First passage time and inverse problem for continuous local martingales

IF 0.8 4区数学 Q3 STATISTICS & PROBABILITY

Journal of Statistical Planning and Inference Pub Date : 2026-07-01 Epub Date: 2026-01-19 DOI:10.1016/j.jspi.2026.106376

Yoann Potiron

引用次数: 0

Abstract

This paper derives an explicit formula for the probability that a continuous local martingale crosses a one or two-sided random constant boundary for a finite time interval. The boundary crossing probability of a continuous local martingale to a constant boundary is equal to the boundary crossing probability of a standard Wiener process, which is time-changed by the martingale quadratic variation, to a constant boundary. This paper also derives an explicit solution to the inverse first passage time problem of quadratic variation. These results are obtained by an application of the Dambis, Dubins–Schwarz theorem. The main elementary idea of the proof is the scale invariant property of the time-changed Wiener process and thus the scale invariant property of the first passage time. This is due to the constancy of the boundary.

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连续局部鞅的首次通过时间与逆问题

本文导出了有限时间区间内连续局部鞅与单侧或双侧随机常数边界相交概率的显式公式。连续局部鞅到常数边界的边界跨越概率等于由鞅二次变分时变的标准Wiener过程到常数边界的边界跨越概率。本文还导出了二次变分首次通过时间逆问题的显式解。这些结果是由Dambis， Dubins-Schwarz定理的应用得到的。证明的主要基本思想是时变维纳过程的尺度不变性质，从而得到第一次通过时间的尺度不变性质。这是由于边界的恒定性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Statistical Planning and Inference 数学-统计学与概率论

CiteScore

2.10

自引率

11.10%

发文量

审稿时长

3-6 weeks

期刊介绍： The Journal of Statistical Planning and Inference offers itself as a multifaceted and all-inclusive bridge between classical aspects of statistics and probability, and the emerging interdisciplinary aspects that have a potential of revolutionizing the subject. While we maintain our traditional strength in statistical inference, design, classical probability, and large sample methods, we also have a far more inclusive and broadened scope to keep up with the new problems that confront us as statisticians, mathematicians, and scientists. We publish high quality articles in all branches of statistics, probability, discrete mathematics, machine learning, and bioinformatics. We also especially welcome well written and up to date review articles on fundamental themes of statistics, probability, machine learning, and general biostatistics. Thoughtful letters to the editors, interesting problems in need of a solution, and short notes carrying an element of elegance or beauty are equally welcome.