Skip to main content Accessibility help
×
Home

Blackwell Optimality for Controlled Diffusion Processes

  • Héctor Jasso-Fuentes (a1) and Onésimo Hernández-Lerma (a1)

Abstract

In this paper we study m-discount optimality (m ≥ −1) and Blackwell optimality for a general class of controlled (Markov) diffusion processes. To this end, a key step is to express the expected discounted reward function as a Laurent series, and then search certain control policies that lexicographically maximize the mth coefficient of this series for m = −1,0,1,…. This approach naturally leads to m-discount optimality and it gives Blackwell optimality in the limit as m → ∞.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Blackwell Optimality for Controlled Diffusion Processes
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Blackwell Optimality for Controlled Diffusion Processes
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Blackwell Optimality for Controlled Diffusion Processes
      Available formats
      ×

Copyright

Corresponding author

Postal address: Department of Mathematics, CINVESTAV-IPN, A. Postal 14-740, Mexico DF 07000, Mexico.
∗∗ Email address: hjasso@math.cinvestav.mx
∗∗∗ Email address: ohernand@math.cinvestav.mx

References

Hide All
[1] Akella, R. and Kumar, P. R. (1986). Optimal control of production rate in a failure prone manufacturing system. IEEE Trans. Automatic Control 31, 116126.
[2] Arapostathis, A., Ghosh, M. K. and Borkar, V. S. (2009). Ergodic Control of Diffusion Processes. To appear.
[3] Blackwell, D. (1962). Discrete dynamic programming. Ann. Math. Statist. 33, 719726.
[4] Borkar, V. S. and Ghosh, M. K. (1990). Ergodic control of multidimensional diffusions. II. Adaptive control. Appl. Math. Optimization 21, 191220.
[5] Dekker, R. and Hordijk, A. (1992). Recurrence conditions for average and Blackwell optimality in denumerable state Markov decision chains. Math. Operat. Res. 17, 271289.
[6] Dynkin, E. B. (1965). Markov Processes, Vol. 1. Springer, Berlin.
[7] Fort, G. and Roberts, G. O. (2005). Subgeometric ergodicity of strong Markov processes. Ann. Appl. Prob. 15, 15651589.
[8] Ghosh, M. K. and Marcus, S. I. (1991). Infinite horizon controlled diffusion problems with nonstandard criteria. J. Math. Systems Estim. Control 1, 4569.
[9] Ghosh, M. K., Arapostathis, A. and Marcus, S. I. (1993). Optimal control of switching diffusions with application to flexible manufacturing systems. SIAM J. Control Optimization 31, 11831204.
[10] Ghosh, M. K., Arapostathis, A. and Marcus, S. I. (1997). Ergodic control of switching diffusions. SIAM J. Control Optimization 35, 19521952.
[11] Glynn, P. W. and Meyn, S. P. (1996). A Liapounov bound for solutions of the Poisson equation. Ann. Prob. 24, 916931.
[12] Has'minskii, R. Z. (1980). Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Germantown, Md.
[13] Hernández-Lerma, O. (1994). Lectures on Continuous-Time Markov Control Processes. Sociedad Matemática Mexicana, Mexico.
[14] Hernández-Lerma, O. and Lasserre, J. B. (1999). Further Topics on Discrete-Time Markov Control Processes (Appl. Math. 42). Springer, New York.
[15] Hilgert, N. and Hernández-Lerma, O. (2003). Bias optimality versus strong 0-discount optimality in Markov control processes with unbounded costs. Acta Appl. Math. 77, 215235.
[16] Hordijk, A. and Yushkevich, A. A. (2002). Blackwell optimality. In Handbook of Markov Decision Processes (Internat. Ser. Operat. Res. Manag. Sci. 40), eds Feinberg, E. A. and Shwartz, A., Kluwer, Boston, MA, pp. 231267.
[17] Jasso-Fuentes, H. (2007). Infinite-horizon optimal control problems for Markov diffusion processes. , Mathematics Department, CINVESTAV-IPN.
[18] Jasso-Fuentes, H. and Hernández-Lerma, O. (2008). Characterizations of overtaking optimality for controlled diffusion processes. Appl. Math. Optimization 57, 349369.
[19] Jasso-Fuentes, H. and Hernández-Lerma, O. (2009). Ergodic control, bias, and sensitive discount optimality for Markov diffusion processes. Stoch. Anal. Appl. 27, 363385.
[20] Leizarowitz, A. (1988). Controlled diffusion processes on infinite horizon with the overtaking criterion. Appl. Math. Optimization 17, 6178.
[21] Leizarowitz, A. (1990). Optimal controls for diffusion in {R}d—min-max max-min formula for the minimal cost growth rate. J. Math. Anal. Appl. 149, 180209.
[22] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time precesses. Adv. Appl. Prob. 25, 518548.
[23] Prieto-Rumeau, T. (2006). Blackwell optimality in the class of Markov policies for continuous-time controlled Markov chains. Acta Appl. Math. 92, 7796.
[24] Prieto-Rumeau, T. and Hernandez-Lerma, O. (2005). Bias and overtaking equilibria for zero-sum continuous-time Markov games. Math. Meth. Operat. Res. 61, 437454.
[25] Prieto-Rumeau, T. and Hernandez-Lerma, O. (2005). The Laurent series, sensitive discount and Blackwell optimality for continuous-time controlled Markov chains. Math. Meth. Operat. Res. 61, 123145.
[26] Prieto-Rumeau, T. and Hernández-Lerma, O. (2006). Bias optimality for continuous-time controlled Markov chains. SIAM J. Control Optimization 45, 5173.
[27] Puterman, M. L. (1974). Sensitive discount optimality in controlled one-dimensional diffusions. Ann. Prob. 2, 408419.
[28] Taylor, H. M. (1976). A Laurent series for the resolvent of a strongly continuous stochastic semi-group. Math. Program. 6, 258263.
[29] Veinott, A. F. Jr. (1969). Discrete dynamic programming with sensitive discount optimality criteria. Ann. Math. Statist. 40, 16351660.
[30] Veretennikov, A. Y. and Klokov, S. A. (2005). On the subexponential rate of mixing for Markov processes. Theory Prob. Appl. 49, 110122.
[31] Yosida, K. (1995). Functional Analysis (Reprint). Springer, Berlin.
[32] Zhu, Q. and Guo, X. (2005). Another set of conditions for strong n (n=−1,0) discount optimality in Markov decision processes. Stoch. Anal. Appl. 23, 953974.

Keywords

MSC classification

Related content

Powered by UNSILO

Blackwell Optimality for Controlled Diffusion Processes

  • Héctor Jasso-Fuentes (a1) and Onésimo Hernández-Lerma (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.