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December  2016, 6(4): 629-651. doi: 10.3934/mcrf.2016018

## An infinite time horizon portfolio optimization model with delays

 1 Department of Mathematics, North Carolina State University, Raleigh, NC 27695-7913, United States 2 Operations Research, North Carolina State University, Raleigh, NC 27695-7913, USA; Current address: Department of Mathematics, Lahore University of Management Sciences, Lahore, 54792, Pakistan

Received  October 2015 Revised  January 2016 Published  October 2016

In this paper we consider a portfolio optimization problem of the Merton's type over an infinite time horizon. Unlike the classical Markov model, we consider a system with delays. The problem is formulated as a stochastic control problem on an infinite time horizon and the state evolves according to a process governed by a stochastic process with delay. The goal is to choose investment and consumption controls such that the total expected discounted utility is maximized. Under certain conditions, we derive the explicit solutions for the associated Hamilton-Jacobi-Bellman (HJB) equations in a finite dimensional space for logarithmic and power utility functions. For those utility functions, verification results are established to ensure that the solutions are equal to the value functions, and the optimal controls are derived, too.
Citation: Tao Pang, Azmat Hussain. An infinite time horizon portfolio optimization model with delays. Mathematical Control and Related Fields, 2016, 6 (4) : 629-651. doi: 10.3934/mcrf.2016018
##### References:
 [1] T. R. Bielecki and S. R. Pliska, Risk sensitive dynamic asset management, Appl. Math. and Optimization, 39 (1999), 337-360. doi: 10.1007/s002459900110. [2] M. H. Chang, T. Pang and M. Pemy, Finite difference approximations for stochastic control systems with delay, Stoch. Anal. Appl., 26 (2008), 451-470. doi: 10.1080/07362990802006980. [3] M. H. Chang, T. Pang and M. Pemy, Optimal control of stochastic functional differential equations with a bounded memory, Stochastics: An International Journal of Probability and Stochastic Processes, 80 (2008), 69-96. doi: 10.1080/17442500701605494. [4] M. H. Chang, T. Pang and Y. Yang, A stochastic portfolio optimization model with bounded memory, Mathematics of Operations Research, 36 (2011), 604-619. doi: 10.1287/moor.1110.0508. [5] R. Cont and D. A. Fournié, Functional Ito Calculus and Stochastic Integral Representation of Martingales, The Annuals of Probability, 41 (2013), 109-133. doi: 10.1214/11-AOP721. [6] B. Dupire, Functional Itô's Calculus,, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS, (): 2009. [7] I. Elsanousi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics and Stochastics Reports, 71 (2000), 69-89. doi: 10.1080/17442500008834259. [8] S. Federico, B. Goldys and F. Gozzi, HJB equations for the optimal control of differential equations with delays and state constraints, I: Regularity of viscosity solutions, SIAM Journal on Control and Optimization, 48 (2010), 4910-4937. doi: 10.1137/09076742X. [9] S. Federico, A stochastic control problem with delay arising in a pension fund model, Finance and Stochastics, 15 (2011), 421-459. doi: 10.1007/s00780-010-0146-4. [10] W. H. Fleming and D. Hernandez-Hernandez, An optimal consumption model with stochastic volatility, Finance and Stochastics, 7 (2003), 245-262. doi: 10.1007/s007800200083. [11] W. H. Fleming and T. Pang, An application of stochastic control theory to financial economics, SIAM J. Control Optimiz., 43 (2004), 502-531. doi: 10.1137/S0363012902419060. [12] W. H. Fleming and T. Pang, A stochastic control model of investment, production and consumption, Quarterly of Applied Mathematics, 63 (2005), 71-87. doi: 10.1090/S0033-569X-04-00941-1. [13] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, 1975. doi: 10.1007/978-1-4612-6380-7. [14] W. H. Fleming and S. J. Sheu, Risk-sensitive control and optimal investment model, Mathematical Finance, 10 (2000), 197-213. doi: 10.1111/1467-9965.00089. [15] J. P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Market with Stochastic Volatility, Cambridge University Press, 2000. [16] F. Gozzi and C. Marinelli, Stochastic optimal control of delay equations arising in advertising models, Stochastic Partial Differential Equations and Applications-VII, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 245 (2006), 133-148. doi: 10.1201/9781420028720.ch13. [17] F. Gozzi, C. Marinelli and S. Savin, On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects, Journal of Optimization Theory and Applications, 142 (2009), 291-321. doi: 10.1007/s10957-009-9524-5. [18] A, J. Koivo, Optimal control of linear stochastic systems described by functional differential equations, Journal of Optimization Theory and Applications, 9 (1972), 161-175. doi: 10.1007/BF00932588. [19] V. B. Kolmanovskiĭ and T. L. Maizenberg, Optimal control of stochastic systems with aftereffect, Avtomat. i Telemeh, 1 (1973), 47-61. [20] V. B. Kolmanovskiĭ and L. E. Shaĭkhet, Control of systems with aftereffect, American Mathematical Society, 157 (1996). [21] B. Larssen, Dynamic programming in stochastic control of systems with delay, Stochastics: An International Journal of Probability and Stochastic Processes, 74 (2002), 651-673. doi: 10.1080/1045112021000060764. [22] B. Larssen and N. h. Risebro, When are HJB-equations in stochastic control of delay systems finite dimensional?, Stochastic Analysis and Applications, 21 (2003), 643-671. doi: 10.1081/SAP-120020430. [23] A. Lindquist, On feedback control of linear stochastic systems, SIAM Journal on Control, 11 (1973), 323-343. doi: 10.1137/0311025. [24] A. Lindquist, Optimal control of linear stochastic systems with applications to time lag systems, Information Sciences, 5 (1973), 81-126. doi: 10.1016/0020-0255(73)90005-4. [25] S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman Publishing, Boston-London-Melbourne, 1984. [26] S. E. A. Mohammed, Stochastic differential equations with memory- theory, examples and applications, Stochastic Analysis and Related Topics VI, the series Progress in Probability, 42 (1998), 1-77. [27] T. Pang, Portfolio optimization models on infinite-time horizon, J. Optim. Theory Appl., 122 (2004), 573-597. doi: 10.1023/B:JOTA.0000042596.26927.2d. [28] T. Pang, Stochastic portfolio optimization with log utility, Int. J. Theor. Appl. Finance, 9 (2006), 869-887. doi: 10.1142/S0219024906003858. [29] T. Pang and A. Hussain, An application of functional Ito's formula to stochastic portfolio optimization with bounded memory, Proceedings of 2015 SIAM Conference on Control and Its Applications (CT15), Paris, France, 2015, 159-166. doi: 10.1137/1.9781611974072.23. [30] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

show all references

##### References:
 [1] T. R. Bielecki and S. R. Pliska, Risk sensitive dynamic asset management, Appl. Math. and Optimization, 39 (1999), 337-360. doi: 10.1007/s002459900110. [2] M. H. Chang, T. Pang and M. Pemy, Finite difference approximations for stochastic control systems with delay, Stoch. Anal. Appl., 26 (2008), 451-470. doi: 10.1080/07362990802006980. [3] M. H. Chang, T. Pang and M. Pemy, Optimal control of stochastic functional differential equations with a bounded memory, Stochastics: An International Journal of Probability and Stochastic Processes, 80 (2008), 69-96. doi: 10.1080/17442500701605494. [4] M. H. Chang, T. Pang and Y. Yang, A stochastic portfolio optimization model with bounded memory, Mathematics of Operations Research, 36 (2011), 604-619. doi: 10.1287/moor.1110.0508. [5] R. Cont and D. A. Fournié, Functional Ito Calculus and Stochastic Integral Representation of Martingales, The Annuals of Probability, 41 (2013), 109-133. doi: 10.1214/11-AOP721. [6] B. Dupire, Functional Itô's Calculus,, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS, (): 2009. [7] I. Elsanousi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics and Stochastics Reports, 71 (2000), 69-89. doi: 10.1080/17442500008834259. [8] S. Federico, B. Goldys and F. Gozzi, HJB equations for the optimal control of differential equations with delays and state constraints, I: Regularity of viscosity solutions, SIAM Journal on Control and Optimization, 48 (2010), 4910-4937. doi: 10.1137/09076742X. [9] S. Federico, A stochastic control problem with delay arising in a pension fund model, Finance and Stochastics, 15 (2011), 421-459. doi: 10.1007/s00780-010-0146-4. [10] W. H. Fleming and D. Hernandez-Hernandez, An optimal consumption model with stochastic volatility, Finance and Stochastics, 7 (2003), 245-262. doi: 10.1007/s007800200083. [11] W. H. Fleming and T. Pang, An application of stochastic control theory to financial economics, SIAM J. Control Optimiz., 43 (2004), 502-531. doi: 10.1137/S0363012902419060. [12] W. H. Fleming and T. Pang, A stochastic control model of investment, production and consumption, Quarterly of Applied Mathematics, 63 (2005), 71-87. doi: 10.1090/S0033-569X-04-00941-1. [13] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, 1975. doi: 10.1007/978-1-4612-6380-7. [14] W. H. Fleming and S. J. Sheu, Risk-sensitive control and optimal investment model, Mathematical Finance, 10 (2000), 197-213. doi: 10.1111/1467-9965.00089. [15] J. P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Market with Stochastic Volatility, Cambridge University Press, 2000. [16] F. Gozzi and C. Marinelli, Stochastic optimal control of delay equations arising in advertising models, Stochastic Partial Differential Equations and Applications-VII, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 245 (2006), 133-148. doi: 10.1201/9781420028720.ch13. [17] F. Gozzi, C. Marinelli and S. Savin, On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects, Journal of Optimization Theory and Applications, 142 (2009), 291-321. doi: 10.1007/s10957-009-9524-5. [18] A, J. Koivo, Optimal control of linear stochastic systems described by functional differential equations, Journal of Optimization Theory and Applications, 9 (1972), 161-175. doi: 10.1007/BF00932588. [19] V. B. Kolmanovskiĭ and T. L. Maizenberg, Optimal control of stochastic systems with aftereffect, Avtomat. i Telemeh, 1 (1973), 47-61. [20] V. B. Kolmanovskiĭ and L. E. Shaĭkhet, Control of systems with aftereffect, American Mathematical Society, 157 (1996). [21] B. Larssen, Dynamic programming in stochastic control of systems with delay, Stochastics: An International Journal of Probability and Stochastic Processes, 74 (2002), 651-673. doi: 10.1080/1045112021000060764. [22] B. Larssen and N. h. Risebro, When are HJB-equations in stochastic control of delay systems finite dimensional?, Stochastic Analysis and Applications, 21 (2003), 643-671. doi: 10.1081/SAP-120020430. [23] A. Lindquist, On feedback control of linear stochastic systems, SIAM Journal on Control, 11 (1973), 323-343. doi: 10.1137/0311025. [24] A. Lindquist, Optimal control of linear stochastic systems with applications to time lag systems, Information Sciences, 5 (1973), 81-126. doi: 10.1016/0020-0255(73)90005-4. [25] S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman Publishing, Boston-London-Melbourne, 1984. [26] S. E. A. Mohammed, Stochastic differential equations with memory- theory, examples and applications, Stochastic Analysis and Related Topics VI, the series Progress in Probability, 42 (1998), 1-77. [27] T. Pang, Portfolio optimization models on infinite-time horizon, J. Optim. Theory Appl., 122 (2004), 573-597. doi: 10.1023/B:JOTA.0000042596.26927.2d. [28] T. Pang, Stochastic portfolio optimization with log utility, Int. J. Theor. Appl. Finance, 9 (2006), 869-887. doi: 10.1142/S0219024906003858. [29] T. Pang and A. Hussain, An application of functional Ito's formula to stochastic portfolio optimization with bounded memory, Proceedings of 2015 SIAM Conference on Control and Its Applications (CT15), Paris, France, 2015, 159-166. doi: 10.1137/1.9781611974072.23. [30] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.
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