I'm looking for a proof for the following proposition, stated in "Brownian Motion, Martingales, and Stochastic Calculus" by Le Gall (page 263):
Let $X=(X_n)_{n\in\mathbb{N}}$ be a supermartingale. For any $n\in\mathbb{N}$ and any $\lambda>0$ –
$$\lambda\mathbb{P}\left(\sup_{k\le n}\left|X_k\right|>\lambda\right)\le
\mathbb{E}\left[\left|X_0\right|\right]+2\mathbb{E}\left[\left|X_n\right|\right]$$
He writes that a proof can be found in "Discrete-Parameter Martingales" by Neveu, but I couldn't find it there.
Not only I've failed proving it, I moreover got results that made me a bit suspicious about the above proposition. For example, I can show that whether $X$ is a supermartingale or a submartingle –
$$\lambda\mathbb{P}\left(\sup_{k\le n}\left|X_k\right|>\lambda\right)\le
12\mathbb{E}\left[|X_0|\right]+9\mathbb{E}\left[|X_n|\right]$$
(note that at the RHS the coefficient of $\mathbb{E}\left[|X_0|\right]$ is larger than the coefficient of $\mathbb{E}\left[|X_n|\right]$, while in the proposition above it's the other way around.)
Best Answer
We can show a little more. Firstly lemma:
Lemma: When $(X_n,\mathcal F_n)$ is supermartingale then for $t>0$ we have $t \mathbb P(Y_n \ge t) \le \mathbb E[X_0] + \mathbb E[(X_n)_{-}]$ where $Y_n = \sup_{k \le n} X_k$ and $(Z)_{-} = \max\{0,-Z\}, (Z)_{+} = \max\{0,Z\}$
Proof (lemma): (I am using $\chi_A$ for indicator function) Note that by Doob stopping theorem we have $\mathbb E[X_0] \ge \mathbb E[X_{\min\{\tau,n\}}] = \mathbb E[X_{\tau} \chi_{\tau \le n}] + \mathbb E[ X_n \chi_{\tau > n}] \ge t \mathbb P(Y_n \ge t) - \mathbb E [ (X_n)_{-} \cdot \chi_{\tau > n}] \ge t \mathbb P(Y_n \ge t) - \mathbb E[(X_n)_{-}]$ where we defined $\tau = \inf \{ m : Y_m \ge t\}$. Some of steps due to $X_{\tau} \ge t$ or $X_n \ge -(X_n)_{-}$
Moreover, when $(X_n, \mathcal F_n)$ is super martingale, then $(-X_n, \mathcal F_n)$ is submartingale
So that we have (again let $\tau$ be the same and $\sup$ means $\sup_{k \le n}$):
$ t \mathbb P( \sup |X_k| \ge t) = t \mathbb P(\sup X_k \ge t) + t \mathbb P( \sup (-X_k) \ge t) \le \mathbb E[X_0] + \mathbb E[(X_n)_{-}] + \mathbb E[ |X_n|]$
Where we used lemma and Doob Inequality. Note that from here you can get your statement, since $\mathbb E[X_0] \le \mathbb E[|X_0|]$ and $\mathbb E[(X_n)_{-}] \le \mathbb E[|X_n|]$
By the way, one can get even better bound, since it is possible to prove Doob Inequality for submartingale in form $t \mathbb P( \sup X_k \ge t) \le \mathbb E[(X_n)_{+}]$, so we can bound:
$t \mathbb P( \sup_{k \le n} |X_k| \ge t) \le \mathbb E[X_0] + 2 \mathbb E[(X_n)_{-}]$