A sharp bound on the expected local time of a continuous \cal L₂-bounded Martingale

2020/02/17 by David Gilat, Gilat, David, Isaac Meilijson +3
Computer Science · Economics, Econometrics and Finance · Mathematics · #60G40 #60G44 #Advanced Mathematical Modeling in Engineering #Applied mathematics #Bounded function #FOS: Mathematics #Local martingale #Martingale (probability theory) #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2002.06978

openalex publication_date 2020/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a continuous \cal L₂-bounded Martingale with no intervals of constancy, starting at 0 and having final variance σ², the expected local time at x ∈ \calR is at most √(σ²+x²)-|x|. This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval (x-√(σ²+x²),x+√(σ²+x²)). 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).

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