Develop (left(epsilon+B_{t}^{2} ight)^{1 / 2}) using It's formula. Letting (epsilon) go to 0, prove that (left|B_{t} ight|=W_{t}+L_{t}^{0}), where (L^{0}) is the local time at 0

Develop \(\left(\epsilon+B_{t}^{2}\right)^{1 / 2}\) using Itô's formula. Letting \(\epsilon\) go to 0, prove that \(\left|B_{t}\right|=W_{t}+L_{t}^{0}\), where \(L^{0}\) is the local time at 0 of \(B\). Prove (6.1.1) using the same method for \(n>1\).

\[\left(\epsilon+B_{t}^{2}\right)^{1 / 2}=\epsilon^{1 / 2}+\frac{1}{2} \int_{0}^{t} 2 B_{s} \frac{d B_{s}}{\sqrt{\epsilon+B_{s}^{2}}}+\frac{\epsilon}{2} \int_{0}^{t} \frac{d s}{\left(\epsilon+B_{s}^{2}\right)^{3 / 2}} .\]

\[\begin{equation*}
d R_{t}=d W_{t}+\frac{n-1}{2} \frac{d t}{R_{t}} \tag{6.1.1}
\end{equation*}\]