Complete corrected diffusion approximations for the maximum of a random walk
成果类型：
Article
署名作者：
Blanchet, Jose; Glynn, Peter
署名单位：
Harvard University; Stanford University
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/105051606000000042
发表日期：
2006
页码：
951-983
关键词：
摘要：
Consider a random walk (S-n : n >= 0) with drift - mu and S-0 = 0. Assuming that the increments have exponential moments. negative mean. and are strongly nonlattice. we provide a complete asymptotic expansion (in powers of mu > 0) that corrects the diffusion approximation of the all time maximum M = max(n >= 0) S-n. Our results extend both the first-order correction of Siegmund [Adv in Appl. Probab. 11 (1979) 701-719] and the full asymptotic expansion provided in the Gaussian case by Chang and Peres [Ann. Probab. 25 (1997) 787-802]. We also show that the Cramer-Lundberg constant (as a function of mu) admits all analytic extension throughout a neighborhood of the origin in the complex plane C. Finally. when the increments of the random walk have nonnegative mean mu. we Show that the Laplace transform. E-mu exp(-bR(infinity)), of the limiting overshoot. call be analytically extended throughout a disc centered at the origin in C x C (jointly for both b and mu). In addition, when the distribution of the increments is continuous and appropriately symmetric. we show that E mu S tau [where tau is the first (strict) ascending ladder epoch] call be analytically extended to a disc centered at the origin in C, generalizing the main result in [Ann. Probab. 25 (1997) 787802] and extending a related result of Chang.