Prove that for (y geq 0) and (y geq x) [mathbf{W}^{(u)}left(X_{t} leq x, M_{t}^{X} geq y ight)=e^{2

Prove that for \(y \geq 0\) and \(y \geq x\)

\[\mathbf{W}^{(u)}\left(X_{t} \leq x, M_{t}^{X} \geq y\right)=e^{2 u y} \mathbb{P}\left(W_{t}+u t \leq x-2 y\right)\]

and that for \(y \leq 0\) and \(y \leq x\)

\[\mathbf{W}^{(u)}\left(X_{t} \geq x, m_{t}^{X} \leq y\right)=e^{2 u y} \mathbb{P}\left(W_{t}+u t \geq x-2 y\right)\]