Simpel Quenstion (maybe) about stoppingtime

Jun 2010
1
0
I have an adaptet, cadlag proces \(\displaystyle M \) with \(\displaystyle M(0)=0\). Then I have that the jumps \(\displaystyle \Delta M \geq 0\) and a stoppingtime

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

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?
 
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