential equation into two
equivalent first-order differential equations. A new time scaled transformation
for better integral calculation will be introduced in Section 3.3. The corre-
sponding transformations of the state functions and the objective function will
also be contained in Section 3.3. The computational algorithms for the finan-
cial oscillator problem will be described in Section 3.4. The computer software
packages for these algorithms will be introduced in Appendix A.2. Differ-
ent patterns of the control might lead the computation to different minimum
The STV Approach to Financial Optimal Control Models 37

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 in-
volves the minimization of deviations of the system variables from their de-
sired path. Such an oscillatory financial model is developed and computed
in the chapter. These type of oscillatory financial models are useful in finan-
cial 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 func-
tion, one control function, and the control taking two constant values sequen-
tially, 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 in-
troduced in Section 3.2 that can represent the oscillatory behavior of the finan-
cial 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
40 OPTIMAL CONTROL MODELS IN FINANCE

the control, state, and the objective function are changed corresponding to the
new time subdivision. The computational methods for these kinds of optimal
control problems are constructed in Section 3.4. In this chapter, the control is
also approximated by a step-function. Additional analysis of the control with
different patterns leading to different minimums is discussed in Section 3.5.
In Section 3.6, two sets of graphical results with different control patterns are
presented.
In economics problems, the dynamic behavior may require a second order
differential equation and more than one control. In Blatt [2, 1976], some gen-
eralizations of multi-state functions and controls were discussed. But Blatt
did not give any explicit proof and explanation of these generalizations. The
computational algorithms in this chapter effectively solve the second order dif-
ferential equation. This technique is also applied to an optimal corporate fi-
nancing model later in Chapter 4.

2. Controlling a Damped Oscillator in a Financial Model
We consider an aggregate dynamic financial system described by a second-
order differential equation as follows:




This financial system second order differential equation can be transformed
into an equivalent pair of first-order differential equations:




We now consider an oscillator problem with the bang-bang optimal control
solution. A damped oscillator financial model for the above aggregate financial
system with forcing function and parameter T, is presented as follows
(see another version of this financial model in Section 5.3.1):
/indexforcing function



subject to:
A Financial Oscillator Model 41




where = stock prices, = the proportion of investment in stocks com-
pared to other form of financial investment.
Here is a given target function showing the desired path to be achieved
by the financial system. Suppose that is restricted by:




Since the dynamics are linear in if an optimum is reached, then bang-
bang control may be expected, with possibly a singular arc for a time interval.
However, singular arc control is not considered here.
The model (3.2) to (3.7) represents a financial decision making problem
where the cost of changing control is added to the objective function, shown as
follows:




Here, K is the cost of switching control, is the number of control jumps. The
modified model calculates the optimal allocation for stocks with an explicit
consideration of changing allocation shares. The cost function becomes the
new objective function to be computed.
The model (3.2) to (3.7) is similar to the model (2.11) to (2.15) and repre-
sent a financial decision making problem of choosing the optimal allocation
of finance in an economy for stocks. The forcing function is an additional
feature of this model compared to the model in (2.11) to (2.15). To make the
model realistic, it is necessary to incorporate the cost of changing control in
this model.

3. Oscillator Transformation of the Financial Model
In this section, a good transformation of time scale is introduced. The com-
puter package “nqq” is also used for solving the differential equations in this
chapter. However, the “nqq” package does a limited job, it only does one
Runge-Kutta step for each subinterval given to it. So when smaller subinter-
vals are needed for accuracy, they must be supplied by the “small subintervals”
42 OPTIMAL CONTROL MODELS IN FINANCE

construction. Although “nqq” package has this limitation, it has a great ben-
efit for solving the differential equation with a non-smooth right-hand side of
the differential equation which means there can be jumps at the end-points of
intervals. Since a time-optimal control problem is not considered in this re-
search, T is a constant. To make it easy, the time period of the optimal control
problem described in last section is simplified to be [0,1] (the transformation
of time scales when T is variable was given in Section 2.3).
First, the time horizon [0, 1] is divided into nb “big subintervals”, with end-
points:




Since the functions change quite rapidly within each big subinterval, further
subdivisions into “small intervals” are needed to get sufficient precision in
solving the differential equations. Then each subinterval
is subdivided into ns “small subintervals” with end-points:




The whole subdivision of the time interval [0, 1] is shown as follows:




The relationship between and can be represented as follows:




The control only jumps from one big interval to the next one. It takes values
as follows:




A scaled time is constructed for the computer package, corresponding to
the total number of subintervals which takes values:
A Financial Oscillator Model 43




Here,
Two vectors which represent two different levels of the subdivision of the
time period [0,1] um and sm are defined for mapping the control and calcu-
lating the integral (see the next section).
The relationship between the real time which is used in the calculation of
the integration and the scaled time is shown as follows:




where time and Here, the
control only jumps at the end-points of the big subintervals. Thus the control
has the same constant value in all the small subintervals which make up a big
subinterval. Another real time pt which represents the real switching times for
the control is defined as follows:




where, nb is the number of big subintervals. This pt is used in differential
equation solving with respect to control Correspondingly the control and
state with respect to are defined:




The two first-order differential equations of the state functions of the finan-
cial system in (3.2) and (3.3) are transformed into two equivalent first-order
equations:
44 OPTIMAL CONTROL MODELS IN FINANCE




The objective function of the financial optimal control model is then trans-
formed to a sum of integrals in successive subintervals:




where h = 1/nn, nn = nb*ns.

4. Computational Algorithm: The Steps
In this approach, time intervals of the financial model are defined at two
levels, with the planning period [0, T] first divided into nb big intervals, repre-
sented by vector um (see also Chen and Craven [10, 2002]):



then each big subinterval is subdivided into ns small subintervals. Thus [0, T]
is subdivided into nb * ns numbers of subintervals, represented by the vector
sm:




At each end-point of the big subintervals, the control jumps.

Algorithm 3.1 Main Program for the oscillator problem (see project2_l.m
in Appendix A.2)
Step 1. Initialization. Set a vector of parameters par = [the number of the state
components, number of control components, nb = the number of big subin-
tervals, ns = the number of small subintervals]; and get the total number
of the subintervals by calculating nn = nb * ns. Set the MATLAB “con-
str” function™s system parameters par(13) = 1 (here, 1 represents only one
equation constraint in the minimization problem), par(14) = the maximum
number of function evaluations; arbitrary starting lengths of the switching
time intervals Set the vectors of upper
A Financial Oscillator Model 45

bounds uu and the vector of lower bounds ul of um, thus
Also set the initial state xinit.
Step 2. Call the MATLAB “constr” function. In turn, “constr” calls the “Min-
imization Program” to calculate the minimization of the calling program
with respect to the optimal vector um.

Step 3. Input the optimal result um into “Minimization Program”, to obtain
the values of the objective function J(nn) (the last value of the integral)
and state vector xm (xm is a vector of all the values of the state functions
take at the grid-points of the switching time intervals).
Step 4. Attach a cost K to nn. Add K * nn to the objective function (3.2) and
calculate

Step 5. Set a bigger nn and new starting time intervals, go back to Step 2;
EXIT when the result of cost function J stops decreasing.
Algorithm 3.2: Minimization Program (see project2_2.m in Appendix A.2)

Step 1. Initialization. Input um, par and initial state xinit. Set the initial state
xm(1, :) = xinit, and nx = par(1), the number of the state components,
nu = par(2), the number of the control components, nb = par(3), the
number of big time intervals, ns = par(4), the number of small time inter-
vals, nn = par(3) * par(4), the number of total time intervals; initialize
the scaled time subinterval counter it = 1, hs = length of each equa-
tion subintervals. Choose the “Input function for second order differential

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