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A consumption-investment problem modelled as a discounted Markov decision process. (English) Zbl 1241.93053
Summary: In this paper a problem of consumption and investment is presented as a model of a discounted Markov decision process with discrete-time. In this problem, it is assumed that the wealth is affected by a production function. This assumption gives the investor a chance to increase his wealth before the investment. For the solution of the problem there is established a suitable version of the Euler Equation (EE) which characterizes its optimal policy completely, that is, there are provided conditions which guarantee that a policy is optimal for the problem if and only if it satisfies the EE. The problem is exemplified in two particular cases: for a logarithmic utility and for a Cobb-Douglas utility. In both cases, explicit formulas for the optimal policy and the optimal value function are supplied.

93E12 Identification in stochastic control theory
62M02 Markov processes: hypothesis testing
91B42 Consumer behavior, demand theory
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[1] Aliprantis, C. D., Burkinshaw, O.: Principles of Real Analysis. Academic Press, San Diego 1998. · Zbl 1006.28001
[2] Angelatos, G. M.: Uninsured idiosyncratic investment risk and aggregate saving. Rev. Econom. Dynam. 10 (2007), 1-30.
[3] Arrow, K. J.: A note on uncertainty and discounting in models of economic growth. J. Risk Unc. 38 (2009), 87-94. · Zbl 1166.91321
[4] Bertsekas, D. P.: Dynamic Programming: Deterministic and Stochastic Models. Prentice-Hall, Belmont 1987. · Zbl 0649.93001
[5] Brock, W., Mirman, L.: Optimal economic growth and uncertainty: the discounted case. J. Econom. Theory 4 (1972), 479-513.
[6] Cruz-Suárez, D., Montes-de-Oca, R., Salem-Silva, F.: Conditions for the uniqueness of optimal policies of discounted Markov decision processes. Math. Meth. Oper. Res. 60 (2004), 415-436. · Zbl 1104.90053
[7] Cruz-Suárez, H., Montes-de-Oca, R.: Discounted Markov control processes induced by deterministic systems. Kybernetika 42 (2006), 647-664. · Zbl 1249.90312
[8] Cruz-Suárez, H., Montes-de-Oca, R.: An envelope theorem and some applications to discounted Markov decision processes. Math. Meth. Oper. Res. 67 (2008), 299-321. · Zbl 1149.90171
[9] Dynkin, E. B., Yushkevich, A. A.: Controlled Markov Processes. Springer-Verlag, New York 1980. · Zbl 0073.34801
[10] Epstein, L., Zin, S.: Substitution, risk aversion, and the temporal behaviour of consumption and asset returns I: Theoretical framework. Econometrica 57 (1989), 937-969. · Zbl 0683.90012
[11] Fuente, A. De la: Mathematical Methods and Models for Economists. Cambridge University Press, Cambridge 2000. · Zbl 0943.91001
[12] Gurkaynak, R. S.: Econometric tests of asset price bubbles: taking stock. J. Econom. Surveys 22 (2008), 166-186.
[13] Heer, B., Maussner, A.: Dynamic General Equilibrium Modelling: Computational Method and Application. Second edition, Springer-Verlag, Berlin 2005.
[14] Hernández-Lerma, O., Lasserre, J. B.: Discrete-Time Markov Control Processes: Basic Optimality Criteria. Springer-Verlag, New York 1996. · Zbl 0840.93001
[15] Hernández-Lerma, O., Lasserre, J. B.: Value iteration and rolling plans for Markov control processes with unbounded rewards. J. Math. Anal. Appl. 177 (1993), 38-55. · Zbl 0781.90093
[16] Jaskiewics, A., Nowak, A. S.: Discounted dynamic programming with unbounded returns: application to economic models. J. Math. Anal. Appl. 378 (2011), 450-462. · Zbl 1254.90292
[17] Korn, R., Kraft, H.: A stochastic control approach to portfolio problems with stochastic interest rates. SIAM J. Control Optim. 40 (2001), 1250-1269. · Zbl 1020.93029
[18] Kamihigashi, T.: Stochastic optimal growth with bounded or unbounded utility and bounded or unbounded shocks. J. Math. Econom. 43 (2007), 477-500. · Zbl 1154.91032
[19] Levhari, D., Srinivasan, T. N.: Optimal savings under uncertainty. Rev. Econom. Stud. 36 (1969), 153-163.
[20] Mirman, L., Zilcha, I.: On optimal growth under uncertainty. J. Econom. Theory 2 (1975), 329-339. · Zbl 0362.90024
[21] Ramsey, F. P.: A Mathematical theory of saving. Econom. J. 38 (1928), 543-559.
[22] Stokey, N., Lucas, R., Prescott, E.: Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge 1989.
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