A sharp bound on the expected number of upcrossings of an -bounded Martingale |
| |
Authors: | David Gilat Isaac Meilijson Laura Sacerdote |
| |
Institution: | 1. School of Mathematical Sciences, R. and B. Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel;2. Department of Mathematics G. Peano, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy |
| |
Abstract: | For a martingale starting at with final variance , and an interval , let be the normalized length of the interval and let be the normalized distance from the initial point to the lower endpoint of the interval. The expected number of upcrossings of by is at most if and at most otherwise. Both bounds are sharp, attained by Standard Brownian Motion stopped at appropriate stopping times. Both bounds also attain the Doob upper bound on the expected number of upcrossings of for submartingales with the corresponding final distribution. Each of these two bounds is at most , with equality in the first bound for . The upper bound on the length covered by during upcrossings of an interval restricts the possible variability of a martingale in terms of its final variance. This is in the same spirit as the Dubins & Schwarz sharp upper bound on the expected maximum of above , the Dubins & Schwarz sharp upper bound on the expected maximal distance of from , and the Dubins, Gilat & Meilijson sharp upper bound on the expected diameter of . |
| |
Keywords: | 60G44 60G40 Brownian Motion Upcrossings Martingale Optimal stopping |
本文献已被 ScienceDirect 等数据库收录! |
|