A positive supermartingale is closed on the right

Question

I would like some help with proving the following statement:

Let $(X_t)_{t \in I}$ be a positive $(F_t)_{t \in I}$-supermartingale, that is, $E[X_t |F_s] \leq X_s$ for all $s \leq t$. Then $(X_t)_{t \in I}$ is closed on the right, that is, there exists some integrable random variable $k$ such that $E[k|F_t] \leq X_t$ for all $t \in I$.

I have tried with the variable $k = \inf_{t \in I}(X_t)$ but couldn't get it to work. Any help would be appreciated.

Answer 1

First we show convergence assuming $E[X_{0}]<\infty$. The $Y_{t}=-X_{t}\leq 0$ is a submartingale that is bounded above by zero and so it is uniformly bounded in $L^{1}$

$$\sup_{t}E[|Y_{t}|]=\sup_{t}(-E[Y_{t}])\leq (-E[Y_{0}])=E[X_{0}]$$

and so by Doob's submartingale convergence, $X_{t}\to X_{+\infty}\in L^{1}$.

We claim that in OP $k:=X_{+\infty}$. By conditional Fatou lemma and submartingale, we have

$$E[X_{\infty}\mid\mathcal F_n]=E(\liminf_{t\to +\infty} X_{n+t}\mid\mathcal F_n)\leq \liminf_{t\to +\infty} E(X_{n+t}\mid\mathcal F_n)\leq X_{n}.$$