Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale/Discrete Time

Theorem

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

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

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

Proof

For each $n \in \Z_{\ge 0}$, we have:

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

if and only if:

$\expect {X_{n + 1} \mid \FF_n} \le X_n$ 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 $\sequence {\FF_n}_{n \mathop \ge 0}$-martingale if and only if it is a $\sequence {\FF_n}_{n \mathop \ge 0}$-supermartingale and a $\sequence {\FF_n}_{n \mathop \ge 0}$-submartingale.

$\blacksquare$

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $10.3$: Martingale, supermartingale, submartingale