I have a problem that goes like this:
Let $(X_n)_{n\geq 0}$ be a submartingale and the constant $\lambda>0$. Show that $\lambda\mathbb{P}(\min_{0\leq k\leq n}X_k<-\lambda)\leq\mathbb{E}[X_n]-\mathbb{E}[X_0]$.
My attempt is the following:
Define a stopping time $\tau = \inf \{ k : X_k < -\lambda\}$.
Now
$\mathbb{P}(\min_{0\leq k\leq n}X_k<-\lambda)=\mathbb{P}(\tau\leq n,X_{\tau}<-\lambda)=\mathbb{E}[\mathbb{I}\{\tau\leq n\}\mathbb{I}\{X_{\tau}< -\lambda\}]\\ \leq\mathbb{E}[\mathbb{I}\{\tau\leq n\}\frac{X_{\tau}}{-\lambda}]=-\frac{1}{\lambda}\mathbb{E}[\mathbb{I}\{\tau\leq n\}X_{\tau}]$.
But how could I proceed from this? I know that on the event $\{\tau\leq n\}$ it holds that $\mathbb{E}[X_{\tau}]\leq\mathbb{E}[X_n]$, since $X_n$ is a submartingale. But I can't see how to use this fact to manipulate the expression in order to get the desired upper bound. I would greatly appreciate if anyone would have any ideas how this could be done.