Chernoff's distribution and differential equations of parabolic and Airy type

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:423

Article

Groeneboom, Piet¹  Lalley, Steven²  Temme, Nico³ 

[1] Delft Univ Technol, Delft, Netherlands

[2] Univ Chicago, Chicago, IL 60637 USA

[3] CWI, Amsterdam, Netherlands

关键词: Parabolic partial differential equations;    Airy functions;    Scorer's functions;    Brownian motion with parabolic drift;    Cameron-Martin-Girsanov;    Feynman-Kac;   

DOI  :  10.1016/j.jmaa.2014.10.051

来源: Elsevier

【 摘 要 】

We give a direct derivation of the distribution of the maximum and the location of the maximum of one-sided and two-sided Brownian motion with a negative parabolic drift. The argument uses a relation between integrals of special functions, in particular involving integrals with respect to functions which can be called incomplete Scorer functions. The relation is proved by showing that both integrals, as a function of two parameters, satisfy the same extended heat equation, and the maximum principle is used to show that these solutions must therefore have the stated relation. Once this relation is established, a direct derivation of the distribution of the maximum and location of the maximum of Brownian motion minus a parabola is possible, leading to a considerable shortening of the original proofs. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free