逆首次通过时间问题的唯一性和Shiryaev边界的形状

Pub Date : 2023-02-01 DOI:10.1137/s0040585x97t991155
A. Klump, M. Kolb
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引用次数: 1

摘要

给定正扩展实线上的一个分布,布朗运动的双面逆首次通过时间问题要求一个函数,使得该函数通过反射布朗运动的首次通过时间具有给定的分布。我们将Ekström和Janson在单侧第一次通过时间逆问题范围内发展起来的思想与De Masi等人在自由边界问题中使用的方法相结合,利用随机顺序关系对双侧第一次通过时间逆问题的唯一性给出了不同的证明。本文给出了逆首通问题解的定性性质判据,该判据适用于对应于指数分布的边界。
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Uniqueness of the Inverse First-Passage Time Problem and the Shape of the Shiryaev Boundary
Given a distribution on the positive extended real line, the two-sided inverse first-passage time problem for Brownian motion asks for a function such that the first passage time of this function by a reflected Brownian motion has the given distribution. We combine the ideas of Ekström and Janson, which were developed within the scope of the one-sided inverse first-passage time problem, with the methods of De Masi et al., which were used in the context of free boundary problems, in order to give a different proof for the uniqueness for the two-sided inverse first-passage time problem by using a stochastic order relation. We provide criteria for qualitative properties of solutions of the inverse first-passage problem, which apply to the boundary corresponding to the exponential distribution.
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