Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale

Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {X_n}_{n \ge 0}$ be a $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.

Then $\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-martingale it is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale and a $\sequence {\FF_n}_{n \ge 0}$-submartingale.

Proof
We have:


 * $\expect {X_{n + 1} \mid \FF_n} = X_n$ almost surely




 * $\expect {X_{n + 1} \mid \FF_n} \le X_n$ almost surely

and:


 * $\expect {X_{n + 1} \mid \FF_n} \ge X_n$ almost surely.

That is:


 * $\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-martingale it is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale and a $\sequence {\FF_n}_{n \ge 0}$-submartingale.