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(supt ϵ[0,1] Bt > b), with (Bt )t ϵ[0,1] a standard BrownianMotion.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.
|Keywords||Brownian Motion, Continuity correction, Discretization error, Hitting time, Optimal discretization, Rare-event simulation|
|Journal||ACM Transactions on Modeling and Computer Simulation|
Bisewski, K.L, Crommelin, D.T, & Mandjes, M.R.H. (2018). Controlling the time discretization bias for the supremum of Brownian Motion. ACM Transactions on Modeling and Computer Simulation, 28(3). doi:10.1145/3177775