Let (tau=inf left{t: M_{t}-W_{t}>a ight}). Prove that (M_{tau}) follows the exponential law with parameter (a^{-1}). The exponential

Let \(\tau=\inf \left\{t: M_{t}-W_{t}>a\right\}\). Prove that \(M_{\tau}\) follows the exponential law with parameter \(a^{-1}\).

The exponential law stems from

\[\mathbb{P}\left(M_{\tau}>x+y \mid M_{\tau}>y\right)=\mathbb{P}\left(\tau>T_{x+y} \mid \tau>T_{y}\right)=\mathbb{P}\left(M_{\tau}>x\right)\]

The value of the mean of \(M_{\tau}\) is obtained by passing to the limit in the equality \(\mathbb{E}\left(M_{\tau \wedge n}\right)=\mathbb{E}\left(M_{\tau \wedge n}-W_{\tau \wedge n}\right)\).