Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale

Theorem

Discrete Time

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.

Continuous Time

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.