Theorem (Stopped Process is also right-continuous Martingale)

Let $(X_{t})_{t\ge 0}$ be a right continuous $(\mathcal{F}_{t})_{t\ge 0}$-martingale and let $T$ be a $(\mathcal{F}_{t})_{t\ge 0}$-stopping time. Then $X^{T}=(X_{t}^{T})_{t\ge 0}=(X_{T\wedge t})_{t\ge 0}$ is a right continuous $(\mathcal{F}_{t})$-martingale.