Abstract
A portfolio optimization problem on an infinite-time horizon is considered. Risky asset prices obey a logarithmic Brownian motion and interest rates vary according to an ergodic Markov diffusion process. The goal is to choose optimal investment and consumption policies to maximize the infinite-horizon expected discounted hyperbolic absolute risk aversion (HARA) utility of consumption. The problem is then reduced to a one-dimensional stochastic control problem by virtue of the Girsanov transformation. A dynamic programming principle is used to derive the dynamic programming equation (DPE). The subsolution/supersolution method is used to obtain existence of solutions of the DPE. The solutions are then used to derive the optimal investment and consumption policies. In addition, for a special case, we obtain the results using the viscosity solution method.
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References
Fleming, W. H., and Soner, H. M., Controlled Markov Processes and Viscosity Solutions, Springer, New York, NY, 1992.
Fleming, W. H., and Stein, J. L., Stochastic Optimal Control in International Finance and Debt, Journal of Banking and Finance (to appear).
Fleming, W. H., and Hernandez-Hernandez, D., An Optimal Consumption Model with Stochastic Volatility, Finance and Stochastics, Vol. 7, pp 245-262, 2003.
Fleming, W. H., and Sheu, S. J., Optimal Long–Term Growth Rate of the Expected Utility of Wealth, Annals of Applied Probability, Vol. 9, pp 871-903, 1999.
Pang, T., Stochastic Control Theory and Its Applications to Financial Economics, PhD Thesis, Brown University, Providence, Rhode Island, 2002.
Bielecki, T. R., and Pliska, S. R., Risk Sensitive Dynamic Asset Management, Applied Mathematics and Optimization, Vol. 39, pp 337-360, 1999.
Zariphopoulou, T., A Solution Approach to Valuation with Unhedgeable Risks, Finance and Stochastics, Vol. 5, pp 61-82, 2001.
Fleming, W. H., and Pang, T., An Application of Stochastic Control Theory to Financial Economics, SIAM Journal on Control and Optimization (to appear).
Üstünel, A. S., and Zakai, M., Transformation of Measure of Weiner Space, Springer, New York, NY, 2000.
Pao, C. V., Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, 1992.
Friedman, A., Stochastic Differential Equations and Applications, Vol. 1, Academic Press, New York, NY, 1975.
Borkar, V. S., Optimal Control of Diffusion Processes, Longman Scientific and Technical, Essex, UK, 1989.
Lions, P. L., Optimal Control of Diffusion Processes and Hamiltion-Jacobi-Bellman Equations, Part 2:Viscosity Solutions and Uniqueness, Communications in Partial Differential Equations, Vol. 8, pp 1229-1276, 1983.
Bailey, P. B., Shampine, L. F., and Waltman, P. E., Nonlinear Two–Point Boundary–Value Problems, Academic Press, New York, NY, 1968.
Crandall, M. G., Ishii, H., and Lions, P. L., User 's Guide to Viscosity Solutions of Second–Order Partial Differential Equations, Bulletin of the American Mathematical Society, Vol. 27, pp 1-67, 1992.
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Pang, T. Portfolio Optimization Models on Infinite-Time Horizon. Journal of Optimization Theory and Applications 122, 573–597 (2004). https://doi.org/10.1023/B:JOTA.0000042596.26927.2d
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DOI: https://doi.org/10.1023/B:JOTA.0000042596.26927.2d