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On the value of a stopped set function process

Author

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  • Schmidt, Klaus D.

Abstract

For certain types of stochastic processes {Xn n [set membership, variant] }, which are integrable and adapted to a nondecreasing sequence of [sigma]-algebras n on a probability space ([Omega], , P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {n n [set membership, variant] }, when issups [integral operator][Omega] X[sigma] dP finite, and when is there a stopping time for which this supremum is attained? In the present paper we set the problem in a measure theoretic framework. This approach turns out to be fruitful since it reveals the root of the problem: It avoids the use of such notions as probability, null set, integral, and even [sigma]-additivity. It thus allows a considerable generalization of known results, simplifies proofs, and opens the door to further research.

Suggested Citation

  • Schmidt, Klaus D., 1980. "On the value of a stopped set function process," Journal of Multivariate Analysis, Elsevier, vol. 10(1), pages 123-134, March.
    • Handle: RePEc:eee:jmvana:v:10:y:1980:i:1:p:123-134
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