Dynamic optimization models in finance: Some extensions to the framework, models, and computation

Previous Article
Optimal ordering policies and sourcing strategies with supply disruption
JIMO Home
This Issue
Next Article
CVaR proxies for minimizing scenario-based Value-at-Risk

October 2014, 10(4): 1129-1146. doi: 10.3934/jimo.2014.10.1129

Dynamic optimization models in finance: Some extensions to the framework, models, and computation

Bruce D. Craven ^1, and Sardar M. N. Islam ^2,

1.	Department of Mathematics & Statistics, University of Melbourne, Victoria 3010, Australia
2.	Victoria University, P.O. Box 14428, Melbourne, Vic. 6001, Australia

Received December 2012 Revised January 2014 Published February 2014

Abstract
Related Papers

Both mathematical characteristics and computational aspects of dynamic optimization in finance have potential for extensions. Various proposed extensions are presented in this paper for dynamic optimization modelling in finance, adapted from developments in other areas of economics and mathematics. They show the need and potential for further areas of study and extensions in financial modelling. The extensions discussed and made concern (a) incorporation of the elements of a dynamic optimization model, (b) an improved model including physical capital, (c) some computational experiments. These extensions make dynamic financial optimisation relatively more organized, coherent and coordinated. These extensions are relevant for applications of financial models to academic and practical exercises. This paper reports initial efforts in providing some useful extensions; further work is necessary to complete the research agenda.

Keywords: Dynamic modelling, sensitivity., elements of an optimization model, financial computation.

Mathematics Subject Classification: 97M3.

Citation: Bruce D. Craven, Sardar M. N. Islam. Dynamic optimization models in finance: Some extensions to the framework, models, and computation. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1129-1146. doi: 10.3934/jimo.2014.10.1129

References:

[1]	P. Brandimarte, Numerical Methods in Finance: A MATLAB-based introduction, Second edition. Statistics in Practice. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. doi: 10.1002/0470080493.
[2]	W. Brock, Sensitivity of optimal growth paths with respect to a change in target stocks, Zeitchrift für National Ökonomie, Supplementum, 1 (1971), 73-89.
[3]	B. D. Craven, Convergence of discrete approximations for constrained minimizatiion, Journal of the Australian Mathematical Society, Series B, 36 (1994), 50-59. doi: 10.1017/S0334270000010237.
[4]	B. D. Craven, Control and Optimization, Chapman & Hall, London, 1995.
[5]	B. D. Craven, Optimal control and invexity, Computers and Mathematics with Applications, 35 (1998), 17-25. doi: 10.1016/S0898-1221(98)00002-9.
[6]	B. D. Craven, Optimal control of an economic model with a small stochastic term, Pacific Journal of Optimization, 1 (2005), 233-241.
[7]	B. D. Craven, K. de Haas and J. Wettenhall, Computing optimal control, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1988), 601-615.
[8]	B. D. Craven and S. M. N. Islam, Computing optimal control on MATLAB: The scom package and economic growth models, in Optimisation and Related Topics, (Ballarat/Melbourne, 1999), 61-70, Appl. Optim., Volume 47 in the Series Applied Optimization, Kluwer Academic Publishers, Dordrecht, 2001.
[9]	B. D. Craven and S. M. N. Islam, Optimization in Economics and Finance, Springer, Dordrecht, 2005.
[10]	K. Cuthbertson, Quantitative Financial Economics: Stocks, Bonds, and Foreign Exchange, John Wiley, Chichester, England, 1996.
[11]	B. Davis and D. Elzinga, The solution of an optimal control problem in financial modeling, Operations Research, 19 (1971), 1419-1433. doi: 10.1287/opre.19.6.1419.
[12]	W. Diewert, Generalized Concavity and Economics, in Generalized Concavity in Optimization and Economics, S. Schaible and W. T. Ziemba (eds.), Academic Press, New York, 1981.
[13]	P. Dutta, On specifying the parameters of a development plan, in Capital, Investment and Development, K. Basu, M. Majumdar and T. Mitra (eds.), Blackwell, Oxford, 1993, 75-98.
[14]	K. Fox, J. K. Sengupta and E. Thorbecke, The Theory of Quantitative Economic Policy with Applications to Economic Growth, Stabilization and Planning, North-Holland, Amsterdam, 1973.
[15]	C. Goh and K. L. Teo, MISER: A FORTRAN program for solving optimal control problems, Advances in Engineering Software, 10 (1988), 90-99. doi: 10.1016/0141-1195(88)90005-8.
[16]	C. Gourieroux and J. Janiak, Financial Econometrics, Princeton University Press, Princeton, 2001. doi: 10.7202/010560ar.
[17]	N. Hakansson, Optimal investment and consumption strategies under risk for a class of utility functions, Econometrica, 38 (1970), 587-607. doi: 10.2307/1912196.
[18]	G. Heal, Valuing the Future: Economic Theory and Sustainability, Columbia University Press, New York, 1998.
[19]	S. M. N. Islam and B. D. Craven, Computation of non-linear continuous capital growth models: Experiments with optimal control algorithms and computer programs, Economic Modelling: The International Journal of Theoretical and Applied Papers on Economic Modelling 18 (2001), 551-586.
[20]	S. M. N. Islam and B. D. Craven, Measuring Sustainable Growth, in Governance and Social Responsibility, J. Batten, (ed.), ELsevier-North-Holland, Amsterdam, 2002.
[21]	K. Judd, Numerical Methods in Economics, MIT Press, Cambridge, 1998.
[22]	D. Leonard D. and N. V. Long, Optimal Control Theory and Static Optimization in Economics, Cambridge University Press, Cambridge, U.K, 1992.
[23]	R. Lucas, Asset prices in an exchange economy, Econometrica, 46 (1978), 1429-1445. doi: 10.2307/1913837.
[24]	A. Malliaris and W. Brock, Stochastic Methods in Economics and Finance, Elsevier Science, Amsterdam, 1982.
[25]	T. Mitra, Sensitivity of optimal programmes with respect to changes in target stocks: The case of irreversible investment, Journal of Economic Theory, 29 (1983), 172-184. doi: 10.1016/0022-0531(83)90128-X.
[26]	T. Mitra and D. Ray, Dynamic optimization on a non-convex feasible set: Some general resulots for non-emooth technologies, Zeitchrift für National Ökonomie, 44 (1984), 151-175. doi: 10.1007/BF01289475.
[27]	M. Ramon and A. Scott, (Ed.), Computational Methods for the Study of Dynamic Economies, Oxford University Press, Oxford, 1999.
[28]	A. L. Schwartz, Theory and implementation of numerical methods based on Runge-Kutta integration for solving optimal control problems, Dissertation, University of California at Berkeley, 1989.
[29]	J. K. Sengupta and P. Fanchon, Control Theory Methods in Economics, Kluwer Academic, Boston, 1997. doi: 10.1007/978-1-4615-6285-6.
[30]	C. Tapiero, Applied Stochastic Models and Control for Insurance and Finance, Kluwer Academic, London, 1998. doi: 10.1007/978-1-4615-5823-1.
[31]	K. L. Teo, C. Goh and K. Wong, A Unified Computational Approach for Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 55. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991.
[32]	G. Thompson and S. Thore, Computational Economics: Economic Modeling with Optimization Software, (Chapters 21 to 22 and Appendices A&B), Scientific Press, 1993.
[33]	R. Vickson and W. Ziemba, Stochastic Optimisation Models in Finance, Academic Press, New York, 1975.
[34]	S. Zenios, (Ed.), Financial Optimization, Cambridge University Press, Cambridge, 1993.

show all references

References:

[1]	P. Brandimarte, Numerical Methods in Finance: A MATLAB-based introduction, Second edition. Statistics in Practice. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. doi: 10.1002/0470080493.
[2]	W. Brock, Sensitivity of optimal growth paths with respect to a change in target stocks, Zeitchrift für National Ökonomie, Supplementum, 1 (1971), 73-89.
[3]	B. D. Craven, Convergence of discrete approximations for constrained minimizatiion, Journal of the Australian Mathematical Society, Series B, 36 (1994), 50-59. doi: 10.1017/S0334270000010237.
[4]	B. D. Craven, Control and Optimization, Chapman & Hall, London, 1995.
[5]	B. D. Craven, Optimal control and invexity, Computers and Mathematics with Applications, 35 (1998), 17-25. doi: 10.1016/S0898-1221(98)00002-9.
[6]	B. D. Craven, Optimal control of an economic model with a small stochastic term, Pacific Journal of Optimization, 1 (2005), 233-241.
[7]	B. D. Craven, K. de Haas and J. Wettenhall, Computing optimal control, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1988), 601-615.
[8]	B. D. Craven and S. M. N. Islam, Computing optimal control on MATLAB: The scom package and economic growth models, in Optimisation and Related Topics, (Ballarat/Melbourne, 1999), 61-70, Appl. Optim., Volume 47 in the Series Applied Optimization, Kluwer Academic Publishers, Dordrecht, 2001.
[9]	B. D. Craven and S. M. N. Islam, Optimization in Economics and Finance, Springer, Dordrecht, 2005.
[10]	K. Cuthbertson, Quantitative Financial Economics: Stocks, Bonds, and Foreign Exchange, John Wiley, Chichester, England, 1996.
[11]	B. Davis and D. Elzinga, The solution of an optimal control problem in financial modeling, Operations Research, 19 (1971), 1419-1433. doi: 10.1287/opre.19.6.1419.
[12]	W. Diewert, Generalized Concavity and Economics, in Generalized Concavity in Optimization and Economics, S. Schaible and W. T. Ziemba (eds.), Academic Press, New York, 1981.
[13]	P. Dutta, On specifying the parameters of a development plan, in Capital, Investment and Development, K. Basu, M. Majumdar and T. Mitra (eds.), Blackwell, Oxford, 1993, 75-98.
[14]	K. Fox, J. K. Sengupta and E. Thorbecke, The Theory of Quantitative Economic Policy with Applications to Economic Growth, Stabilization and Planning, North-Holland, Amsterdam, 1973.
[15]	C. Goh and K. L. Teo, MISER: A FORTRAN program for solving optimal control problems, Advances in Engineering Software, 10 (1988), 90-99. doi: 10.1016/0141-1195(88)90005-8.
[16]	C. Gourieroux and J. Janiak, Financial Econometrics, Princeton University Press, Princeton, 2001. doi: 10.7202/010560ar.
[17]	N. Hakansson, Optimal investment and consumption strategies under risk for a class of utility functions, Econometrica, 38 (1970), 587-607. doi: 10.2307/1912196.
[18]	G. Heal, Valuing the Future: Economic Theory and Sustainability, Columbia University Press, New York, 1998.
[19]	S. M. N. Islam and B. D. Craven, Computation of non-linear continuous capital growth models: Experiments with optimal control algorithms and computer programs, Economic Modelling: The International Journal of Theoretical and Applied Papers on Economic Modelling 18 (2001), 551-586.
[20]	S. M. N. Islam and B. D. Craven, Measuring Sustainable Growth, in Governance and Social Responsibility, J. Batten, (ed.), ELsevier-North-Holland, Amsterdam, 2002.
[21]	K. Judd, Numerical Methods in Economics, MIT Press, Cambridge, 1998.
[22]	D. Leonard D. and N. V. Long, Optimal Control Theory and Static Optimization in Economics, Cambridge University Press, Cambridge, U.K, 1992.
[23]	R. Lucas, Asset prices in an exchange economy, Econometrica, 46 (1978), 1429-1445. doi: 10.2307/1913837.
[24]	A. Malliaris and W. Brock, Stochastic Methods in Economics and Finance, Elsevier Science, Amsterdam, 1982.
[25]	T. Mitra, Sensitivity of optimal programmes with respect to changes in target stocks: The case of irreversible investment, Journal of Economic Theory, 29 (1983), 172-184. doi: 10.1016/0022-0531(83)90128-X.
[26]	T. Mitra and D. Ray, Dynamic optimization on a non-convex feasible set: Some general resulots for non-emooth technologies, Zeitchrift für National Ökonomie, 44 (1984), 151-175. doi: 10.1007/BF01289475.
[27]	M. Ramon and A. Scott, (Ed.), Computational Methods for the Study of Dynamic Economies, Oxford University Press, Oxford, 1999.
[28]	A. L. Schwartz, Theory and implementation of numerical methods based on Runge-Kutta integration for solving optimal control problems, Dissertation, University of California at Berkeley, 1989.
[29]	J. K. Sengupta and P. Fanchon, Control Theory Methods in Economics, Kluwer Academic, Boston, 1997. doi: 10.1007/978-1-4615-6285-6.
[30]	C. Tapiero, Applied Stochastic Models and Control for Insurance and Finance, Kluwer Academic, London, 1998. doi: 10.1007/978-1-4615-5823-1.
[31]	K. L. Teo, C. Goh and K. Wong, A Unified Computational Approach for Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 55. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991.
[32]	G. Thompson and S. Thore, Computational Economics: Economic Modeling with Optimization Software, (Chapters 21 to 22 and Appendices A&B), Scientific Press, 1993.
[33]	R. Vickson and W. Ziemba, Stochastic Optimisation Models in Finance, Academic Press, New York, 1975.
[34]	S. Zenios, (Ed.), Financial Optimization, Cambridge University Press, Cambridge, 1993.

[1]	Caglar S. Aksezer. On the sensitivity of desirability functions for multiresponse optimization. Journal of Industrial and Management Optimization, 2008, 4 (4) : 685-696. doi: 10.3934/jimo.2008.4.685
[2]	Krzysztof Fujarewicz, Krzysztof Łakomiec. Adjoint sensitivity analysis of a tumor growth model and its application to spatiotemporal radiotherapy optimization. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1131-1142. doi: 10.3934/mbe.2016034
[3]	Alireza Bahiraie, A.K.M. Azhar, Noor Akma Ibrahim. A new dynamic geometric approach for empirical analysis of financial ratios and bankruptcy. Journal of Industrial and Management Optimization, 2011, 7 (4) : 947-965. doi: 10.3934/jimo.2011.7.947
[4]	Qiong Liu, Ahmad Reza Rezaei, Kuan Yew Wong, Mohammad Mahdi Azami. Integrated modeling and optimization of material flow and financial flow of supply chain network considering financial ratios. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 113-132. doi: 10.3934/naco.2019009
[5]	Francesca Biagini, Jacopo Mancin. Financial asset price bubbles under model uncertainty. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 14-. doi: 10.1186/s41546-017-0026-3
[6]	Sergey V Lototsky, Henry Schellhorn, Ran Zhao. An infinite-dimensional model of liquidity in financial markets. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 117-138. doi: 10.3934/puqr.2021006
[7]	Alex James, Simon Green, Mike Plank. Modelling the dynamic response of oxygen uptake to exercise. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 361-370. doi: 10.3934/dcdsb.2009.12.361
[8]	M.A.J Chaplain, G. Lolas. Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Networks and Heterogeneous Media, 2006, 1 (3) : 399-439. doi: 10.3934/nhm.2006.1.399
[9]	Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1145-1160. doi: 10.3934/jimo.2021013
[10]	Michael Herty, S. Moutari, M. Rascle. Optimization criteria for modelling intersections of vehicular traffic flow. Networks and Heterogeneous Media, 2006, 1 (2) : 275-294. doi: 10.3934/nhm.2006.1.275
[11]	Dimitra C. Antonopoulou, Marina Bitsaki, Georgia Karali. The multi-dimensional stochastic Stefan financial model for a portfolio of assets. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 1955-1987. doi: 10.3934/dcdsb.2021118
[12]	Avner Friedman, Bei Hu, Chuan Xue. A three dimensional model of wound healing: Analysis and computation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2691-2712. doi: 10.3934/dcdsb.2012.17.2691
[13]	Jinghuan Li, Shuhua Zhang, Yu Li. Modelling and computation of optimal multiple investment timing in multi-stage capacity expansion infrastructure projects. Journal of Industrial and Management Optimization, 2022, 18 (1) : 297-314. doi: 10.3934/jimo.2020154
[14]	Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial and Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381
[15]	Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial and Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1
[16]	Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations and Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022
[17]	Behrouz Kheirfam, Kamal mirnia. Comments on ''Generalized support set invariancy sensitivity analysis in linear optimization''. Journal of Industrial and Management Optimization, 2008, 4 (3) : 611-616. doi: 10.3934/jimo.2008.4.611
[18]	Francesco Sanna Passino, Nicholas A. Heard. Modelling dynamic network evolution as a Pitman-Yor process. Foundations of Data Science, 2019, 1 (3) : 293-306. doi: 10.3934/fods.2019013
[19]	Prabhu Manyem. A note on optimization modelling of piecewise linear delay costing in the airline industry. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1809-1823. doi: 10.3934/jimo.2020047
[20]	Alessia Marigo, Benedetto Piccoli. A model for biological dynamic networks. Networks and Heterogeneous Media, 2011, 6 (4) : 647-663. doi: 10.3934/nhm.2011.6.647

2021 Impact Factor: 1.411

Tools

Metrics

Other articles
by authors

on AIMS
Bruce D. Craven Sardar M. N. Islam
on Google Scholar
Bruce D. Craven Sardar M. N. Islam

[Back to Top]