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Explicit construction of a dynamic Bessel bridge of dimension 3

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Date
2013
Link to item file
http://hal.archives-ouvertes.fr/hal-00534273/fr/
Dewey
Probabilités et mathématiques appliquées
Sujet
Brownian motion; dimension 3; Bessel bridge
Journal issue
Electronic Journal of Probability
Volume
18
Publication date
2013
Article pages
25
DOI
http://dx.doi.org/10.1214/EJP.v18-1907
URI
https://basepub.dauphine.fr/handle/123456789/5187
Collections
  • CEREMADE : Publications
Metadata
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Author
Danilova, Albina
Cetin, Umut
Campi, Luciano
Type
Article accepté pour publication ou publié
Abstract (EN)
Given a deterministically time-changed Brownian motion $Z$ starting from $1$, whose time-change $V(t)$ satisfies $V(t) > t$ for all $t\geq 0$, we perform an explicit construction of a process $X$ which is Brownian motion in its own filtration and that hits zero for the first time at $V(\tau)$, where $\tau := \inf\{t>0: Z_t =0\}$. We also provide the semimartingale decomposition of $X$ under the filtration jointly generated by $X$ and $Z$. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process $X$ may be viewed as the analogue of a $3$-dimensional Bessel bridge starting from $1$ at time $0$ and ending at $0$ at the random time $V(\tau)$. We call this {\em a dynamic Bessel bridge} since $V(\tau)$ is not known in advance. Our study is motivated by insider trading models with default risk.

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