Integrated Brownian motion, conditioned to be positive
成果类型：
Article
署名作者：
Groeneboom, P; Jongbloed, G; Wellner, JA
署名单位：
Delft University of Technology; Vrije Universiteit Amsterdam; University of Washington; University of Washington Seattle
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
发表日期：
1999
页码：
1283-1303
关键词：
摘要：
We study the two-dimensional process of integrated Brownian motion and Brownian motion, where integrated Brownian motion is conditioned to be positive. The transition density of this process is derived from the asymptotic behavior of hitting times of the unconditioned process. Explicit expressions for the transition density in terms of confluent hypergeometric functions are derived, and it is shown how our results on the hitting time distributions imply previous results of Isozaki-Watanabe and Goldman, The conditioned process is characterized by a system of stochastic differential equations (SDEs) for which we prove an existence and unicity result. Some sample path properties are derived from the SDEs and it is shown that t --> t(9/10) is a critical curve for the conditioned process in the sense that the expected time that the integral part of the conditioned process spends below any curve t --> t(alpha) is finite for alpha < 9/10 and infinite for alpha greater than or equal to 9/10.