At start, we have a stochastic process $X_t$, which is a submartingale. Furthermore, we have that the expression $\mathbb{E}[(X_t-x)_+] = C(t,x)$. $C$ is a function from $\mathbb{R}_+\times\mathbb{R}$ to $\mathbb{R}$ with the following properties (Call-functions):
$C(t,x)$ is convex in $x$ and continuous and increasing in $t$,
$C(t,x) \rightarrow_{x\rightarrow \infty} 0$ for every $t$,
There exists a $a \in \mathbb{R}$ such that $C(t,x)+x \rightarrow_{x\rightarrow -\infty} a$ for every $t$.
The article states that under these conditions, $X_t$ is a martingale. I do not see the way to say this... Should one show the independence of $t$ of $\mathbb{E}[X_t]$? But why? Thank you all in advance!!
The real issue is somewhat camouflaged here. Notice that if $Y$ is an integrable random variable, then $\lim_{x\to -\infty}\Bbb E[(Y-x)_+]+x=\Bbb E[Y]$, by dominated convergence. Properties 1. and 2. of $C(t,x)$ (with the exception of the continuity-in-$t$ in 1.) follow from the definition of $C(t,x)$ and the submartingale property of $(X_t)$. Property 3. implies that $\Bbb E[X_t]=a$ for all $t$. This constancy of expectation means that the submartingale $(X_t)$ is in fact a martingale.