Notes on Martingale Theory
UNIVERSITY OF ILLINOIS URBANA United States
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Although several writers, for example Bernisteini, Levy, and Ville, had used what would now be identified as martingale concepts, the first systematic studies appeared in 2 and 3. Since then, martingale theory has been applied extensively, but little progress has been made in the theory itself. The purpose of the present paper is to point out how much spade work remains to be done in the theory, by deriving new theorems without the use of deep technical apparatus. Throughout this paper, the more appropriate nomenclature submartingale,supermartingale is used, rather than the semimartingale, lower semimartingale found in 3. The unifying thread in the following work will be the fact that certain simple operations on submartingales transform them into submartingales. This leads to a new submartingale convergence theorem, to a sharpening of the upcrossing inequality, and thereby into an examination of apparently hitherto unnoticed interrelations between martingale and potential theory.