I have a question:
Let $T$ be a bounded stopping time and let $(\mathcal{F}_t)_{t\geq 0}$ be a filtration satisfying the usual conditions. Define $\mathcal{G}_t:=\mathcal{F}_{T+t}$, $t\geq 0$. Then $(\mathcal{G}_t)_{t\geq 0}$ satisfies also the usual conditions.
If $\widetilde{X}=(\widetilde{X}_t)_{t\geq 0}$ is a right-continuous submartingale with respect to $(\mathcal{G}_t)_{t\geq 0}$ with $\widetilde{X}_0=0$ then show that $Y:=(Y_t:=\widetilde{X}_{\max(t-T,0)})_{t\geq 0}$ is a submartingale with respect to $(\mathcal{F}_t)_{t\geq 0}$.
Does someone have an idea, many thanks!