Abstract

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.