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) := P(suptε [0,1] Bt > b), with (Bt)t∈ [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.