A stochastic control model of investment, production and consumption
Authors:
Wendell H. Fleming and Tao Pang
Journal:
Quart. Appl. Math. 63 (2005), 71-87
MSC (2000):
Primary 60H30, 91B28, 93E20
DOI:
https://doi.org/10.1090/S0033-569X-04-00941-1
Published electronically:
December 17, 2004
MathSciNet review:
2126570
Full-text PDF Free Access
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Abstract: We consider a stochastic control model in which an economic unit has productive capital and also liabilities in the form of debt. The worth of capital changes over time through investment as well as through random Brownian fluctuations in the unit price of capital. Income from production is also subject to random Brownian fluctuations. The goal is to choose investment and consumption controls which maximize total expected discounted HARA utility of consumption. Optimal control policies are found using the method of dynamic programming. In case of logarithmic utility, these policies have explicit forms.
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[F]Fl Fleming, W. H. (1995), Optimal investment models and risk-sensitive stochastic control. IMA Vol. Math. Appl., 65, 35-45, Springer, New York.
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[F-So]FlSo Fleming, W. H. and Soner, H. M. (1992), Controlled Markov Processes and Viscosity Solutions. Springer, New York.
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[B-P]BiPl Bielecki, T. R. and Pliska, S. R. (1999), Risk-sensitive dynamic asset management. Appl. Math. Optim. 39, 337-360.
[F]Fl Fleming, W. H. (1995), Optimal investment models and risk-sensitive stochastic control. IMA Vol. Math. Appl., 65, 35-45, Springer, New York.
[F-HH]FlHH Fleming, W. H. and Hernandez-Hernandez, D.(2003), An optimal consumption model with stochastic volatility, Finance Stoch., Vol. 7, 245-262.
[F-P]FlP Fleming, W. H. and Pang, T. (2004), An Application of Stochastic Control Theory to Financial Economics, SIAM J. of Control and Optimization, Vol. 43, 502-531.
[F-R]FlR Fleming, W. H. and Rishel, R. W. (1975), Deterministic and Stochastic Optimal Control. Springer, New York.
[F-Sh1]FlSh1 Fleming, W. H. and Sheu, S. J. (1999), Optimal long term growth rate of expected utility of wealth. Ann. Appl. Probab., Vol. 9, No. 3, 871-903.
[F-Sh2]FlSh2 Fleming, W. H. and Sheu, S. J. (2000), Risk-sensitive control and an optimal investment model. Math. Finance, Vol. 10, No. 2, 197-213.
[F-So]FlSo Fleming, W. H. and Soner, H. M. (1992), Controlled Markov Processes and Viscosity Solutions. Springer, New York.
[F-St1]FlSt1 Fleming, W. H. and Stein, J. L. (2001), Stochastic inter-temporal optimization in discrete time, in Negishi, Takashi, Rama Ramachandran and Kazuo Mino (ed) Economic Theory, Dynamics and Markets: Essays in Honor of Ryuzo Sato, Kluwer.
[F-St2]FlSt2 Fleming, W. H. and Stein, J. L. (2004), Stochastic optimal control in international finance and debt, J. Banking and Finance, Vol. 28, 979-996.
[F-P-S]FPS Fouque, J. P., Papanicolaou, G. and Sircar, R. (2000): Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press.
[P1]P Pang, T. (2002), Stochastic Control Theory and Its Applications to Financial Economics, Ph.D. thesis, Brown University, Providence, RI.
[P2]P2 Pang, T. (2004), Portfolio optimization models on infinite time horizon, Journal of Optimization Theory and Applications, Vol. 22, 119-143.
[P-R]PR Platen, E. and Rebolledo, R. (1996), Principles for modelling financial markets. J. Appl. Probab. 33, 601-613.
[Z]Z Zariphopoulou, T. (2001), A solution approach to valuation with unhedgeable risks. Finance Stoch., Vol. 5(1), 61-82.
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Additional Information
Wendell H. Fleming
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912
Email:
whf@cfm.brown.edu
Tao Pang
Affiliation:
Department of Mathematics, NC State University, Raleigh, NC 27695
Email:
tpang@math.ncsu.edu
Received by editor(s):
April 22, 2004
Published electronically:
December 17, 2004
Article copyright:
© Copyright 2004
Brown University