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

Given a geometric Brownian motion S=(St)t[set membership, variant][0,T] and a Borel measurable function such that g(ST)[set membership, variant]L2, we approximate bywhere 0=[tau]0[less-than-or-equals, slant]...[less-than-or-equals, slant][tau]n=T is an increasing sequence of stopping times and the vi-1 are -measurable random variables such that ( is the augmentation of the natural filtration of the underlying Brownian motion). In case that g is not almost surely linear, we show that one gets a lower bound for the L2-approximation rate of 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.