Hello, guys!
Can anybody give a hint on how to prove that (on continuous time setting with t in [0,T]) if M is a supermartingale such that E[M(T)] = M(0), then M is in fact a martingale.
Would appreciate any help. Thanks!
Hello, guys!
Can anybody give a hint on how to prove that (on continuous time setting with t in [0,T]) if M is a supermartingale such that E[M(T)] = M(0), then M is in fact a martingale.
Would appreciate any help. Thanks!
Tried to use your advice.
This is what I get:
$\displaystyle E_s\[M(T)\] < M(s) \Rightarrow E\[E_s\[M(T)\] \] < E\[M(s)\] \Rightarrow E\[M(T)\] < E\[M(s)\] \Rightarrow M(0) < E\[M(s)\]$
So if this is correct last inequality is a contradiction to M being supermartingale. So strict inequallity does not hold, but equality must hold.
Am I right now?