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October  2014, 1(4): 639-659. doi: 10.3934/jdg.2014.1.639

A limit theorem for Markov decision processes

Mathias Staudigl 1,

1. 

Center for Mathematical Economics, University Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

Received  January 2014 Revised  June 2014 Published  November 2014

In this paper I prove a deterministic approximation theorem for a sequence of Markov decision processes with finitely many actions and general state spaces as they appear frequently in economics, game theory and operations research. Using viscosity solution methods no a-priori differentiabililty assumptions are imposed on the value function.
Keywords: Markov decision processes, optimal control, viscosity solutions, weak convergence methods..
Mathematics Subject Classification: Primary: 49L20, 49L25, 90C40; Secondary: 60J20, 60F1.
Citation: Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics & Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639
References:
[1]

R. J. Aumann and M. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, MA, 1995.  Google Scholar

[2]

M. Bardi and I. Capuzzo-Dolcetta, Optimal control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser - Systems & Control: Foundations & Applications, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[3]

D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete Time Case, Academic Press, 1978.  Google Scholar

[4]

P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[5]

I. Capuzzo-Dolcetta and H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory, Applied Mathematics & Optimization, 11 (1984), 161-181. doi: 10.1007/BF01442176.  Google Scholar

[6]

P. Cardaliaguet, C. Rainer, D. Rosenberg and N. Vieille, Markov games with frequent actions and incomplete information,, 2013. arXiv:1307.3365v1 [math.OC]., ().   Google Scholar

[7]

N. El Karoui, D. Nguyen and M. Jeanblanc-Picqué, Compactification methods in the control of degenerate diffusions: Existence of an optimal control, Stochastics, 20 (1987), 169-219. doi: 10.1080/17442508708833443.  Google Scholar

[8]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.  Google Scholar

[9]

M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Applied Mathematics & Optimization, 15 (1987), 1-13. doi: 10.1007/BF01442644.  Google Scholar

[10]

W. H. Fleming, The convergence problem for differential games, Journal of Mathematical Analysis and Applications, 3 (1961), 102-116. doi: 10.1016/0022-247X(61)90009-9.  Google Scholar

[11]

N. Gast, B. Gaujal and J.-Y. Le Boudec, Mean field for markov decision processes: From discrete to continuous optimization, IEEE Transactions on Automatic Control, 57 (2012), 2266-2280. doi: 10.1109/TAC.2012.2186176.  Google Scholar

[12]

F. Gensbittel, Continuous-time Limit of Dynamic Games with Incomplete Information and a More Informed Player, hal-00910970, version 1, 2013. Google Scholar

[13]

R. Gonzalez and E. Rofman, On deterministic control problems: An approximation procedure for the optimal cost I. the stationary problem, SIAM Journal on Control and Optimization, 23 (1985), 242-266. doi: 10.1137/0323018.  Google Scholar

[14]

X. Guo and O. Hernández-Lerma, Nonzero-sum games for continuous-time markov chains with unbounded discounted payoffs, Journal of Applied Probability, 42 (2005), 303-320. doi: 10.1239/jap/1118777172.  Google Scholar

[15]

O. Hernández-Lerma and J. B. Laserre, Discrete-Time Markov Control Processes: Basic Optimality criteria, Springer-Verlag, 1996. doi: 10.1007/978-1-4612-0729-0.  Google Scholar

[16]

J. Hörner, T. Sugaya, S. Takahashi and N. Vieille, Recursive methods in discounted stochastic games: An algorithm for $\delta\rightarrow 1$ and a folk theorem, Econometrica, 79 (2011), 1277-1318. doi: 10.3982/ECTA9004.  Google Scholar

[17]

O. Kallenberg, Foundations of Modern Probability, Springer, New York [u.a.], 2nd edition, 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[18]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, 2nd edition, 2000. Google Scholar

[19]

H. J. Kushner, Weak convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser - Systems & Control: Foundations & Applications, Boston, MA, 1990. doi: 10.1007/978-1-4612-4482-0.  Google Scholar

[20]

H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, New York, 2nd edition, 2001. doi: 10.1007/978-1-4613-0007-6.  Google Scholar

[21]

A. Neyman, Stochastic games with short-stage duration, Dynamic Games and Applications, 3 (2013), 236-278. doi: 10.1007/s13235-013-0083-x.  Google Scholar

[22]

A. S. Nowak and T. E. S. Raghavan, Existence of stationary correlated equilibria with symmetric information for discounted stochastic games, Mathematics of Operations Research, 17 (1992), 519-526. doi: 10.1287/moor.17.3.519.  Google Scholar

[23]

W. H. Sandholm and M. Staudigl, Stochastic stability in the small noise double limit, I: Theory, Unpublished manuscript, University of Wisconsin and Bielefeld University, 2014. Google Scholar

[24]

W. H. Sandholm and M. Staudigl, Stochastic stability in the small noise double limit, II: The logit model, Unpublished manuscript, University of Wisconsin and Bielefeld University, 2014. Google Scholar

[25]

Y. Sannikov, Games with imperfectly observable actions in continuous time, Econometrica, 75 (2007), 1285-1329. doi: 10.1111/j.1468-0262.2007.00795.x.  Google Scholar

[26]

S. Soulaimani, M. Quincampoix and S. Sorin, Repeated games and qualitative differential games: Approachability and comparison of strategies, SIAM Journal on Control and Optimization, 48 (2009), 2461-2479. doi: 10.1137/090749098.  Google Scholar

[27]

M. Staudigl, Stochastic stability in asymmetric binary choice coordination games, Games and Economic Behavior, 75 (2012), 372-401. doi: 10.1016/j.geb.2011.11.003.  Google Scholar

[28]

M. Staudigl and J.-H. Steg, On repeated games with imperfect public monitoring: From discrete to continuous time, Bielefeld University, unpublished manuscript, 2014. Google Scholar

[29]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York-London, 1972.  Google Scholar

show all references

References:
[1]

R. J. Aumann and M. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, MA, 1995.  Google Scholar

[2]

M. Bardi and I. Capuzzo-Dolcetta, Optimal control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser - Systems & Control: Foundations & Applications, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[3]

D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete Time Case, Academic Press, 1978.  Google Scholar

[4]

P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[5]

I. Capuzzo-Dolcetta and H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory, Applied Mathematics & Optimization, 11 (1984), 161-181. doi: 10.1007/BF01442176.  Google Scholar

[6]

P. Cardaliaguet, C. Rainer, D. Rosenberg and N. Vieille, Markov games with frequent actions and incomplete information,, 2013. arXiv:1307.3365v1 [math.OC]., ().   Google Scholar

[7]

N. El Karoui, D. Nguyen and M. Jeanblanc-Picqué, Compactification methods in the control of degenerate diffusions: Existence of an optimal control, Stochastics, 20 (1987), 169-219. doi: 10.1080/17442508708833443.  Google Scholar

[8]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.  Google Scholar

[9]

M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Applied Mathematics & Optimization, 15 (1987), 1-13. doi: 10.1007/BF01442644.  Google Scholar

[10]

W. H. Fleming, The convergence problem for differential games, Journal of Mathematical Analysis and Applications, 3 (1961), 102-116. doi: 10.1016/0022-247X(61)90009-9.  Google Scholar

[11]

N. Gast, B. Gaujal and J.-Y. Le Boudec, Mean field for markov decision processes: From discrete to continuous optimization, IEEE Transactions on Automatic Control, 57 (2012), 2266-2280. doi: 10.1109/TAC.2012.2186176.  Google Scholar

[12]

F. Gensbittel, Continuous-time Limit of Dynamic Games with Incomplete Information and a More Informed Player, hal-00910970, version 1, 2013. Google Scholar

[13]

R. Gonzalez and E. Rofman, On deterministic control problems: An approximation procedure for the optimal cost I. the stationary problem, SIAM Journal on Control and Optimization, 23 (1985), 242-266. doi: 10.1137/0323018.  Google Scholar

[14]

X. Guo and O. Hernández-Lerma, Nonzero-sum games for continuous-time markov chains with unbounded discounted payoffs, Journal of Applied Probability, 42 (2005), 303-320. doi: 10.1239/jap/1118777172.  Google Scholar

[15]

O. Hernández-Lerma and J. B. Laserre, Discrete-Time Markov Control Processes: Basic Optimality criteria, Springer-Verlag, 1996. doi: 10.1007/978-1-4612-0729-0.  Google Scholar

[16]

J. Hörner, T. Sugaya, S. Takahashi and N. Vieille, Recursive methods in discounted stochastic games: An algorithm for $\delta\rightarrow 1$ and a folk theorem, Econometrica, 79 (2011), 1277-1318. doi: 10.3982/ECTA9004.  Google Scholar

[17]

O. Kallenberg, Foundations of Modern Probability, Springer, New York [u.a.], 2nd edition, 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[18]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, 2nd edition, 2000. Google Scholar

[19]

H. J. Kushner, Weak convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser - Systems & Control: Foundations & Applications, Boston, MA, 1990. doi: 10.1007/978-1-4612-4482-0.  Google Scholar

[20]

H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, New York, 2nd edition, 2001. doi: 10.1007/978-1-4613-0007-6.  Google Scholar

[21]

A. Neyman, Stochastic games with short-stage duration, Dynamic Games and Applications, 3 (2013), 236-278. doi: 10.1007/s13235-013-0083-x.  Google Scholar

[22]

A. S. Nowak and T. E. S. Raghavan, Existence of stationary correlated equilibria with symmetric information for discounted stochastic games, Mathematics of Operations Research, 17 (1992), 519-526. doi: 10.1287/moor.17.3.519.  Google Scholar

[23]

W. H. Sandholm and M. Staudigl, Stochastic stability in the small noise double limit, I: Theory, Unpublished manuscript, University of Wisconsin and Bielefeld University, 2014. Google Scholar

[24]

W. H. Sandholm and M. Staudigl, Stochastic stability in the small noise double limit, II: The logit model, Unpublished manuscript, University of Wisconsin and Bielefeld University, 2014. Google Scholar

[25]

Y. Sannikov, Games with imperfectly observable actions in continuous time, Econometrica, 75 (2007), 1285-1329. doi: 10.1111/j.1468-0262.2007.00795.x.  Google Scholar

[26]

S. Soulaimani, M. Quincampoix and S. Sorin, Repeated games and qualitative differential games: Approachability and comparison of strategies, SIAM Journal on Control and Optimization, 48 (2009), 2461-2479. doi: 10.1137/090749098.  Google Scholar

[27]

M. Staudigl, Stochastic stability in asymmetric binary choice coordination games, Games and Economic Behavior, 75 (2012), 372-401. doi: 10.1016/j.geb.2011.11.003.  Google Scholar

[28]

M. Staudigl and J.-H. Steg, On repeated games with imperfect public monitoring: From discrete to continuous time, Bielefeld University, unpublished manuscript, 2014. Google Scholar

[29]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York-London, 1972.  Google Scholar

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