In Tomas Bjork's Arbitrage Theory Continous Time (2009), Proposition 21.20 claimed that, in discrete time, the (pointwise) infinum of an arbitrary family of supermartingales is still a supermartingale. And the proof is left as an exercise.
Specifically, let's consider a family of $F_t$-supermartingales $\{D^i\}_{i\in I}$ in discrete time $\Bbb N$, how to prove that:
$$X_n:=\inf_{i\in I}D_n^i,\quad n=0,1,2,\cdots$$
is still an $F_t$-supermartingale?
The closest thing I can think of is Fatou's lemma, but it only applies to limit inferior, not infinum.
Best Answer
You have that $\inf_{i\in I} D_n^i \leq D^j_n$ almost surely for all $j\in I$. Then,
$E(\inf_{i\in I} D_n^i\left|\mathcal{F}_{n-1}\right.) \leq E(D_n^j\left|\mathcal{F}_{n-1}\right.)$ for all $j$. In particular, this is useful for the first inequality below
$E(X_n\left|\mathcal{F}_{n-1}\right.)=E(\inf_{i\in I} D_n^i\left|\mathcal{F}_{n-1}\right.) \leq \inf_{j\in I}E(D_n^j\left|\mathcal{F}_{n-1}\right.)\leq \inf_{j\in I}D_{n-1}^j=X_{n-1}$