On the expected uniform error of geometric Brownian motion approximated by the Lévy-Ciesielski construction

Abstract

It is known that the Brownian bridge or Lévy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is on the error. In particular, we show for geometric Brownian motion that at level <em>N</em> , at which there are <em>d</em> = 2<em>^N</em> points evaluated on the Brownian path, the expected uniform error has an upper bound of order <em>O</em>(&Sqrt;<em>N</em>/2<em>^N</em> ), or equivalently, <em>O</em>(&Sqrt;ln<em>d</em>/<em>d</em>). This upper bound matches the known order for the expected uniform error of the standard Brownian motion. We apply the result to an option pricing example.