Introduction

Let $(Y, \mathscr {F})$ be a martingale, and show that $\mathbb {E}\left[Y_{n+m} \mid \mathcal{F}_{n}\right] = Y_{n}$ for all $n, m \geq 0$.

Let $Z_{n}$ be the size of the $n$th generation of a branching process with $Z_{0} = 1$, having mean family size $\mu$ and extinction probability $\eta$. Show that $Z_{n} \mu^{-n}$ and $\eta^{Z_{n}}$ define martingales.

Let $\left\{ S_{n}: n \geq 0\right\}$ be a simple symmetric random walk on the integers with $S_{0} = k$. Show that $S_{n}$ and $S_{n}^{2}-n$ are martingales. Making assumptions similar to those of de Moivre (see Example (12.1.4)), find the probability of ruin and the expected duration of the game for the gambler's ruin problem.

Let $(Y, \mathscr {F})$ be a martingale with the property that $\mathbb {E}\left[Y_{n}^{2}\right] < \infty$ for all $n$. Show that, for $i \leq j \leq k$, $\mathbb {E}\left[\left(Y_{k}-Y_{j}\right) Y_{i}\right] = 0$, and $\mathbb {E}\left[\left(Y_{k}-Y_{j}\right)^{2} \mid \mathcal{F}_{i}\right] = \mathbb {E}\left[Y_{k}^{2} \mid \mathcal{F}_{i}\right]-\mathbb {E}\left[Y_{j}^{2} \mid \mathcal{F}_{i}\right]$. Suppose there exists $K$ such that $\mathbb {E}\left[Y_{n}^{2}\right] \leq K$ for all $n$. Show that the sequence $\left\{ Y_{n}\right\}$ converges in mean square as $n \rightarrow \infty$.

Let $Y$ be a martingale and let $u$ be a convex function mapping $\mathbb {R}$ to $\mathbb {R}$. Show that $\left\{ u\left(Y_{n}\right): n \geq 0\right\}$ is a submartingale provided that $\mathbb {E}\left[u\left(Y_{n}\right)^{+}\right] < \infty$ for all $n$.

Show that $\left|Y_{n}\right|, Y_{n}^{2}$, and $Y_{n}^{+}$constitute submartingales whenever the appropriate moment conditions are satisfied.

Let $Y$ be a submartingale and let $u$ be a convex non-decreasing function mapping $\mathbb {R}$ to $\mathbb {R}$. Show that $\left\{ u\left(Y_{n}\right): n \geq 0\right\}$ is a submartingale provided that $\mathbb {E}\left[u\left(Y_{n}\right)^{+}\right] < \infty$ for all $n$.

Show that (subject to a moment condition) $Y_{n}^{+}$constitutes a submartingale, but that $\left|Y_{n}\right|$ and $Y_{n}^{2}$ need not constitute submartingales.

Let $X$ be a discrete-time Markov chain with countable state space $S$ and transition matrix $\mathbf{P}$. Suppose that $\psi : S \rightarrow \mathbb {R}$ is bounded and satisfies $\sum_{j \in S} p_{i j} \psi (j) \leq \lambda \psi (i)$ for some $\lambda > 0$ and all $i \in S$. Show that $\lambda^{-n} \psi \left(X_{n}\right)$ constitutes a supermartingale.

Let $G_{n}(s)$ be the probability generating function of the size $Z_{n}$ of the $n$th generation of a branching process, where $Z_{0} = 1$ and $\operatorname {Var}\left[Z_{1}\right] > 0$. Let $H_{n}$ be the inverse function of the function $G_{n}$, viewed as a function on the interval $[0,1]$, and show that $M_{n} = \left\{ H_{n}(s)\right\}^{Z_{n}}$ defines a martingale with respect to the sequence $Z$.