For a continuous $L^2$-bounded Martingale with no intervals of constancy, starting at $0$ and having final variance $sigma^2$, the expected local time at $x in cal{R}$ is at most $sqrt{sigma^2+x^2}-|x|$. This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval $(x-sqrt{sigma^2+x^2},x+sqrt{sigma^2+x^2})$. 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 (1988), Dubins, Gilat and Meilijson (2009) and by the authors (2017).

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

Sacerdote L.
2021

Abstract

For a continuous $L^2$-bounded Martingale with no intervals of constancy, starting at $0$ and having final variance $sigma^2$, the expected local time at $x in cal{R}$ is at most $sqrt{sigma^2+x^2}-|x|$. This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval $(x-sqrt{sigma^2+x^2},x+sqrt{sigma^2+x^2})$. 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 (1988), Dubins, Gilat and Meilijson (2009) and by the authors (2017).
Published online July 31, 2021
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https://www.researchgate.net/publication/339324339_A_sharp_bound_on_the_expected_local_time_of_a_continuous_cal_L_2-bounded_Martingale
https://link.springer.com/content/pdf/10.1007/s10959-021-01118-0.pdf
Local time, Brownian Motion, Martingale, Upcrossings
Gilat D.,Meilijson I., Sacerdote L.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2318/1829944
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