Let \(X_{t}\) be the geometric Brownian motion given by the stochastic differential equation

\[d X_{t}=r X_{t} d t+\sigma X_{t} d W_{t}\]

a) Compute the Euler discretization \(\left(\widehat{X}_{t_{k}}^{N}\right)_{k=0,1, \ldots, N}\) of \(\left(X_{t}\right)_{t \in \mathbb{R}_{+}}\).

b) Compute the Milshtein discretization \(\left(\widehat{X}_{t_{k}}^{N}\right)_{k=0,1, \ldots, N}\) of \(\left(X_{t}\right)_{t \in \mathbb{R}_{+}}\).