\(\displaystyle

\tau_{x}=inf \{ t \geq 0 : M(t) - \mu t <x \}

\)

But why is it that

\(\displaystyle M( \tau_{x} )= \mu \tau_{x} + x\)

on \(\displaystyle (\tau_{x} < \infty) \)?

And not \(\displaystyle M( \tau_{x} ) < \mu \tau_{x} + x\)

Can anyone explain this?