I have an adaptet, cadlag proces  M with M(0)=0. Then I have that the jumps \Delta M \geq 0 and a stoppingtime

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

But why is it that

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

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

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

Can anyone explain this?