  In this paper we introduce a new method for the simulation of the exit time
and position of a $\delta$-dimensional Brownian motion from a domain. The main
interest of our method is that it avoids splitting time schemes as well as
inversion of complicated series. The idea is to use the connexion between the
$\delta$-dimensional Bessel process and the $\delta$-dimensional Brownian
motion thanks to an explicit Bessel hitting time distribution associated with a
particular curved boundary. This allows to build a fast and accurate numerical
scheme for approximating the hitting time. Numerical comparisons with existing
methods are performed.
