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.

Corollary

$\expect {X_n^T} \ge \expect {X_0}$ for each $n \in \Z_{\ge 0}$.

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.

$\blacksquare$