Martingales and super-martingales problem

Question

Let $(\Omega, F, P)$ be a probability space, equipped with a filtration $(F_t)_{t=0}^T$ , where $F_0=\{\emptyset, \Omega\}$ (i.e. trivial) and let $(X_t)_{t=0}^T$ be a supermartingale. Show that if $E[X_T]=X_0$ $\implies$ $(X_t)_{t=0}^T$ is a martingale.

Answer 1

Def. Supermartingale:

$$ E(X_t | F_s) ≤ X_s, \quad \forall s ≤ t $$

Using the Tower property, for any t ≤ T:

$$ E(X_t) = E(X_t | F_0) ≤ X_0 = E(X_T) = E( E(X_T |F_t)) ≤ E(X_t) $$

thus all inequalities are equalities.

EDIT: The rest of the details are desired:

By the definition $E(X_T | F_t) - X_t≤ 0$, so we subtract $E(X_t)$ above :

$$ E(E(X_T|F_t) - X_t) = 0, \; \Rightarrow \; E(X_T|F_t) = X_t \;a.s. $$

Tower property for conditional expectation yields the martingale property.