Explicit Construction of a Dynamic Bessel Bridge of Dimension 3

17 June 2011
Dr Albina Danilova
Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V (t) satisfies $V (t) > t$ for all $t>=0$, 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 (s), where $s:= inf {t > 0 : Z_t = 0}$. 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 $V (s)$. We call this a dynamic Bessel bridge since V(s) is not known in advance. Our study is motivated by insider trading models with default risk.(this is a joint work with Luciano Campi and Umut Cetin)