Exercises in Analysis by Leszek Gasińksi & Nikolaos S. Papageorgiou

Author:Leszek Gasińksi & Nikolaos S. Papageorgiou
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

If the index in a martingale (respectively, submartingale, supermartingale) is replaced by two stopping times, does the main defining the process equality (respectively, inequality) remain true? The answer is “yes”, provided that two stopping times are bounded.

Theorem 3.173 (Stopping Time Theorem)

is a martingale (respectively, submartingale, supermartingale) and σ,τ are two bounded stopping times, (respectively , ). Hence

for every bounded stopping time σ.

Remark 3.174.

In this theorem, the assumption of the boundedness of the stopping times is essential and cannot be removed without additional hypotheses.

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