> [!theorem] > > Let $\bracs{X_t}$ be a continuous time $\bracs{\cf_t}$-martingale/non-negative submartingale with [[RCLL]] sample paths. Then for any $\bracs{\cf_t}$-[[Stopping Time|stopping time]] $\tau$, $X_{t \wedge \tau}$ is a $\bracs{\cf_t}$-martingale/non-negative submartingale. > > *Proof*. Let $0 \le s \le t < \infty$ and $B \in \cf_s$, then since $B \cap \bracs{\tau > s} \in \cf_{s \wedge \tau}$ and $s \wedge \tau \le t \wedge \tau \le t$, by [[Hunt's Theorem]], > $ > \ev\braks{X_{t \wedge \tau}|\cf_{s \wedge \tau}} = (\ge) X_{s \wedge \tau} > $ > Taking the integral > $ > \begin{align*} > \int_B X_{t \wedge \tau}d\bp&=\int_{B \cap \bracs{\tau \le s}} X_{t \wedge \tau}d\bp + \int_{B \cap \bracs{\tau > s}}X_{t \wedge \tau}d\bp \\ > &= \int_{B \cap \bracs{\tau \le s}} X_{s \wedge \tau}d\bp + \int_{B \cap \bracs{\tau > s}}X_{s \wedge \tau}d\bp \\ > &= \int_{B}X_{s \wedge \tau}d\bp > \end{align*} > $ Task: Hunt's theorem (continuous) for unbounded stopping times under uniform integrability of the process. 1. $X_{\tau}$ with $\tau < \infty$ almost surely is uniformly integrable. 2. If $\tau_1 \le \tau_2 < \infty$, then $\ev\braks{X_{\tau_2}|\cf_{\tau_1}} = X_{\tau_1}$.