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Controlled jump processes

Author

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  • Pliska, Stanley R.

Abstract

Finite and infinite planning horizon Markov decision problems are formulated for a class of jump processes with general state and action spaces and controls which are measurable functions on the time axis taking values in an appropriate metrizable vector space. For the finite horizon problem, the maximum expected reward is the unique solution, which exists, of a certain differential equation and is a strongly continuous function in the space of upper semi-continuous functions. A necessary and sufficient condition is provided for an admissible control to be optimal, and a sufficient condition is provided for the existence of a measurable optimal policy. For the infinite horizon problem, the maximum expected total reward is the fixed point of a certain operator on the space of upper semi-continuous functions. A stationary policy is optimal over all measurable policies in the transient and discounted cases as well as, with certain added conditions, in the positive and negative cases.

Suggested Citation

  • Pliska, Stanley R., 1975. "Controlled jump processes," Stochastic Processes and their Applications, Elsevier, vol. 3(3), pages 259-282, July.
    • Handle: RePEc:eee:spapps:v:3:y:1975:i:3:p:259-282
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    Cited by:

    1. Qingda Wei, 2017. "Finite approximation for finite-horizon continuous-time Markov decision processes," 4OR, Springer, vol. 15(1), pages 67-84, March.
    2. Bandini, Elena & Fuhrman, Marco, 2017. "Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1441-1474.
    3. Naveed Chehrazi & Peter W. Glynn & Thomas A. Weber, 2019. "Dynamic Credit-Collections Optimization," Management Science, INFORMS, vol. 67(6), pages 2737-2769, June.

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