Weak limits of perturbed random walks and the equation Y-t=B-t+alpha sup{Y-s:s<=t}+beta inf{Y-s:s<=t}

• 成果类型: Article
• 署名作者: Davis, B
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1996
• 页码: 2007-2023
• 关键词: reinforced random-walk; brownian-motion

摘要

Let alpha and beta be real numbers and f is an element of C-0[O,infinity). We study the existence and uniqueness of solutions g of the equation g(t) =f(t)f alpha sup(0 less than or equal to s less than or equal to) t g(s) + beta info 0 (less than or equal to s less than or equal to) t g(S). Carmona, Petit, Le Gall, and Yor have shown existence (or noxexistence) and uniqueness for some alpha, beta. We settle the remaining cases. We study the nearest neighbor walk on the integers, which behaves just like fair random walk unless one neighbor has been visited and the other has not, when it jumps to the unvisited neighbor with probability p. if p < 2/3, we show these processes, scaled, converge to the solution of the equation above for Brownian paths, with alpha = beta = (2p - 1)/p.