On an approximation problem for stochastic integrals where random time nets do not help

Abstract. Given a geometric Brownian motion S = (St)t∈[0,T] and a Borel function g: (0,∞) → IR such that g(ST) ∈ L2, we approximate g(ST) − IEg(ST) by ∑n i=1 vi−1(Sτi − Sτi−1) where 0 = τ0 ≤ · · · ≤ τn = T is an increasing sequence of stopping times and the vi are Fτi-measurable random variables such that IEv2i−1(Sτi−Sτi−1)2 <∞. In case that g is not almost surely linear, we show that one gets a lower bound for the L2-approximation rate of 1/ n if one optimizes over all nets consisting of n+ 1 stopping times. This lower bound coincides with the upper bound for all reasonable functions g, in case deterministic time-nets are used. Hence random time-nets do not improve the rate of convergence in this case. The same result holds true for the Brownian motion instead of the geometric Brownian motion. 1. Introduction an