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In this paper we derive weak limits for the discretization errors of samplingbarrier-hitting and extreme events of Brownian motion by using the Eulerdiscretization simulation method. Specifically, we consider the Eulerdiscretization approximation of Brownian motion to sample barrier-hittingevents, i.e. hitting for the first time a deterministic "barrier" function; andto sample extreme events, i.e. attaining a minimum on a given compact timeinterval or unbounded closed time interval. For each case we study thediscretization error between the actual time the event occurs versus the timethe event occurs for the discretized path, and also the discretization error onthe position of the Brownian motion at these times. We show limits indistribution for the discretization errors normalized by their convergencerate, and give closed-form analytic expressions for the limiting randomvariables. Additionally, we use these limits to study the asymptotic behaviourof Gaussian random walks in the following situations: (1.) the overshoot of aGaussian walk above a barrier that goes to infinity; (2.) the minimum of aGaussian walk compared to the minimum of the Brownian motion obtained wheninterpolating the Gaussian walk with Brownian bridges, both up to the same timehorizon that goes to infinity; and (3.) the global minimum of a Gaussian walkcompared to the global minimum of the Brownian motion obtained wheninterpolating the Gaussian walk with Brownian bridges, when both have the samepositive drift decreasing to zero. In deriving these limits in distribution weprovide a unified framework to understand the relation between several paperswhere the constant $-\zeta(1/2)/\sqrt{2 \pi}$ has appeared, where $\zeta$ isthe Riemann zeta function. In particular, we show that this constant is themean of some of the limiting distributions we derive.

On the Euler discretization error of Brownian motion about random times