A sharp bound on the expected number of upcrossings of an L2-bounded Martingale

David Gilat, Isaac Meilijson*, Laura Sacerdote

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

For a martingale M starting at x with final variance σ2, and an interval (a,b), let Δ=[Formula presented] be the normalized length of the interval and let δ=[Formula presented] be the normalized distance from the initial point to the lower endpoint of the interval. The expected number of upcrossings of (a,b) by M is at most [Formula presented] if Δ2≤1+δ2 and at most [Formula presented] 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 (a,b) for submartingales with the corresponding final distribution. Each of these two bounds is at most [Formula presented], with equality in the first bound for δ=0. The upper bound [Formula presented] on the length covered by M 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 σ on the expected maximum of M above x, the Dubins & Schwarz sharp upper bound σ2 on the expected maximal distance of M from x, and the Dubins, Gilat & Meilijson sharp upper bound σ3 on the expected diameter of M.

Original languageEnglish
Pages (from-to)1849-1856
Number of pages8
JournalStochastic Processes and their Applications
Volume128
Issue number6
DOIs
StatePublished - Jun 2018

Keywords

  • Brownian Motion
  • Martingale
  • Optimal stopping
  • Upcrossings

Fingerprint

Dive into the research topics of 'A sharp bound on the expected number of upcrossings of an L2-bounded Martingale'. Together they form a unique fingerprint.

Cite this