Hi everyone, please consider the following problem:


Let $(M\_t)\_{t\geq 0}$ be a continuous and positive submartingale and $S\_t=\sup\_{0\leq s\leq t}M\_s$. Please prove that for any $\lambda>0$ we have


$$\lambda P(S\_t>2\lambda)\leq E[M\_t1\_{\{M\_t>\lambda\}}]$$


This inequality makes me remember the Doob inequality


$$\lambda P(S\_t>\lambda)\leq E[M\_t1\_{\{S\_t>\lambda\}}]$$


So it is enough to show that 


$$E[M\_t1\_{\{S\_t>\lambda\}}]\leq 2E[M\_t1\_{\{M\_t>\lambda\}}]$$


But I have no idea to deal about the term $1\_{\{M\_t>\lambda\}}$, even by introducing a stopping time $T\_{\lambda}$. Could someone help me prove this inequality or give some idea? Many thanks!



modification of Doob inequality
Nice little exercise, it should go to math.stackexchange.com, though. I'll give you a hint: Let $T$ be the hitting time of $2\lambda$. Then, $E[M\_t 1\_{M\_t\ge \lambda}] \ge E[M\_t 1\_{M\_t\ge \lambda,\ T\le t}] \ge E[M\_t1\_{T\le t}]- E[M\_t1\_{M\_t\le \lambda}1\_{T\le t}].$


