Open Access
July 1999 Integrated Brownian Motion, Conditioned to be Positive
Piet Groeneboom, Geurt Jongbloed, Jon A. Wellner
Ann. Probab. 27(3): 1283-1303 (July 1999). DOI: 10.1214/aop/1022677447


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 $t \to t^{9/10}$ 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 $t \to t^{\alpha}$ is finite for $\alpha < 9 /10$ and infinite for $\alpha \geq 9/10$.


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Piet Groeneboom. Geurt Jongbloed. Jon A. Wellner. "Integrated Brownian Motion, Conditioned to be Positive." Ann. Probab. 27 (3) 1283 - 1303, July 1999.


Published: July 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0983.60078
MathSciNet: MR1733148
Digital Object Identifier: 10.1214/aop/1022677447

Primary: 60G40 , 60J25 , 60J65

Keywords: Conditioning , confluent hypergeometric functions , hitting times , Integrated Brownian motion , Kolmogorov diffusion , Stochastic differential equations

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 3 • July 1999
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