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An Introduction to Singular Stochastic Control

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Stochastic Differential Systems, Stochastic Control Theory and Applications

Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 10))

Abstract

Bounded variation stochastic control may be defined to include any stochastic control problem in which one restricts the cumulative displacement of the state caused by control to be of bounded variation on finite time intervals. In classical control problems, this cumulative displacement is the integral of the control process (or some function of it), and so is absolutely continuous. In impulse control (see Bensoussan & Lions (1978)), this cumulative displacement has jumps, between which it is either constant or absolutely continuous. Bounded variation control admits both these possibilities and also the possibility that the displacement of the state caused by the optimal control is singularly continuous, at least with positive probability over some interval of time. Problems which exhibit this feature will be called singular, and these are the objects of interest of the present paper.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Steven E. Shreve

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  1. Steven E. Shreve
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Editor information

Editors and Affiliations

  1. Division of Applied Mathematics, Brown University, 02912, Providence, Rhode Island, USA

    Wendell Fleming

  2. Ceremade, Universite Paris-Dauphine, Place de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Pierre-Louis Lions

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© 1988 Springer-Verlag New York Inc.

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Shreve, S.E. (1988). An Introduction to Singular Stochastic Control. In: Fleming, W., Lions, PL. (eds) Stochastic Differential Systems, Stochastic Control Theory and Applications. The IMA Volumes in Mathematics and Its Applications, vol 10. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8762-6_30

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  • DOI: https://doi.org/10.1007/978-1-4613-8762-6_30

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8764-0

  • Online ISBN: 978-1-4613-8762-6

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