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An infinite time horizon portfolio optimization model with delays. (English) Zbl 1348.91259

Summary: 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.

MSC:

91G10 Portfolio theory
49L20 Dynamic programming in optimal control and differential games
93E20 Optimal stochastic control
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[1] T. R. Bielecki, Risk sensitive dynamic asset management,, Appl. Math. and Optimization, 39, 337 (1999) · Zbl 0984.91047 · doi:10.1007/s002459900110
[2] M. H. Chang, Finite difference approximations for stochastic control systems with delay,, Stoch. Anal. Appl., 26, 451 (2008) · Zbl 1147.93027 · doi:10.1080/07362990802006980
[3] M. H. Chang, Optimal control of stochastic functional differential equations with a bounded memory,, Stochastics: An International Journal of Probability and Stochastic Processes, 80, 69 (2008) · Zbl 1138.93065 · doi:10.1080/17442500701605494
[4] M. H. Chang, A stochastic portfolio optimization model with bounded memory,, Mathematics of Operations Research, 36, 604 (2011) · Zbl 1244.34101 · doi:10.1287/moor.1110.0508
[5] R. Cont, Functional Ito Calculus and Stochastic Integral Representation of Martingales,, The Annuals of Probability, 41, 109 (2013) · Zbl 1272.60031 · doi:10.1214/11-AOP721
[6] B. Dupire, Functional Itô’s Calculus,, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS, 2009
[7] I. Elsanousi, Some solvable stochastic control problems with delay,, Stochastics and Stochastics Reports, 71, 69 (2000) · Zbl 0999.93072 · doi:10.1080/17442500008834259
[8] S. Federico, 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, 4910 (2010) · Zbl 1208.49048 · doi:10.1137/09076742X
[9] S. Federico, A stochastic control problem with delay arising in a pension fund model,, Finance and Stochastics, 15, 421 (2011) · Zbl 1302.93238 · doi:10.1007/s00780-010-0146-4
[10] W. H. Fleming, An optimal consumption model with stochastic volatility,, Finance and Stochastics, 7, 245 (2003) · Zbl 1035.60028 · doi:10.1007/s007800200083
[11] W. H. Fleming, An application of stochastic control theory to financial economics,, SIAM J. Control Optimiz., 43, 502 (2004) · Zbl 1101.93085 · doi:10.1137/S0363012902419060
[12] W. H. Fleming, A stochastic control model of investment, production and consumption,, Quarterly of Applied Mathematics, 63, 71 (2005) · Zbl 1080.91031 · doi:10.1090/S0033-569X-04-00941-1
[13] W. H. Fleming, <em>Deterministic and Stochastic Optimal Control</em>,, Springer (1975) · Zbl 0323.49001 · doi:10.1007/978-1-4612-6380-7
[14] W. H. Fleming, Risk-sensitive control and optimal investment model,, Mathematical Finance, 10, 197 (2000) · Zbl 1039.93069 · doi:10.1111/1467-9965.00089
[15] J. P. Fouque, <em>Derivatives in Financial Market with Stochastic Volatility</em>,, Cambridge University Press (2000) · Zbl 0954.91025
[16] F. Gozzi, Stochastic optimal control of delay equations arising in advertising models,, Stochastic Partial Differential Equations and Applications-VII, 245, 133 (2006) · Zbl 1107.93035 · doi:10.1201/9781420028720.ch13
[17] F. Gozzi, On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects,, Journal of Optimization Theory and Applications, 142, 291 (2009) · Zbl 1175.90245 · doi:10.1007/s10957-009-9524-5
[18] A, Optimal control of linear stochastic systems described by functional differential equations,, Journal of Optimization Theory and Applications, 9, 161 (1972) · Zbl 0228.93030 · doi:10.1007/BF00932588
[19] V. B. Kolmanovskiĭ, Optimal control of stochastic systems with aftereffect,, Avtomat. i Telemeh, 1, 47 (1973) · Zbl 0274.93068
[20] V. B. Kolmanovskiĭ, Control of systems with aftereffect,, American Mathematical Society, 157 (1996) · Zbl 0937.93001
[21] B. Larssen, Dynamic programming in stochastic control of systems with delay,, Stochastics: An International Journal of Probability and Stochastic Processes, 74, 651 (2002) · Zbl 1022.34075 · doi:10.1080/1045112021000060764
[22] B. Larssen, When are HJB-equations in stochastic control of delay systems finite dimensional?,, Stochastic Analysis and Applications, 21, 643 (2003) · Zbl 1052.60051 · doi:10.1081/SAP-120020430
[23] A. Lindquist, On feedback control of linear stochastic systems,, SIAM Journal on Control, 11, 323 (1973) · Zbl 0256.93065 · doi:10.1137/0311025
[24] A. Lindquist, Optimal control of linear stochastic systems with applications to time lag systems,, Information Sciences, 5, 81 (1973) · Zbl 0268.93034 · doi:10.1016/0020-0255(73)90005-4
[25] S. E. A. Mohammed, <em>Stochastic Functional Differential Equations</em>,, Pitman Publishing (1984) · Zbl 0584.60066
[26] S. E. A. Mohammed, Stochastic differential equations with memory- theory, examples and applications,, Stochastic Analysis and Related Topics VI, 42, 1 (1998) · Zbl 0901.60030
[27] T. Pang, Portfolio optimization models on infinite-time horizon,, J. Optim. Theory Appl., 122, 573 (2004) · Zbl 1082.91051 · doi:10.1023/B:JOTA.0000042596.26927.2d
[28] T. Pang, Stochastic portfolio optimization with log utility,, Int. J. Theor. Appl. Finance, 9, 869 (2006) · Zbl 1138.91468 · doi:10.1142/S0219024906003858
[29] T. Pang, 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), 159 (2015) · doi:10.1137/1.9781611974072.23
[30] J. Yong, <em>Stochastic Controls: Hamiltonian Systems and HJB Equations</em>,, Springer-Verlag (1999) · Zbl 0943.93002 · doi:10.1007/978-1-4612-1466-3
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