Stopped Martingale is Martingale

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 $\sequence {\FF_n}_{n \ge 0}$-martingale.

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 $\sequence {\FF_n}_{n \ge 0}$-martingale.

Corollary

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

Proof

From Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale:

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

From Stopped Supermartingale is Supermartingale and Stopped Submartingale is Submartingale:

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

From Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale:

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

$\blacksquare$