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Martingale calculus and a maximal inequality for supermartingales

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In the first hour of this two-part presentation, the calculus of semimartingales, which includes martingales with both continuous and discrete compotents, will be reviewed. In the second hour of the presentation, a tight upper bound is given involving the maximum of a supermartingale. Specifically, it is shown that if Y is a semimartingale with initial value zero and quadratic variation process [Y, Y] such that Y + [Y, Y] is a supermartingale, then the probability the maximum of Y is greater than or equal to a positive constant is less than or equal to 1/(1+a). The proof uses the semimartingale calculus and is inspired by dynamic programming. If Y has stationery independent increments, the bounds of JFC Kingman apply to this situation. Complements and extensions will also be given.

This talk is part of the Optimization and Incentives Seminar series.

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