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

Theorem

Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.

Let $\sequence {X_t}_{t \ge 0}$ be a continuous-time $\sequence {\FF_t}_{t \ge 0}$-adapted stochastic process.

Then $\sequence {X_t}_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-martingale if and only if it is a $\sequence {\FF_t}_{t \ge 0}$-supermartingale and a $\sequence {\FF_t}_{t \ge 0}$-submartingale.

Proof

For each $t, s \in \hointr 0 \infty$ with $0 \le s < t$, we have:

$\expect {X_t \mid \FF_s} = X_s$ almost surely

if and only if:

$\expect {X_t \mid \FF_s} \le X_s$ almost surely

and:

$\expect {X_t \mid \FF_s} \ge X_s$ almost surely.

That is:

$\sequence {X_t}_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-martingale if and only if it is a $\sequence {\FF_t}_{t \ge 0}$-supermartingale and a $\sequence {\FF_t}_{t \ge 0}$-submartingale.

$\blacksquare$