Find out if a variable is a martingale (discrete time)

Question

Let X be a stochastic variable on a finite, filtered probability space. If $$ M_1(t) = E(X| \mathcal{F}_t) $$ Is $M_1(t)$ a martingale?

I have a lot of trouble checking if variables are martingales or not. I know about the $(\Omega, \mathcal{F}, P)$ notation, and I am allowed to use this to answer the question.

I am pretty sure you need to use the tower law to solve this problem, which says that

$$ E(E(X_u|\mathcal{F}_t)|\mathcal{F_d}) = E(X_u|\mathcal{F}_d)\text{ } whenever\text{ } d \leq t \leq u $$

But I just dont see how its done? Can someone help me out? What is the general method to find out if a variable is a martingale or not?

Answer 1

For $s\le t$, we have by the tower rule : $$\mathbb E[M_1(t)\mid \mathcal F_s]=\mathbb E[\mathbb E[X\mid \mathcal F_t]\mid\mathcal F_s]=\mathbb E[X\mid\mathcal F_s]=M_1(s)$$ Furthermore, since for all $t$ we have $\mathbb E[M_1(t)]=\mathbb E[X]$ and $X$ is integrable because the probability space is finite, then $(M_1(t))_t$ is integrable and hence a martingale. This form of martingale is called a closed martingale.