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Optimal Stopping Problems

  • Xiaoqiang Cai
  • Xianyi Wu
  • Xian Zhou
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 207)

Abstract

This chapter provides the foundations for the general theory of stochastic processes and optimal stopping problems. In Section 5.1, we elaborate on the concepts of s-algebras and information, probability spaces, uniform integrability, conditional expectations and essential supremum or infimum at an advanced level of probability theory. Then stochastic processes and the associated filtrations are introduced in Section 5.2, which intuitively explains the meaning of information flow. Section 5.3 deals with the concept of stopping times, with focus on the s-algebras at stopping times. Section 5.4 provides a concise introduction to the concept and fundamental results of martingales. The emphasis is focused on Doob’s stopping theorem and the convergence theorems of martingales, as well as their applications in studying the path properties of martingales. The materials in this chapter are essential in the context of stochastic controls, especially for derivation of dynamic policies.

Keywords

Stochastic Process Probability Space Conditional Expectation Path Property Uniform Integrability 
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References

  1. Dellacherie, C., & Meyer, P. (1978). Probabilities and potential A. Amsterdam/New York: North-Holland.Google Scholar
  2. Dellacherie, C., & Meyer, P. (1982). Probabilities and potential B: Theory of martingales. Amsterdam/Oxford: North-Holland.Google Scholar
  3. Karatzas, I., & Shreve, S. E. (1998). Methods of mathematical finance. New York: Springer.Google Scholar
  4. Peskir, G., & Shiryaev, A. N. (2006). Optimal stopping and free-boundary problems (Lectures in mathematics). Basel: ETH Zürich/Birkhäuser.Google Scholar
  5. Stoyanov, J. M. (1997). Counterexamples in probability (2nd ed.). Chichester/New York: Wiley.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Xiaoqiang Cai
    • 1
  • Xianyi Wu
    • 2
  • Xian Zhou
    • 3
  1. 1.Department of Systems Engineering and Engineering ManagementThe Chinese University of Hong KongShatin, N.T.Hong Kong SAR
  2. 2.Department of Statistics and Actuarial ScienceEast China Normal UniversityShanghaiPeople’s Republic of China
  3. 3.Department of Applied Finance and Actuarial StudiesMacquarie UniversityNorth RydeAustralia

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