LINEBURG

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Appendix F
A Sample Test Problem

The following example has two controls and one state,

subs=cell(1,9);
subs={â€™t3xâ€™,â€™t3jâ€™,â€™t3fâ€™,â€™t3câ€™,>t3kâ€™,â€™t3lâ€™,â€™t3gâ€™,â€™t3aâ€™,â€™t3pâ€™};
par=[1, 2, 20, 0, 1, 0.25]; % Parameters
% nx, nu, nn, npa, grad, c

nnb=par(3); nx=par(1); nu=par(2);
xinit=[0.5];
% Initial value for the state

u0=zeros(nn,nu);
% Starting values for computing the control

ul=zeros(nn,nu);
% Lower bound (vector or matrix) for the control

uu=ones(nn,nu);
% Upper bound (vector or matrix) for the control

figure

Control=constr(â€™fqqâ€™,u0,[],ul,uu,â€™gqqâ€™,par,subs,xinit)
% Calls constr package

cqq(Control,par,subs,xinit) % Right hand side of differential
equation

function yy=t3x(t,it,z,yin,hs,um,xm,lm,ps)
192 OPTIMAL CONTROL MODELS IN FINANCE

yy(1)=z(1)*um(floor(it),1)+um(floor(it),2);
% Dependent variables on the right hand side
of a differential equation are coded as
z(1), z(2), etc.

function ff=t3j(t,it,z,yin,hs,um,xm,lm,ps)
% Integrand of objective function

c=ps(6)
ff(1)=(um(floor(it),1)-1)*li3(xm,hs,t);
ff(1)=ff(1)+c*um(floor(it),2);

function ff=t3f(xf,um,xm,ps)
% Endpoint term

ff(1)=0; % Control constraint

function gg=t3c(ii,hs,um,xm,lm,ps)
gg=um(ii,1) + um(ii,1) - 1;

function dg=t3k(ii,hs,um,xm,lm,ps)

dg=[eye(nn);eye(nn)];

function yy=t3g(t,hs,um,xm,lm,nn)

temp= 0.5*(lm(t,1)+lm(t+1,1));
t2=t/nn;
yy=[(1+temp)*li3(xm,hs,t2), 0.25+temp];

function yy=t3a(nn,xf,um,xm,ps)
%Boundary condition for adjoint equation (at t=1)

yy=0;

function yy=t3l(t,it,z,yin,hs,um,xm,lm,ps)
% Right hand side of adjoint equation

yy=-(1+z(1))*um(floor(it),1)+1;
References

[1] Ahmed, N.U. (1988). Elements of Finite-Dimensional Systems and Control Theory. Long-
man Scientific and Technical, Harlow, Essex, England.

[2] Blatt, John M. (1976). Optimal control with a cost of switching control. J. Austral. Math.
Soc., 19:316-332.

[3] Brekke, K.A. and Ã˜ksendal, B. (1991). A verification theorem for combined stochastic
control and impulse control. Stochastic Anal. Related Topics, 6:211-220.

[4] Brigham, E. and Houston, J. (2000) Fundamentals of Financial Management, 9th edn. The
Dryden Press, Harcourt Brace College Publishers, Orlando.

[5] Bryson, A.E.Jr. and Ho, Y.G. (1975). Applied Optimal Control. Halsted Press, New York.

[6] Cadenillas, A. and Zapatero, F. (2000). Classical and impulse stochastic control of the
exchange rate using interest rates and reserves. Math. Finance, 10:141-156.

[7] Campbell, J. Y., Lo, A. W. and MacKinlay, A. C. (1997). The Econometrics of Financial
Markets. Princeton University Press, Princeton, New Jersey.

[8] Cesari, L. (1983). Optimization - Theory and Applications. Springer-Verlag, New York.

[9] Chakravarty, S. (1969). Capital and Economic Development Planning. MIT Press, Cam-
bridge.

[10] Chen, P. and Craven, B.D. (2002). Computing switching times in bang-bang control.
Mimeo, University of Melbourne.

[11] Chen, P. and Islam, S.M.N. (2002). Optimal financing for corporations: optimal control
with switching time and computational experiments using CSTVA. Paper presented at the
Financial Modeling seminar, Victoria University of Technology, Melbourne, Australia.

[12] Clarke, C.W. (1976). Mathematical Bioeconomics: The Optimal Management of Renew-
able Resources. John Wiley, New York.

[13] Craven, B.D. (1978). Mathematical Programming and Control Theory. Chapman and
Hall, London.
194 OPTIMAL CONTROL MODELS IN FINANCE

[14] Craven, B.D. (1995). Control and Optimization. Chapman and Hall Mathematics, London.

[15] Craven, B.D., HAAS, K.De. and Wettenhall, J.M. (1998). Computing optimal control.
Dynamics of Continuous, Discrete and Impulsive Systems, pp. 601-615.

[16] Craven, B.D. (1999). Computing optimal control on MATLAB, Optimization Day.
Mimeo, University of Ballarat, Melbourne, Australia.

[17] Craven, B.D. (1999). Optimal control for an obstruction problem. Journal of Optimization
Theory and Applications, 100.

[18] Craven, B.D. and Islam, S.M.N. (2001). Computing optimal control on Matlab - The
SCOM package and economic growth models. Optimization and Related Topics, 61-70,

[19] Cuthbertson, K. (1997). Quantitative Financial Economics, Stocks, Bonds and Foreign
Exchange. John Wiley and Sons Ltd., West Sussex.

[20] Dadebo, S.A., McCauley, K.B. and McLellan, P.J. (1998). On the computation of optimal
singular and bang-bang controls. Optimal Control Applications and Methods, 19:287-297.

[21] Dasgupta, S. and Titman, S. (1998). Pricing strategy and financial policy. The Review of
Financial Studies, 11:705-737.

[22] Davis, B.E. and Elzinga, D. Jack. (1970). The solution of an optimal control problem in
financial modeling. Operations Research, 19:1419-1433.

[23] Davis, B.E. (1970). Investment and rate of return for the regulated firm. The Bell Journal
of Economics and Management Science, 1:245-270.

[24] Dolezal, J. (1981). On the solution of optimal control problems involving parameters and
general boundary conditions, Kybernetika, 17:71-81.

[25] Eatwell, J., Milgate, M. and Nuemann, P. (1989). The New Palgrave Dictionary of Fi-
nance. MacMillan Press, London.

[26] Elton, E. and Gruber, M. (1975). Finance as a Dynamic Process. Prentice-Hall, Engle-
wood Cliffs, NJ.

[27] Evans, L.C. and Friedman, A. (1979). Optimal stochastic switching and the Dirichlet prob-
lem for the bellman equation, Trans, Amer. Math. Soc., 253:365-389.

[28] Fletcher, R. (1980). Practical Methods of Optimization. Vol.1. Unconstrained Optimiza-
tion. Wiley-Interscience Publication, New York.

[29] Fleming, W.H. and Rishel, R.W. (1975). Deterministic and Stochastic Optimal Control.
Application of Mathematics, No. 1. Springer-Verlag, Berlin-New York.

[30] FrÃ¶berg, Carl-Erick. (1965). Introduction to Numerical Analysis. Addison-Wesley Pub-

[31] Garrad, W.L. and Jordan, J.M. (1977). Design of nonlinear automatic flight control sys-
tem. Automatica, 19:497-505.

[32] Giannessi, F. (1996). Private Communication. University of Pisa, Pisa, Italy.
REFERENCES 195

[33] Goh, G.J. and Teo, K.L. (1987). MISER, an Optimal Control Software. Department of
Industrial and Systems Engineering, National University of Singapore.

[34] Hasdorff.L. (1976). Gradient Optimization and Nonlinear Control. John Wiley and Sons,
New York.

[35] Isaacs, R. (1965). Differential Games. John Wiley, New York.

[36] Islam, S.M.N. (2001). Optimal Growth Economics: An Investigation of the Contempo-
rary Issues, and Sustainability Implications, Contributions to Economic Analysis. North
Holland Publishing, Amsterdam.

[37] Islam, S. M. N. and Craven B.D. (2001). Computation of non-Linear continuous optimal
growth models: Experiments with Optimal control algorithms and computer programs,
Economic Modeling. 18:551-586.

[38] Islam, S. M. N. and Craven B.D. (2002). Dynamic optimization models in finance: some
extensions to the framework, models, and computation. Research Monograph, CSES, Vic-
toria University, Melbourne.

[39] Islam, S. M. N. and Oh, K.B. (2003). Applied Financial Econometrics in E-Commerce,
Series Contributions to Economic Analysis. North Holland Publishing, Amsterdam.

[40] Jennings, L.S., Fisher, M.E., Teo, K.L. and Goh, G.J. (1991). MISER3.0: Solving optimal
control problems - an update. Advances in Engineering Software, 13.

[41] Jennings, L.S. and Teo, K.L. (1990). A numerical algorithm for constrained optimal con-
trol problem with applications to harvesting. Dynamics of Complex Interconnected Biolog-
ical Systems, 218-234.

[42] Jennings, L.S. and Teo, K.L. (1991). A computational algorithm for functional inequality
constrained optimization problems. Automatica, 26:371-376.

[43] Kaya, C.Y. and Noakes, J.L. (1994). A global control law with implications in time op-
timal control. In Proceedings of the 33rd IEEE Conference on Decision and Control, pp.
3823-3824, Orlando, Florida.

[44] Kaya, C.Y. and Noakes, J.L. (1996). Computations and time-optimal controls. Optimal
Control Applications and Methods, 17:171-185.

[45] Kaya, C.Y. and Noakes, J.L. (1997). Geodesics and an optimal control algorithm. In Pro-
ceedings of the 36th IEEE, pp. 4918-4919, San Diego, California.

[46] Kaya, C.Y. and Noakes, J.L. (1998). The leap-frog algorithm and optimal control: Back-
ground and demonstration. In Proceedings of International Conference on Optimization
Techniques and Applications (ICOTA â€™98), pp. 835-842, Perth, Australia.

[47] Kaya, C.Y. and Noakes, J.L. (1998). The leap-frog algorithm and optimal control: theoret-
ical aspects. In Proceedings of International Conference on Optimization Techniques and
Applications (ICOTA â€™98), pp. 843-850, Perth, Australia.

[48] Kendrick, D.A. and Taylor, L. (1971). Numerical Methods and Nonlinear Optimizing
Models for Economic Planning. Studies in Development Planning, Cambridge, Mass.
196 OPTIMAL CONTROL MODELS IN FINANCE

[49] Krouse, C.G. and Lee, W.Y. (1973). Optimal equity financing of the corporation. Journal
of Financial and Quantitative Analysis, 8:539-563.

[50] Lee, E.B. and Markus, L. (1967). Foundations of Optimal Control Theory. John Wiley and
Sons, New York.

[51] Lee, H.W.J., Teo, K.L., Rehbock, V. and Jennings, L.S. (1997). Control parameterization
enhancing technique for time optimal control problems. Dynamic Systems and Applica-
tions, 6:243-262.

[52] Leonard, D. and Long, N.V. (1992). Optimal Control Theory and Static Optimization in
Economics. Cambridge University Press, Melbourne.

[53] Li, X. and Yong, J. (1995). Optimal Control Theory for Infinite Dimensional Systems.
Birkauser Boston, Boston.

[54] Liu, L. and Teo, K.L. (2000). Computational Method For a Class of Optimal Switching
Control Problems. Progress in Optimization, 221-237.

[55] Matula, J. (1987). An extreme problem. Journal of Australian Mathematical Society,
28:376-392.

[56] Miele, J. (1975). Recent advances in gradient algorithms for optimal control problems.
Journal of Optimization Theory and Applications, 17:361-430.

[57] Miele, A. and Wang, T. (1986). Primal-dual properties of sequential gradient-restoration
algorithms for optimal control problems, Part 1, Basic Problem. Integral Methods in Sci-
ence and Engineering, 577-607.

[58] Miele, A. and Wang, T. (1986). Primal-dual properties of sequential gradient-restoration
algorithms for optimal control problems, Part 2, General Problem. Journal of Mathemati-
cal Analysis and Applications, 119:21-54.

[59] Modigliani, F. and Miller, M.H. (1958). The cost of capital, corporation finance, and the
theory of investment. American Economic Review, 48:261-97.

[60] Modigliani, F. and Miller, M.H. (1963). Corporation income taxes and the cost of capital.
American Economic Review, 53:433-43.

[61] Morellec, E. (2001). Asset liquidity, capital structure, and secured debt. Journal of Finan-
cial Economics, 61:173-206.

[62] Mundaca, G. and Ã˜ksendal, B. (1998). Optimal stochastic intervention control with ap-
plication to the exchange rate. J.Math. Economy, 29:225-243.

[63] Nerlove, M. and Arrow, K.J. (1962). Optimal advertising policy under dynamic condi-
tions. Economica, 39:129-142.

[64] Noakes, J. Lyle (1997). A global algorithm for geodesics. Journal of the Australian Math-
ematical Society, Series A, 37-50.

[65] Nordecai Avriel (1976). Nonlinear Programming Analysis and Methods. Technion-Israel
Institute of Technology, Haifa, Israel.
REFERENCES 197

[66] Noussair, E.S. (1977). On the existence of piecewise continuous optimal controls. Journal
of Australia Mathematical Society, Series B, 20:31-37.

[67] Oh, K.B. and Islam, S.M.N. (2001). Empirical Finance of E-Commerce: A Quantitative
Study of the Financial Issues of the Knowledge Economy. CSES Research Monograph,
Victoria University, Melbourne.

[68] Perthame, B. (1984). Continuous and impulse control of diffusion process in Non-
linear Anal, 8:1227-1239.

[69] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V. and Mishchenko, E.F. (1962). The
Mathematical Theory of Optimal Processes. John Wiley, New York.

[70] Powell, M.J.D. Private communication.

[71] Redbock, V., Teo, K.L. and Jennings, L.S. (1994). Suboptimal feedback control for a
class of nonlinear systems using spline interpolation. Discrete and Continuous Dynamical
Systems, 1:223-236.

[72] Redbock V., Teo, K.L., Jennings, L.S. and lEE, H.W.J. (1999). A survey of the control
parameterization and control parameterization enhancing methods for constrained optimal
control problems. Progress in Optimization, 247-275.

[73] Richard, R. Lumley and Mihall Zervos (2001). A model for investments in the material
resource industry with switching costs. Mathematics of Operations Research, 26:637-653.

[74] Rockafellar, R.T. (1974). Lagrange multiplier functions and duality in non-convex pro-
gramming. SIAM J. Control, 12:268-287.

[75] Sakama, Y. and Shindo, Y. (1980). On global convergence of an algorithm for optimal
control. IEEE Transaction Automatic Control, AC-25:1149-1153.

[76] Sakama, Y. (1981). On local convergence of an algorithm for optimal control. Numerical
Functional Analysis and Optimization, 3:301-319.

[77] Schwartz, A., Polak, E. and Chen Y. (1997). Recursive Integration Optimal Trajectory
Solver 95. A MATLAB TOOLBOX for solving Optimal control problems, Version 1.0,
Stanford University, California.

[78] Sethi, S.P. (1978). Optimal equity financing model of Krouse and Lee: corrections and
extensions. Journal of Financial and Quantitative Analysis, 13:487-505.

[79] Sethi, S.P. and Thompson, G.L. (2000). Optimal Control Theory. Kluwer Academic Pub-
lishers, Amsterdam.

[80] Sengupta, J.K. and Fanchon, P. (1997). Control Theory Methods in Economics. Kluwer

[81] Tapiero, C.S. (1998). Applied Stochastic Models and Control for Insurance and Finance.

[82] Teo, K.L. (1992). A computational approach to an optimal control problem with a cost on
changing control. Optimization, 1:397-413.
198 OPTIMAL CONTROL MODELS IN FINANCE

[83] Teo, K.L. and Goh, C.J. (1988). On constrained optimization problems with nonsmooth
cost functionals. Applied Mathematics and Optimization, 18:181-190.

[84] Teo, K.L., Goh, G.J. and Wong, K.H. (1991). A Unified Computational Approach to Op-
timal Control Problems. Longman Scientific and Technical, London.

[85] Teo, K.L. and Jennings, G.J. (1991). Optimal control with a cost on changing control.
Journal of Optimization Theory and Applications, 68:335-357.

[86] Vidale, M.L. and Wolfe, H.B. (1957). An operations research study of sales response to

[87] Yong, J. (1989). Systems governed by ordinary differential equations with continuous,
switching and impulse controls. Appl. Math.Optim., 20:223-235.

[88] Yong, J. (1991). Existence of the value for a differential game with switching strategies in
a Banach space. System Sci. Math. Sci., 4:321-340.

[89] Ziemba, W.T. and Vickson, R.G. (1975). Stochastic Optimization Models in Finance. Aca-
demic Press, New York.
Index

control policy, 10, 29, 46, 54, 61, 67, 88, 101,
103, 109
aggregate dynamic financial system, 40
control restraint, 95
analytical solution, xv, 1, 91, 104â€“108, 143
control strategy, 16
approximate solution methods, 10
control system, 11, 17
approximation methods, 11
control variables, 93, 139
approximation problem, 12
convergence properties, 12
associated problems, 6, 7
convex function, 96
augmented Lagrangian algorithm, 11, 15, 98
cost analysis, 10
cost function, 2, 11â€“13, 19, 27, 28, 30, 35, 36,
bang-bang control, 1, 4, 7, 16, 17, 19, 26, 41,
41, 45, 91, 142
47, 97, 101
cost of changing control, 8, 19, 20, 27, 30, 36,
bang-bang optimal control problems, 26, 27,
41, 86, 89, 142
143
CPET, 16, 19
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