A 3 part question.
Suppose $X:[0,1]\times\Omega \longrightarrow \mathbb{R}$ is a bounded and progressively measurable (with respect to the filtration $\left\{\mathscr{F}_t\right\}_{t\in[0,1]}$) stochastic process.
Note that by bounded I mean: \begin{equation*} \sup_{(t,\omega)\in [0,1]\times\Omega}\left\vert X(t,\omega)\right\vert <\infty \end{equation*} Define the process $X_n:[0,1]\times\Omega$ by: \begin{equation*} X_n(t,\omega):= n\int_{0\vee t-\tfrac{1}{n}}^t X(s,\omega) ds \end{equation*} Then is:
1) $X_n$ continuous in $t$ (I know this is true but can't show it...I've been stuck on it for a long while now).
2) If 1. is true, then is $X_n$ is Lipschitz Continuous in $t$? (I originally tried to prove this statement, but got horribly stuck so I thought I would try to prove a weaker statement first....which I still couldn't prove unfortunately.
3) Show that $X_n$ is adapted with respect to the filtration $\left\{\mathscr{F}_t\right\}_{t\in[0,1]}$
Many thanks.