QUICKEST DETECTION OF A HIDDEN TARGET AND EXTREMAL SURFACES

• 成果类型: Article
• 署名作者: Peskir, Goran
• 署名单位: University of Manchester
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP979
• 发表日期: 2014
• 页码: 2340-2370
• 关键词: brownian-motion; Local Time; maximum; diffusion; inequalities; PRINCIPLE

摘要

Let Z = (Z(t))(t >= 0) be a regular diffusion process started at 0, an independent random variable with a strictly increasing and continue:.stribution function F, and let tau(l) = inf{t >= 0 vertical bar Z(t) = l} be the first entry, ... of Z at the level l. We show that the quickest detection problem tau sup E[R-tau - (0)integral(tau) c(R-t) dt], where R = S I is the range process of X = 2F(Z) 1 (i.e., the difference between the running maximum and the running minimum of X) and c(r) = cr with c > 0. Solving the latter problem we find that the following stopping time is optimal: tau* = inf { t >= 0 vertical bar f(*) (I-t, S-t) <= X-t <= g(*) (I-t, S-t)}, where the surfaces f* and g* can be characterised as extremal solutions to a couple of first-order nonlinear PDEs expressed in terms of the infinitesimal characteristics of X and c. This is done by extending the arguments associated with the maximality principle [Ann. Probab. 26 (1998) 1614-1640] to the three-dimensional setting of the present problem and disclosing the general structure of the solution that is valid in all particular cases. The key arguments developed in the proof should be applicable in similar multi-dimensional settings.