Optional stopping

Let $(Y, \mathcal{F})$ be a martingale and $T$ a stopping time such that $\mathbb {P}\left(T < \infty \right) = 1$. Show that $\mathbb {E}\left[Y_{T}\right] = \mathbb {E}\left[Y_{0}\right]$ if either of the following holds:

(a) $\mathbb {E}\left[\sup_{n}\left|Y_{T \wedge n}\right|\right] < \infty$,

(b) $\mathbb {E}\left[\left|Y_{T \wedge n}\right|^{1+\delta }\right] \leq c$ for some $c, \delta > 0$ and all $n$.

Let $(Y, \mathscr {F})$ be a martingale. Show that $\left(Y_{T \wedge n}, \mathcal{F}_{n}\right)$ is a uniformly integrable martingale for any finite stopping time $T$ such that either:

Let $(Y, \mathcal{F})$ be a uniformly integrable martingale, and let $S$ and $T$ be finite stopping times satisfying $S \leq T$. Let $Y_{\infty }$ be the almost sure limit as $n \rightarrow \infty$ of $Y_{n}$. Prove that

Let $\left\{ S_{n}: n \geq 0\right\}$ be a simple symmetric random walk with $0 < S_{0} < N$ and with absorbing barriers at 0 and $N$. Use the optional stopping theorem to show that the mean time until absorption is $\mathbb {E}\left[S_{0}\left(N-S_{0}\right)\right]$.

Let $\left\{ S_{n}: n \geq 0\right\}$ be a simple symmetric random walk with $S_{0} = 0$. Show that

constitutes a martingale if $\cos \lambda \neq 0$.

Let $a$ and $b$ be positive integers. Show that the time $T$ until absorption at one of two absorbing barriers at $-a$ and $b$ satisfies

Let $\left\{ S_{n}: n \geq 0\right\}$ be a simple symmetric random walk on the positive and negative integers, with $S_{0} = 0$. For each of the following random variables, determine whether or not it is a stopping time and find its mean:

Let $S_{n} = a+\sum_{r = 1}^{n} X_{r}$ be a simple symmetric random walk. The walk stops at the earliest time $T$ when it reaches either of the two positions 0 or $K$ where $0 < a < K$. Show that $M_{n} = \sum_{r = 0}^{n} S_{r}-\frac{1}{3} S_{n}^{3}$ is a martingale and deduce that $\mathbb {E}\left[\sum_{r = 0}^{T} S_{r}\right] = \frac{1}{3}\left(K^{2}-a^{2}\right) a+a$.

Let $X_{i}$ be independent random variables each equally likely to take the values $\pm 1$, and let $T = \min \left\{ n: S_{n} \in \left\{ -a, b\right\} \right\}$. Verify the conditions of the optional stopping theorem (12.5.1) for the martingale $S_{n}^{2}-n$ and the stopping time $T$.

Children are either female or male. Their sexes are independent random variables, being female with probability $q$ or male with probability $p = 1-q$. A woman ceases childbearing at stage $T$, and we write $G_{n}$ and $B_{n}$ for the numbers of girls and boys born to her up to and including stage $n$. Assume that $T$ is a finite stopping time for the sequence $\left\{ \left(G_{n}, B_{n}\right): n \geq 1\right\}$. Show that, no matter the stopping rule that yields $T$, we have $\mathbb {E}\left[G_{T}\right] / \mathbb {E}\left[B_{T}\right] = q / p$. What can be said about $\mathbb {E}\left[G_{T} / B_{T}\right]$?

Let $X = \left\{ X_{n}: n \geq 0\right\}$ be a Markov chain with state space $\left\{ 0,1, \ldots , b\right\}$, such that $i \rightarrow 0, b$ for $i \in \left\{ 1,2, \ldots , b-1\right\}$. If $X$ is also a martingale, show that 0 and $b$ are absorbing, and that, given $X_{0}$, the probability of absorption at $b$ is $X_{0} / b$.