EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE INVERSE BOUNDARY CROSSING PROBLEM FOR DIFFUSIONS

• 成果类型: Article
• 署名作者: Chen, Xinfu; Cheng, Lan; Chadam, John; Saunders, David
• 署名单位: Pennsylvania Commonwealth System of Higher Education (PCSHE); University of Pittsburgh; State University of New York (SUNY) System; SUNY Fredonia; University of Waterloo
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP714
• 发表日期: 2011
• 页码: 1663-1693

摘要

We study the inverse boundary crossing problem for diffusions. Given a diffusion process X-t, and a survival distribution p on [0,8), we demonstrate that there exists a boundary b(t) such that p(t) = P [tau > t], where t is the first hitting time of X-t to the boundary b(t). The approach taken is analytic, based on solving a parabolic variational inequality to find b. Existence and uniqueness of the solution to this variational inequality were proven in earlier work. In this paper, we demonstrate that the resulting boundary b does indeed have p as its boundary crossing distribution. Since little is known regarding the regularity of b arising from the variational inequality, this requires a detailed study of the problem of computing the boundary crossing distribution of X-t to a rough boundary. Results regarding the formulation of this problem in terms of weak solutions to the corresponding Kolmogorov forward equation are presented.