Controlling the time discretization bias for the supremum of Brownian Motion

We consider the bias arising from time discretization when estimating the threshold crossing probability $w(b) := \mathbb{P}(\sup_{t\in[0,1]} B_t > b)$, with $(B_t)_{t\in[0,1]}$ a standard Brownian Motion. We prove that if the discretization is equidistant, then to reach a given target value of the relative bias, the number of grid points has to grow quadratically in $b$, as $b$ grows. When considering non-equidistant discretizations (with threshold-dependent grid points), we can substantially improve on this: we show that for such grids the required number of grid points is independent of $b$, and in addition we point out how they can be used to construct a strongly efficient algorithm for the estimation of $w(b)$. Finally, we show how to apply the resulting algorithm for a broad class of stochastic processes; it is empirically shown that the threshold-dependent grid significantly outperforms its equidistant counterpart.