论文标题
连续$ {\ cal l} _2 $结合的martingale的预期当地时间的急剧限制
A sharp bound on the expected local time of a continuous ${\cal L}_2$-bounded Martingale
论文作者
论文摘要
对于连续的$ {\ cal l} _2 $结合的martingale，没有恒定的间隔，起点为$ 0 $，最终方差$σ^2 $，预期的当地时间为$ x \ in \ cal {r} $最多是$ \ sqrt $ \ sqrt {σ^2+x^2+x^2} - x^2+x^2} - x | x | $。标准布朗运动在第一个退出时间停止了从间隔$（x- \ sqrt {σ^2+x^2}，x+\ sqrt {σ^2+x^2}）$实现。 Dubins和Schwarz（1988），Dubins，Gilat和Meilijson（2009）和作者（2017）建立了预期最大，最大绝对值，最大直径和最大间隔数量的急剧界限。
For a continuous ${\cal L}_2$-bounded Martingale with no intervals of constancy, starting at $0$ and having final variance $σ^2$, the expected local time at $x \in \cal{R}$ is at most $\sqrt{σ^2+x^2}-|x|$. This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval $(x-\sqrt{σ^2+x^2},x+\sqrt{σ^2+x^2})$. 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).
