EMBEDDING LAWS IN DIFFUSIONS BY FUNCTIONS OF TIME

• 成果类型: Article
• 署名作者: Cox, A. M. G.; Peskir, G.
• 署名单位: University of Bath; University of Manchester
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP941
• 发表日期: 2015
• 页码: 2481-2510
• 关键词: stopping-times

摘要

We present a constructive probabilistic proof of the fact that if B = (B-t)(t >= 0) is standard Brownian motion started at 0, and mu is a given probability measure on R such that mu({0}) = 0, then there exists a unique left-continuous increasing function b: (0, infinity) -> R boolean OR {+infinity} and a unique left-continuous decreasing function c: (0, infinity) -> R boolean OR (-infinity) such that B stopped at tau(b,c) = inf{t > 0 vertical bar B-t >= b(t) or B-t <= c(t)} has the law p,. The method of proof relies upon weak convergence arguments arising from Helly's selection theorem and makes use of the Levy metric which appears to be novel in the context of embedding theorems. We show that tau(b,c) is minimal in the sense of Monroe so that the stopped process B-tau b,B-c = (B-t boolean AND tau b,B-c)(t >= 0) satisfies natural uniform integrability conditions expressed in terms of mu. We also show that tau(b,c) has the smallest truncated expectation among all stopping times that embed mu into B. The main results extend from standard Brownian motion to all recurrent diffusion processes on the real line.