Matkowski, Janusz and Nowak, Andrzej S. (2008): On Discounted Dynamic Programming with Unbounded Returns.

PDF
MPRA_paper_12215.pdf Download (167kB)  Preview 
Abstract
In this paper, we apply the idea of $k$local contraction of \cite{zec, zet} to study discounted stochastic dynamic programming models with unbounded returns. Our main results concern the existence of a unique solution to the Bellman equation and are applied to the theory of stochastic optimal growth. Also a discussion of some subtle issues concerning klocal and global contractions is included.
Item Type:  MPRA Paper 

Original Title:  On Discounted Dynamic Programming with Unbounded Returns 
Language:  English 
Keywords:  Stochastic dynamic programming, Bellman functional equation, contraction mapping, stochastic optimal growth 
Subjects:  D  Microeconomics > D9  Intertemporal Choice > D90  General D  Microeconomics > D9  Intertemporal Choice > D91  Intertemporal Household Choice ; Life Cycle Models and Saving C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61  Optimization Techniques ; Programming Models ; Dynamic Analysis 
Item ID:  12215 
Depositing User:  Andrzej Nowak 
Date Deposited:  16. Dec 2008 18:34 
Last Modified:  09. Jan 2014 11:46 
References:  Berge, C.(1963) Topological Spaces. MacMillan, New York. Bertsekas, D.P., Shreve, S.E.(1978) Stochastic Optimal Control: the Discrete Time Case. Academic Press, New York Blackwell, D. (1965) Discounted dynamic programming. Annals of Mathematical Statistics 36: 226235. Boyd III, J.H. (1990) Recursive utility and the Ramsey problem. Journal of Economic Theory 50: 326345. Boyd III, J.H., Becker, R.A. (1997) Capital Theory, Equilibrium Analysis and Recursive Utility. Blackwell Publishers, New York. Brown, L.D., Purves, R.(1973) Measurable selections of extrema. Annals of Statistics 1: 902912. Brock, W.A., Mirman, L.J.(1972) Optimal economic growth and uncertainty: the discounted case. Journal of Economic Theory 4: 479513. Dana, R.A., Le Van, C., Mitra, T., Nishimura, K., (Eds) (2006) Handbook of Optimal Growth 1. Springer, Berlin. Dutta, P.K., Mitra, T. (1989). On continuity of the utility function in intertemporal allocation models: an example. International Economic Review 30: 527536. Filipe MartinsdaRocha, V., Vailakis, Y. (2008) Existence and uniqueness of fixedpoint for local contractions. Personal communication, submitted for publication. HernandezLerma, O., Lasserre, J.B. (1999) Further Topics on DiscreteTime Markov Control Processes. SpringerVerlag, New York. Himmelberg, C.J., (1975) Measurable relations. Fundamenta Mathematicae 87: 5372. Kuratowski, K., RyllNardzewski, C. (1965) A general theorem on selectors. Bulletin de l'Academie Polonaise des Sciences (Ser. Mathematique) 13: 397403. Le Van, C., Morhaim, L. (2002) Optimal growth models with bounded or unbounded returns: a unifying approach. Journal of Economic Theory 105: 158187. Le Van, C., Vailakis, Y. (2005) Recursive utility and optimal growth with bounded or unbounded returns. Journal of Economic Theory 123: 18720 Neveu, J. (1965) Mathematical Foundations of the Calculus of Probability. HoldenDay, San Francisco. Nowak, A.S. (1984) On zerosum stochastic games with general state space I. Probability and Mathematical Statistics 4: 1332. Nowak, A.S. (1985) Universally measurable strategies in zerosum stochastic games. Annals of Probability 13: 269287. Puterman, M. (2005) Markov Decision Processes: Discrete Stochastic Dynamic Programming. WileyInterscience, New York. Rinc\'{o}nZapatero, J. P., RodriguesPalmero, C. (2003) Existence and uniqueness of solutions to the Bellman equation in the unbounded case. Econometrica 71: 15191555. Rinc\'{o}nZapatero, J. P., RodriguesPalmero, C. (2007) Recursive utility with unbounded aggregators. Economic Theory 33: 381391. Schal, M. (1975) Conditions for optimality in dynamic programming and for the limit of nstage optimal policies to be optimal. Zeischrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 32: 179196. Stokey, N.L., Lucas, R.E. with Prescott, E. (1989) Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge, MA. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/12215 