A Brownian motion observed at equidistant sampling points renders a random walk with normally distributed increments. For the difference between the expected maximum of the Brownian mo- tion and its sampled version, an expansion is derived with coefficients in terms of the drift, the Riemann zeta function and the normal distribution function.
"Equidistant sampling for the maximum of a Brownian motion with drift on a finite horizon." Electron. Commun. Probab. 14 143 - 150, 2009. https://doi.org/10.1214/ECP.v14-1453