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Optimal control of a Brownian storage system


  • Harrison, J. Michael
  • Taylor, Allison J.


Consider a storage system (such as an inventory or bank account) whose content fluctuates as a Brownian Motion X = {X(t), t [greater-or-equal, slanted] 0} in the absence of any control. Let Y = {Y(t), t [greater-or-equal, slanted] 0} and Z = {Z(t), t [greater-or-equal, slanted] 0} be non-decreasing, non-anticipating functionals representing the cumulative input to the system and cumulative output from the system respectively. Theproblem is to choose Y and Z so as to minimize expected discounted cost subject to the requirement that X(t) + Y(t) - Z(t) [greater-or-equal, slanted] 0 for all t [greater-or-equal, slanted] 0 almost surely. In our first formulation, we assume a proportional input cost, a linear holding cost, and a proportional output reward (or cost). We explicitly compute an optimal policy involving a single critical number. In our second formulation, the cumulative input Y is required to be a step function, and an additional fixed charge is incurred each time that an input jump occurs. We explicitly compute an optimal policy involving two critical numbers. Applications to inventory/production control and stochastic cash management are discussed.

Suggested Citation

  • Harrison, J. Michael & Taylor, Allison J., 1978. "Optimal control of a Brownian storage system," Stochastic Processes and their Applications, Elsevier, vol. 6(2), pages 179-194, January.
  • Handle: RePEc:eee:spapps:v:6:y:1978:i:2:p:179-194

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    Cited by:

    1. Peng, Xiaofan & Chen, Mi & Guo, Junyi, 2012. "Optimal dividend and equity issuance problem with proportional and fixed transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 576-585.
    2. Joachim Gahungu and Yves Smeers, 2012. "A Real Options Model for Electricity Capacity Expansion," RSCAS Working Papers 2012/08, European University Institute.
    3. Shah, Sudhir A., 2005. "Optimal management of durable pollution," Journal of Economic Dynamics and Control, Elsevier, vol. 29(6), pages 1121-1164, June.
    4. Shah, Sudhir A., 1996. "Controlling inventory when prices fluctuate randomly," Journal of Economic Dynamics and Control, Elsevier, vol. 20(1-3), pages 145-171.
    5. Sudhir A. Shah, 2003. "Optimal Management of Durable Pollution," Working papers 113, Centre for Development Economics, Delhi School of Economics.
    6. Grossman, Sanford J & Laroque, Guy, 1990. "Asset Pricing and Optimal Portfolio Choice in the Presence of Illiquid Durable Consumption Goods," Econometrica, Econometric Society, vol. 58(1), pages 25-51, January.
    7. Sudhir A. Shah, 2000. "Optimal Pollution Regulation in a Dynamic Stochastic Model," Working papers 84, Centre for Development Economics, Delhi School of Economics.

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