Stopped Submartingale is 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 an $\sequence {\FF_n}_{n \ge 0}$-submartingale.

Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

Let $\sequence {X_n^T}_{n \ge 0}$ be the stopped process.

Then $\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-submartingale.

Proof
From Adapted Stochastic Process is Supermartingale iff Negative is Submartingale, we have:


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

From Stopped Supermartingale is Supermartingale, we have:


 * $\sequence {-X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale.

From Adapted Stochastic Process is Supermartingale iff Negative is Submartingale, we have:


 * $\sequence {X_n^T}$ is a $\sequence {\FF_n}_{n \ge 0}$-submartingale.