  We present an iterative sampling method which delivers upper and lower
bounding processes for the Brownian path. We develop such processes with
particular emphasis on being able to unbiasedly simulate them on a personal
computer. The dominating processes converge almost surely in the supremum and
$L_1$ norms. In particular, the rate of converge in $L_1$ is of the order
$\mathcal {O}(\mathcal{K}^{-1/2})$, $\mathcal{K}$ denoting the computing cost.
The a.s. enfolding of the Brownian path can be exploited in Monte Carlo
applications involving Brownian paths whence our algorithm (termed the
$\varepsilon$-strong algorithm) can deliver unbiased Monte Carlo estimators
over path expectations, overcoming discretisation errors characterising
standard approaches. We will show analytical results from applications of the
$\varepsilon$-strong algorithm for estimating expectations arising in option
pricing. We will also illustrate that individual steps of the algorithm can be
of separate interest, giving new simulation methods for interesting Brownian
distributions.
