Let (left(X_{t} ight)) be a standard Brownian motion with initial state 0 , and for a point

Let \(\left(X_{t}\right)\) be a standard Brownian motion with initial state 0 , and for a point \(M>0\) let \(T_{M}\) be the first time that the Brownian motion achieves a value of at least \(M\). Let \(Y_{t}=\max _{u \leq t} X_{u}\). The process \(\left(Y_{t}\right)\) is called the maximum process for \(X_{t}\). By relating the maximum process to hitting times \(T_{M}\), show that the c.d.f. of \(Y_{t}\) is \(G(y)=1-2 \int_{y}^{\infty} \frac{1}{\sqrt{2 \pi t}} e^{-x^{2} / 2 t} d x\).