Let \(d X_{t}=b_{t} d t+d B_{t}\) where \(B\) is a Brownian motion and \(b\) a given bounded \(\mathbf{F}^{B}\)-adapted process. Let

\[L_{t}=\exp \left(-\int_{0}^{t} b_{s} d B_{s}-\frac{1}{2} \int_{0}^{t} b_{s}^{2} d s\right) .\]

Show that \(L\) and \(L X\) are local martingales.