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!