Martingale method

Use the exponential martingale to find $\mathbb {E}\left[e^{-\lambda T}\right]$ where $T = \inf \left\{ t : t \geq 0, B(t) \geq \alpha +\beta t\right\} , \lambda > 0, \alpha > 0, \beta > 0$ and $B(\cdot )$ SBM.

Suppose $W$ is a Brownian motion and $b>0$. If $S=\inf \left\{ t>0: W_{t}=b\right\}$, show that the Laplace transform of the density of $S$ is given by

If $W$ is a $d$-dimensional Brownian motion started at 0 , find $\mathbb {E}\left[T\right]$, where $T$ is the first time $W$ exits the ball of radius $r$ centered at the origin.

Hint: Use the fact that $\left|W_{t}\right|^{2}-d t$ is a martingale.

Let $\left\{ \xi_{n}\right\}_{n \geqslant 1}$ be a sequence of independent real random variables. Set $S_{n} = \sum_{k = 1}^{n} \xi_{k}, n \geqslant 1$. Using martingale methods, prove that the following statements about the series $\sum_{k = 1}^{\infty } \xi_{k}$ are equivalent:

1. the series converges a.s.,

2. the series converges in probability,

3. the series converges in distribution.

Note that by Kolmogorov's zero-one law (Theorem 6 of Chapter III) this series converges or diverges with probability one.

Theorem 6 of Chapter III (zero-one law). Let a family $\left\{ X_{t}, t \in T\right\}$, $T \subset \mathbb {R}$, of independent random elements with values in measurable spaces $(\mathrm{S}_{t}, \mathscr {B}_{t})$, i.e. $X_{t}: \Omega \rightarrow \mathrm{S}_{t}$, $X_{t} \in \mathscr {F} \mid \mathscr {B}_{t}$, $t \in T$, be given on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$. Denote

Then the "tail" $\sigma$-algebra, i.e. the $\sigma$-algebra $\mathscr {G} = \bigcap_{t \in T} \mathscr {F}_{\geqslant t}^{X}$, is degenerate. In other words, for every $A \in \mathscr {G}$ the probability $\mathbb {P}\left(A\right)$ equals zero or one (if $T \cap [t, \infty ) = \varnothing$, then $\mathscr {F}_{\geqslant t}^{X}: = \varnothing$).

(see [12; p. 182]). Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be an $m$-dimensional Brownian motion, where $m \geqslant 3$. Prove that $\left\| W_{t}\right\| \rightarrow \infty$ a.s. as $t \rightarrow \infty$. Prove that with probability 1 the process $W$ does not return to the point 0 in any finite time. Explain why for $m = 1$ these statements are false. What does the corresponding result look like in the case $m = 2$ ?

Let $\left\{ Z_{n} = \exp \left(rS_{n} - n\gamma (r)\right); n \geq 1\right\}$ be the generating function martingale of (9.77), where $S_{n} = X_{1} + \cdots + X_{n}$ and $X_{1}, \ldots , X_{n}$ are IID with mean $\overline{X} < 0$. Let $J$ be the possibly-defective stopping trial for which the process stops after crossing a threshold at $\alpha > 0$ (there is no negative threshold). Show that $\exp \left(r^{*}\alpha \right)$ is an upper bound to the probability of threshold crossing by considering the stopped process $\left\{ Z_{n}^{*}; n \geq 1\right\}$.

This exercise uses a martingale to find the expected number of successive trials $\mathbb {E}\left[J\right]$ until some fixed pattern, $a_{1}, a_{2}, \ldots , a_{k}$, of successive binary digits occurs within a sequence of IID binary rv's $X_{1}, X_{2}, \ldots$ (see Example 4.5.1 and Exercise 5.35 for alternative approaches). We take the stopping time $J$ to be the smallest $n$ for which $(X_{n-k+1}, \ldots , X_{n}) = (a_{1}, \ldots , a_{k})$. A mythical casino and sequence of gamblers who follow a prescribed strategy will be used to determine $\mathbb {E}\left[J\right]$. The outcomes of the plays (trials), $\left\{ X_{n}; n \geq 1\right\}$ at the casino is a binary IID sequence for which $\mathbb {P}\left(X_{n} = i\right) = p_{i}$ for $i \in \left\{ 0,1\right\}$.

If a gambler places a bet $s$ on 1 at play $n$, the return is $s/p_{1}$ if $X_{n} = 1$ and $0$ otherwise. With a bet $s$ on 0, the return is $s/p_{0}$ if $X_{n} = 0$ and $0$ otherwise; i.e., the game is fair.

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 $\left\{ S_{n}\right\}$ 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 $X = \left\{ X_{n}: n \geq 0\right\}$ 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 $X_{0} / b$.

Let $X = \left\{ X(t): t \geq 0\right\}$ be a birth-death process with parameters $\lambda_{i}, \mu_{i}$, where $\lambda_{i} = 0$ if and only if $i = 0$. Define $h(0) = 0, h(1) = 1$, and

Here is a gambling system for playing a fair game. Choose a sequence $x_{1}, x_{2}, \ldots , x_{n}$ of positive numbers.

Wager the sum of the first and last numbers on an evens bet. If you win, delete those two numbers; if you lose, append their sum as an extra term $x_{n+1}\left( = x_{1}+x_{n}\right)$ at the right-hand end of the sequence.

You play iteratively according to the above rule. If the sequence ever contains one term only, you wager that amount on an evens bet. If you win, you delete the term, and if you lose you append it to the sequence to obtain two terms.

Show that, with probability $1$, the game terminates with a profit of $\sum_{1}^{n} x_{i}$, and that the time until termination has finite mean.

This looks like another clever strategy. Show that the mean size of your maximum deficit is infinite. (When Henry Labouchere was sent down from Trinity College, Cambridge, in 1852, his gambling debts exceeded $£ 6000$.)

Here is a martingale approach to the question of determining the mean number of tosses of a coin before the first appearance of the sequence HHH. A large casino contains infinitely many gamblers $G_{1}, G_{2}, \ldots$, each with an initial fortune of $\$ 1$. A croupier tosses a coin repeatedly. For each $n$, gambler $G_{n}$ bets as follows. Just before the $n$th toss he stakes his $\$ 1$ on the event that the $n$th toss shows heads. The game is assumed fair, so that he receives a total of $\$ p^{-1}$ if he wins, where $p$ is the probability of heads. If he wins this gamble, then he repeatedly stakes his entire current fortune on heads, at the same odds as his first gamble. At the first subsequent tail he loses his fortune and leaves the casino, penniless. Let $S_{n}$ be the casino's profit (losses count negative) after the $n$th toss. Show that $S_{n}$ is a martingale. Let $N$ be the number of tosses before the first appearance of HHH ; show that $N$ is a stopping time and hence find $\mathbb {E}\left[N\right]$.

Now adapt this scheme to calculate the mean time to the first appearance of the sequence HTH.

The game 'Red Now' may be played by a single player with a well shuffled conventional pack of 52 playing cards. At times $n = 1,2, \ldots , 52$ the player turns over a new card and observes its colour. Just once in the game he must say, just before exposing a card, "Red Now". He wins the game if the next exposed card is red. Let $R_{n}$ be the number of red cards remaining face down after the $n$th card has been turned over. Show that $X_{n} = R_{n} /(52-n), 0 \leq n < 52$, defines a martingale. Show that there is no strategy for the player which results in a probability of winning different from $\frac{1}{2}$.

In a cloakroom there are $K$ coats belonging to $K$ people who make an attempt to leave by picking a coat at random. Those who pick their own coat leave, the rest return the coats and try again at random. Let $N$ be the number of rounds of attempts until everyone has left. Show that $\mathbb {E}\left[N\right] = K$ and $\operatorname {Var}\left[N\right] \leq K$.

An insurance company receives premiums (net of costs) at rate $\rho$ per unit time. Claims $X_{1}, X_{2}, \ldots$ are independent random variables with the exponential distribution with parameter $\mu$, and they arrive at the times of a Poisson process of rate $\lambda$ (assume the usual independence, as well as $\lambda , \mu , \rho > 0)$. Let $Y(t)$ be the assets of the company at time $t$ where $Y(0) = y > 0$. Show that

where $\theta = \mu -(\lambda / \rho )$.