Optimal stopping of the maximum process: The maximality principle

• 成果类型: Article
• 署名作者: Peskir, G
• 署名单位: Aarhus University; University of Zagreb
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1022855875
• 发表日期: 1998
• 页码: 1614-1640
• 关键词: INEQUALITIES

摘要

The solution is found to the optimal stopping problem with payoff sup(tau) E(S-tau - integral(0)(tau) c(Xt)dt), where S = (S-t)(t greater than or equal to 0) is the maximum process associated with the one-dimensional time-homogeneous diffusion X = (X-t)(t greater than or equal to 0), the function x /--> c(x) is positive and continuous, and the supremum is taken over all stopping times tau of X for which the integral has finite expectation. It is proved, under no extra conditions, that this problem has a solution; that is, the payoff is finite and there is an optimal stopping time, if and only if the following maximality principle holds: the first-order nonlinear differential equation g'(s) = sigma(2)(g(s))L'(g(s))/2c(g(s))(L(s) - L(g(s))) admits a maximal solution s /--> g*(s) which stays strictly below the diagonal in R-2. [In this equation x /--> sigma(x) is the diffusion coefficient and x /--> L(x) the scale function of X.] In this case the stopping time tau* = inf{t > 0 \ X-t less than or equal to g*(S-t)} is proved optimal, and explicit formulas for the payoff are given. The result has a large number of applications and may be viewed as the cornerstone in a general treatment of the maximum process.