Integrated Brownian motion, conditioned to be positive

Citation
P. Groeneboom et al., Integrated Brownian motion, conditioned to be positive, ANN PROBAB, 27(3), 1999, pp. 1283-1303
Citations number
16
Categorie Soggetti
Mathematics
Journal title
ANNALS OF PROBABILITY
ISSN journal
00911798 → ACNP
Volume
27
Issue
3
Year of publication
1999
Pages
1283 - 1303
Database
ISI
SICI code
0091-1798(199907)27:3<1283:IBMCTB>2.0.ZU;2-N
Abstract
We study the two-dimensional process of integrated Brownian motion and Brow nian motion, where integrated Brownian motion is conditioned to be positive . The transition density of this process is derived from the asymptotic beh avior of hitting times of the unconditioned process. Explicit expressions f or the transition density in terms of confluent hypergeometric functions ar e derived, and it is shown how our results on the hitting time distribution s imply previous results of Isozaki-Watanabe and Goldman, The conditioned p rocess is characterized by a system of stochastic differential equations (S DEs) for which we prove an existence and unicity result. Some sample path p roperties 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 expecte d time that the integral part of the conditioned process spends below any c urve t --> t(alpha) is finite for alpha < 9/10 and infinite for alpha great er than or equal to 9/10.