A note on the maximal expected local time of L2-bounded martingales

For an L 2 -bounded martingale starting at 0 and having final variance σ 2 , the expected local time at a ∈ R is at most σ 2 + a 2 - | a | . This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval ( a - σ 2 + a 2 , a + σ 2 + a 2 ) . In particular, th...

Full description

Saved in:
Bibliographic Details
Published inJournal of theoretical probability Vol. 35; no. 3; pp. 1952 - 1955
Main Authors Gilat, David, Meilijson, Isaac, Sacerdote, Laura
Format Journal Article
LanguageEnglish
Published New York Springer US 2022
Springer Nature B.V
Subjects
Brownian motion
Martingales
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Statistics
Brownian motion
Upcrossings
60G44
Local time
60G40
Martingale
Online AccessGet full text

Cover

More Information
Summary:For an L 2 -bounded martingale starting at 0 and having final variance σ 2 , the expected local time at a ∈ R is at most σ 2 + a 2 - | a | . This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval ( a - σ 2 + a 2 , a + σ 2 + a 2 ) . In particular, the maximal expected local time anywhere is at most σ , and this bound is sharp. Sharp bounds for the expected maximum, maximal absolute value, maximal diameter and maximal number of upcrossings of intervals have been established by Dubins and Schwarz (Societé Mathématique de France, Astérisque 157(8), 129–145 1988), by Dubins et al. (Ann Probab 37(1), 393–402 2009) and by the authors (2018).
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-021-01118-0