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Derivative Estimates from Simulation of Continuous-Time Markov Chains

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  • Paul Glasserman

    (Columbia University, New York, New York)

Abstract

Countable-state, continuous-time Markov chains are often analyzed through simulation when simple analytical expressions are unavailable. Simulation is typically used to estimate costs or performance measures associated with the chain and also characteristics like state probabilities and mean passage times. Here we consider the problem of estimating derivatives of these types of quantities with respect to a parameter of the process. In particular, we consider the case where some or all transition rates depend on a parameter. We derive derivative estimates of the infinitesimal perturbation analysis type for Markov chains satisfying a simple condition, and argue that the condition has significant scope. The unbiasedness of these estimates may be surprising—a “naive” estimator would fail in our setting. What makes our estimates work is a special construction of specially structured parameteric families of Markov chains. In addition to proving unbiasedness, we consider a variance reduction technique and make comparisions with derivative estimates based on likelihood ratios.

Suggested Citation

  • Paul Glasserman, 1992. "Derivative Estimates from Simulation of Continuous-Time Markov Chains," Operations Research, INFORMS, vol. 40(2), pages 292-308, April.
    • Handle: RePEc:inm:oropre:v:40:y:1992:i:2:p:292-308
    DOI: 10.1287/opre.40.2.292
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    Cited by:

    1. Do Van, Phuc & Barros, Anne & Bérenguer, Christophe, 2008. "Reliability importance analysis of Markovian systems at steady state using perturbation analysis," Reliability Engineering and System Safety, Elsevier, vol. 93(11), pages 1605-1615.
    2. Özdemir, Deniz & Yücesan, Enver & Herer, Yale T., 2013. "Multi-location transshipment problem with capacitated production," European Journal of Operational Research, Elsevier, vol. 226(3), pages 425-435.
    3. Ozdemir, Deniz & Yucesan, Enver & Herer, Yale T., 2006. "Multi-location transshipment problem with capacitated transportation," European Journal of Operational Research, Elsevier, vol. 175(1), pages 602-621, November.
    4. Trueck, Stefan & Rachev, Svetlozar T., 2008. "Rating Based Modeling of Credit Risk," Elsevier Monographs, Elsevier, edition 1, number 9780123736833.
    5. Gong, Y. & Yucesan, E., 2006. "The Multi-Location Transshipment Problem with Positive Replenishment Lead Times," ERIM Report Series Research in Management ERS-2006-048-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.

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