  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 $V(\tau)$, 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, where the
insider observes the firm's value continuously on time.

Probability