Stopping times

If $T_{1}$ and $T_{2}$ are stopping times with respect to a filtration $\mathcal{F}$, show that $T_{1}+T_{2}, \max \left\{ T_{1}, T_{2}\right\}$, and $\min \left\{ T_{1}, T_{2}\right\}$ are stopping times also.

Let $X_{1}, X_{2}, \ldots$ be a sequence of non-negative independent random variables and let $N(t) = \max \left\{ n: X_{1}+X_{2}+\cdots +X_{n} \leq t\right\}$. Show that $N(t)+1$ is a stopping time with respect to a suitable filtration to be specified.

Let $(Y, \mathcal{F})$ be a submartingale and $x > 0$. Show that

Let $(Y, \mathscr {F})$ be a non-negative supermartingale and $x > 0$. Show that

Let $(Y, \mathscr {F})$ be a submartingale and let $S$ and $T$ be stopping times satisfying $0 \leq S \leq T \leq N$ for some deterministic $N$. Show that $\mathbb {E}\left[Y_{0}\right] \leq \mathbb {E}\left[Y_{S}\right] \leq \mathbb {E}\left[Y_{T}\right] \leq \mathbb {E}\left[Y_{N}\right]$.

Let $\left\{ S_{n}\right\}$ be a simple random walk with $S_{0} = 0$ such that $0 < p = \mathbb {P}\left(S_{1} = 1\right) < \frac{1}{2}$. Use de Moivre's martingale to show that $\mathbb {E}\left[\sup_{m} S_{m}\right] \leq p /(1-2 p)$. Show further that this inequality may be replaced by an equality.

Let $\mathcal{F}$ be a filtration. For any stopping time $T$ with respect to $\mathcal{F}$, denote by $\mathcal{F}_{T}$ the collection of all events $A$ such that, for all $n, A \cap \left\{ T \leq n\right\} \in \mathcal{F}_{n}$. Let $S$ and $T$ be stopping times.