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

An infinite time horizon portfolio optimization model with delays

Tao Pang 1, and  Azmat Hussain 2,

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.
Keywords: Hamilton-Jacobi-Bellman equation, Portfolio optimization, stochastic delay equation., dynamic programming, stochastic control.
Mathematics Subject Classification: Primary: 91G80, 49L20; secondary: 91G10, 93E2.
Citation: Tao Pang, Azmat Hussain. An infinite time horizon portfolio optimization model with delays. Mathematical Control & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. Dupire, Functional Itô's Calculus,, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS, (): 2009.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Market with Stochastic Volatility, Cambridge University Press, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. B. Kolmanovskiĭ and T. L. Maizenberg, Optimal control of stochastic systems with aftereffect, Avtomat. i Telemeh, 1 (1973), 47-61.  Google Scholar

[20]

V. B. Kolmanovskiĭ and L. E. Shaĭkhet, Control of systems with aftereffect, American Mathematical Society, 157 (1996). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. Lindquist, On feedback control of linear stochastic systems, SIAM Journal on Control, 11 (1973), 323-343. doi: 10.1137/0311025.  Google Scholar

[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.  Google Scholar

[25]

S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman Publishing, Boston-London-Melbourne, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

T. Pang, Stochastic portfolio optimization with log utility, Int. J. Theor. Appl. Finance, 9 (2006), 869-887. doi: 10.1142/S0219024906003858.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. Dupire, Functional Itô's Calculus,, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS, (): 2009.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Market with Stochastic Volatility, Cambridge University Press, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. B. Kolmanovskiĭ and T. L. Maizenberg, Optimal control of stochastic systems with aftereffect, Avtomat. i Telemeh, 1 (1973), 47-61.  Google Scholar

[20]

V. B. Kolmanovskiĭ and L. E. Shaĭkhet, Control of systems with aftereffect, American Mathematical Society, 157 (1996). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. Lindquist, On feedback control of linear stochastic systems, SIAM Journal on Control, 11 (1973), 323-343. doi: 10.1137/0311025.  Google Scholar

[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.  Google Scholar

[25]

S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman Publishing, Boston-London-Melbourne, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

T. Pang, Stochastic portfolio optimization with log utility, Int. J. Theor. Appl. Finance, 9 (2006), 869-887. doi: 10.1142/S0219024906003858.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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