Part VIII

Conditional Expectation. Martingales

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Section 29Conditional Expectation[28/86%]

Problem 29.1

Let $\mathcal{A}$ be the algebra consisting of the two elements $\emptyset$ and the whole space $\Omega$ . Find $\mathbb {E}\left[\xi \mid \mathcal{A}\right]$ .

Problem 29.2

The joint distribution of a pair of integer-valued random variables $\xi$ and $\eta$ is given by the table

	$\xi = -1$	$\xi = 0$	$\xi = 1$
$\eta = -1$	$1/8$	$1/12$	$7/24$
$\eta = 1$	$5/24$	$1/6$	$1/8$

, where the entry at the intersection of column $\xi = i$ and row $\eta = j$ is the probability $\mathbb {P}\left(\xi =i, \eta =j\right)$ . Find:

(a)

$\mathbb {E}\left[\xi \mid \eta \right]$ ;

(b)

$\mathbb {E}\left[\eta \mid \xi \right]$ ;

(c)

$\mathbb {E}\left[\xi \mid \eta^{2}\right]$ ;

(d)

$\mathbb {E}\left[\xi \mid \eta^{3}\right]$ ;

(e)

$\mathbb {E}\left[\eta \mid \xi^{2}\right]$ ;

(f)

$\mathbb {E}\left[\eta \mid \xi^{3}\right]$ .

Problem 29.3

Solve problem 29.2 for the table

	$\xi = -2$	$\xi = 0$	$\xi = 1$
$\eta = -1$	$1/6$	$1/6$	$1/6$
$\eta = 2$	$1/6$	$1/6$	$1/6$

(a)

(b)

(c)

(d)

(e)

(f)

Problem 29.4

Solve problem 29.2 for the table

	$\xi = -1$	$\xi = 0$	$\xi = 1$
$\eta = -2$	$1/8$	$1/12$	$7/24$
$\eta = 0$	$1/12$	$1/12$	$1/8$
$\eta = 1$	$1/8$	$1/12$	$0$

(a)

(b)

(c)

(d)

(e)

(f)

Problem 29.5

On the probability space $(\Omega , \mathcal{F}, \mathbb {P}\left(\right))$ , where $\Omega = [0,1]$ , $\mathcal{F}$ is the $\sigma$ -algebra of Borel subsets, and $\mathbb {P}\left(\right)$ is Lebesgue measure, a random variable $\xi$ is given. Let $\mathcal{A}$ be the $\sigma$ -algebra generated by the sets $[0,1/3)$ , $\{ 1/3\}$ , and $(1/3,1/2)$ . Find $\mathbb {E}\left[\xi \mid \mathcal{A}\right]$ if:

(a)

$\xi (\omega ) = \omega$ ;

(b)

$\xi (\omega ) = \sin \pi \omega$ ;

(c)

$\xi (\omega ) = \begin{cases} 1, & \omega \in [0, 1/3], \\ 2, & \omega \in (1/3, 1]; \end{cases}$

(d)

$\xi (\omega ) = 1 - \omega$ ;

(e)

$\xi (\omega ) = \omega^{2}$ ;

(f)

$\xi (\omega ) = \begin{cases} 0, & \omega \in [0, 2/3], \\ 1, & \omega \in (2/3, 1]. \end{cases}$

Problem 29.6

(a)(b)(c)

On the probability space $(\Omega , \mathcal{F}, \mathbb {P}\left(\right))$ , where $\Omega = [0,1]$ , $\mathcal{F}$ is the $\sigma$ -algebra of Borel subsets, and $\mathbb {P}\left(\right)$ is Lebesgue measure, a random variable $\xi (\omega ) = \omega$ is given. Find $\mathbb {E}\left[\xi \mid \mathcal{A}\right]$ if $\mathcal{A}$ is the $\sigma$ -algebra:

(a)

of all Borel subsets of the segment $[0,1]$ that are symmetric about the point $1/2$ ;

(b)

generated by the sets $[0,1/3]$ and $[1/3,2/3]$ ;

(c)

generated by the random variable $\eta = \min \{ 2\omega , 1\}$ .

Problem 29.7

(a)(b)

\eta (\omega ) = \begin{cases} -1, & \omega < 1/2, \\ 1, & \omega \geq 1/2 \end{cases}

are given. Sketch the graphs of:

(a)

$\mathbb {E}\left[\xi \mid \eta \right]$ ;

(b)

$\mathbb {E}\left[\xi \mid \eta^{2}\right]$ .

Problem 29.8

Suppose the random variable $\xi$ takes at most $n$ values. Is it true that $\mathbb {E}\left[\xi \mid \mathcal{A}\right]$ also takes at most $n$ values?

Problem 29.9

Must the random variable $\mathbb {E}\left[\xi \mid \mathcal{A}\right]$ be measurable with respect to $\sigma (\xi )$ ?

Problem 29.10

Let $\xi$ and $\eta$ be independent, identically distributed random variables with finite expectation. Find $\mathbb {E}\left[\xi \mid \xi +\eta \right]$ .

Problem 29.11

Is the following statement true: if $\xi_{n} \to \xi$ almost surely, then $\mathbb {E}\left[\eta \mid \xi_{n}\right] \xrightarrow {p} \mathbb {E}\left[\eta \mid \xi \right]$ for every random variable $\eta$ ?

Problem 29.12

(a)(b)

Let the random variable $\xi$ have the exponential distribution with parameter $1$ , and let $t>0$ . Find:

(a)

$\mathbb {E}\left[\xi \mid \min (\xi ,t)\right]$ ;

(b)

$\mathbb {E}\left[\xi \mid \max (\xi ,t)\right]$ .

Problem 29.13

(a)(b)

Solve problem 29.12 if the random variable $\xi$ has the uniform distribution on the segment $[0,2]$ .

(a)

(b)

Problem 29.14

Let the random variable $\xi$ have the standard normal distribution. Find $\mathbb {E}\left[\xi \mid \xi^{2}\right]$ .

Problem 29.15

Let $\xi$ and $\eta$ be independent random variables, each with the standard normal distribution. Find $\mathbb {E}\left[\xi^{2}+\eta^{2} \mid \xi +\eta \right]$ .

Problem 29.16

Let the independent random variables $\xi$ and $\eta$ each have the geometric distribution with parameter $p$ . Find the conditional distribution $\mathbb {P}\left(\xi =k \mid \xi +\eta =n\right)$ .

Problem 29.17

(a)(b)

Let the independent random variables $\xi$ and $\eta$ each have the Poisson distribution with parameter $\lambda$ . Find:

(a)

$\mathbb {E}\left[\xi^{2} \mid \xi +\eta =k\right]$ ;

(b)

the conditional distribution $\mathbb {P}\left(\xi =k \mid \xi +\eta =n\right)$ .

Problem 29.18

(a)(b)

Let the random variables $\xi$ and $\eta$ be independent and have the binomial distribution with parameters $m$ and $p$ . Find:

(a)

$\mathbb {E}\left[\xi^{2} \mid \xi +\eta =1\right]$ ;

(b)

the conditional distribution $\mathbb {P}\left(\xi =k \mid \xi +\eta =1\right)$ .

Problem 29.19

(a)(b)

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables having the uniform distribution on the segment $[0,1]$ . Find:

(a)

$\mathbb {E}\left[\xi_{1} \mid \max (\xi_{1}, \ldots , \xi_{n})\right]$ ;

(b)

$\mathbb {E}\left[\xi_{1} \mid \min (\xi_{1}, \ldots , \xi_{n})\right]$ .

Problem 29.20

Solve problem 29.19 if the random variables $\xi_{k}$ have the exponential distribution with parameter $1$ .

Problem 29.21

(a)(b)(c)(d)

Let $\xi$ and $\eta$ be independent random variables having the uniform distribution on the segment $[0,1]$ . Find:

(a)

$\mathbb {E}\left[\xi \mid \xi +\eta \right]$ ;

(b)

$\mathbb {E}\left[\xi -\eta \mid \xi +\eta \right]$ ;

(c)

$\mathbb {E}\left[\xi^{2}-\eta^{2} \mid \xi +\eta \right]$ ;

(d)

$\mathbb {E}\left[\xi +\eta \mid \xi \right]$ .

Problem 29.22

Let $\mathbb {E}\left[\xi^{2}\right] < \infty$ . Prove that the random variable $\xi$ is measurable with respect to the $\sigma$ -algebra $\mathcal{A}$ if and only if

\mathbb {E}\left[\left(\xi - \mathbb {E}\left[\xi \mid \mathcal{A}\right]\right)^{2}\right] = 0.

Problem 29.23

Let $\mathbb {E}\left[\xi^{2}\right] < \infty$ and $\mathbb {E}\left[\eta^{2}\right] < \infty$ . Prove that if the equalities $\mathbb {E}\left[\xi \mid \eta \right] = \eta$ and $\mathbb {E}\left[\eta \mid \xi \right] = \xi$ hold, then $\xi = \eta$ almost surely.

Problem 29.24

Find $\mathbb {E}\left[\xi \mid \eta \right]$ if the joint density of the random vector $(\xi ,\eta )$ equals:

(a)

$p(x,y) = \begin{cases} 4xy & \text{for } 0 \leq x,y \leq 1, \\ 0 & \text{otherwise}; \end{cases}$

(b)

$p(x,y) = \begin{cases} x+y & \text{for } 0 \leq x,y \leq 1, \\ 0 & \text{otherwise}; \end{cases}$

(c)

$p(x,y) = \begin{cases} ye^{-x}+e^{-2x} & \text{for } x \geq 0,\ 0 \leq y \leq 1, \\ 0 & \text{otherwise}; \end{cases}$

(d)

$p(x,y) = \begin{cases} \dfrac {1+9x^{2}y^{2}}{8} & \text{for } -1 \leq x,y \leq 1, \\ 0 & \text{otherwise}; \end{cases}$

(e)

the two-dimensional normal density with correlation coefficient $\rho$ .

Problem 29.25

Let the joint density of the random vector $(\xi ,\eta )$ equal

p(x,y) = \begin{cases} xe^{-x(y+1)}, & x,y \geq 0, \\ 0, & \text{otherwise}. \end{cases}

Find:

(a)

the one-dimensional (marginal) distributions of $\xi$ and $\eta$ ;

(b)

the conditional density of $\xi$ given $\eta$ ;

(c)

the conditional density of $\eta$ given $\xi$ ;

(d)

the conditional mean $\mathbb {E}\left[\xi \mid \eta \right]$ ;

(e)

the conditional mean $\mathbb {E}\left[\eta \mid \xi \right]$ .

Problem 29.26

(a)(b)(c)

Let $\xi$ have the Poisson distribution with parameter $\lambda$ , and let the random variable $\mathbb {E}\left[\eta \mid \xi \right]$ have a normal density with zero mean and variance $\xi$ .

(a)

Find the characteristic function of $\eta$ .

(b)

Prove that the distribution of $\eta$ has no density.

(c)

Prove that $\eta /\sqrt{\lambda }$ converges weakly, as $\lambda \to \infty$ , to the standard normal law.

Problem 29.27

Let the random variable $\xi$ have the exponential distribution with parameter $1$ , and let the random variable $\mathbb {E}\left[\eta \mid \xi \right]$ have the Poisson distribution with parameter $\xi$ . Find the distribution of the random variable $\eta$ .

Problem 29.28

Let $\xi_{1}, \xi_{2}, \ldots$ be independent, identically distributed random variables with finite expectation, and let $S_{n} = \xi_{1} + \ldots + \xi_{n}$ . Let $\mathcal{A}_{n}$ be the $\sigma$ -algebra generated by the sums $S_{n}, S_{n+1}, \ldots$ . Prove that

\mathbb {E}\left[\xi _{1} \mid \mathcal{A}_{n}\right] = \frac{S_{n}}{n}.

Section 30Martingales[16/6%]

Problem 30.1

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with $\mathbb {P}\left(\xi_{i}=0\right) = \mathbb {P}\left(\xi_{i}=2\right) = 1/2$ . Prove that the sequence of products $\eta_{n} = \prod_{i=1}^{n} \xi_{i}$ forms a martingale.

Remark.

Let $(\Omega , \mathcal{F}, \mathbb {P}\left(\right))$ be a probability space, $T \subseteq \mathbb {R}$ , and let $\{ \mathcal{A}_{t}\}$ be a filtration of $\sigma$ -algebras, i.e. $\mathcal{A}_{t} \subseteq \mathcal{F}$ and $\mathcal{A}_{t} \subseteq \mathcal{A}_{s}$ for all $t,s \in T$ , $t<s$ . A random process $\xi (t)$ , $t \in T$ , is called a martingale with respect to the filtration $\{ \mathcal{A}_{t}\}$ if $\xi (t)$ is measurable with respect to $\mathcal{A}_{t}$ , $\mathbb {E}\left[\left|\xi (t)\right|\right] < \infty$ , and $\mathbb {E}\left[\xi (s) \mid \mathcal{A}_{t}\right] = \xi (t)$ almost surely for all $t<s$ . If for every $t$ , $\mathcal{A}_{t}$ is generated by the family $\{ \xi (s), s \leq t\}$ and $\xi (t)$ is a martingale with respect to the filtration $\{ \mathcal{A}_{t}\}$ , one simply says that $\xi (t)$ is a martingale. The process $\xi (t)$ is called a submartingale (supermartingale) with respect to $\{ \mathcal{A}_{t}\}$ if the first two properties in the definition of a martingale hold and $\mathbb {E}\left[\xi (s) \mid \mathcal{A}_{t}\right] \geq \xi (t)$ almost surely (respectively, $\mathbb {E}\left[\xi (s) \mid \mathcal{A}_{t}\right] \leq \xi (t)$ almost surely) for all $t<s$ .

Problem 30.2

(a)(b)

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with $\mathbb {P}\left(\xi_{i}=1\right) = 1 - \mathbb {P}\left(\xi_{i}=-1\right) = p$ , $p \neq 1/2$ . Set $S_{n} = \xi_{1} + \ldots + \xi_{n}$ and $\mathcal{A}_{n} = \sigma (S_{1}, \ldots , S_{n})$ . Prove that the following sequence is a martingale with respect to the filtration $\{ \mathcal{A}_{n}\}$ :

(a)

$\eta_{n} = S_{n} - n(2p-1)$ ;

(b)

$\eta_{n} = \left(\dfrac {1-p}{p}\right)^{S_{n}}$ .

Problem 30.3

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with $\mathbb {P}\left(\xi_{i}=1\right) = \mathbb {P}\left(\xi_{i}=-1\right) = 1/2$ . Set $S_{n} = \xi_{1} + \ldots + \xi_{n}$ and $\mathcal{A}_{n} = \sigma (S_{1}, \ldots , S_{n})$ . Prove that for any numbers $A \in (-\infty ,\infty )$ and $\lambda \in (0,\pi /2)$ , the sequence

\eta _{n} = \frac{e^{i\lambda (S_{n}+A)}}{\cos ^{n}\lambda }

is a martingale with respect to the filtration $\{ \mathcal{A}_{n}\}$ .

Problem 30.4

(a)(b)

Let $\xi_{1}, \xi_{2}, \ldots$ be independent, identically distributed random variables with zero mean. Set $S_{n} = \xi_{1} + \ldots + \xi_{n}$ and $\mathcal{A}_{n} = \sigma (S_{1}, \ldots , S_{n})$ . Prove that the following sequence is a martingale with respect to the filtration $\{ \mathcal{A}_{n}\}$ :

(a)

$\eta_{n} = (S_{n})^{2} - n\mathbb {E}\left[\xi_{1}^{2}\right]$ , if $\mathbb {E}\left[\xi_{1}^{2}\right] < \infty$ ;

(b)

$\eta_{n} = \dfrac {e^{\lambda S_{n}}}{(\mathbb {E}\left[e^{\lambda \xi_{1}}\right])^{n}}$ , if $\mathbb {E}\left[e^{\lambda \xi_{1}}\right] < \infty$ .

Problem 30.5

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with zero expectations. Denote $S_{n} = \xi_{1} + \ldots + \xi_{n}$ . Prove that the sequence of sums $\{ S_{n}\}$ forms a martingale.

Problem 30.6

Let $\xi_{1}, \xi_{2}, \ldots$ be nonnegative random variables with finite mean values, $S_{n} = \xi_{1} + \ldots + \xi_{n}$ . Prove that the sequence of sums $\{ S_{n}\}$ forms a submartingale.

Problem 30.7

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with zero expectations and finite variances. Denote $S_{n} = \xi_{1} + \ldots + \xi_{n}$ . Prove that the sequence $\{ S_{n}^{2} - B_{n}\}$ forms a martingale, where $B_{n} = \operatorname {Var}\left[\xi_{1}\right] + \ldots + \operatorname {Var}\left[\xi_{n}\right]$ .

Problem 30.8

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with $\mathbb {E}\left[\xi_{n}\right] = 1$ for every $n$ . Prove that the sequence of products $\eta_{n} = \prod_{i=1}^{n} \xi_{i}$ forms a martingale.

Problem 30.9

Let $\xi$ be a random variable with finite expectation, and let $\{ \mathcal{A}_{n}\}_{n \geq 0}$ be a nondecreasing sequence of $\sigma$ -algebras. Prove that the sequence $\xi_{n} = \mathbb {E}\left[\xi \mid \mathcal{A}_{n}\right]$ forms a martingale.

Problem 30.10

Let $\xi_{1}, \xi_{2}, \ldots$ be independent, identically distributed random variables with finite expectation, and let $S_{n} = \xi_{1} + \ldots + \xi_{n}$ . Prove that the sequence

\ldots , \frac{S_{n}}{n}, \ldots , \frac{S_{3}}{3}, \frac{S_{2}}{2}, S_{1}

forms a martingale.

Problem 30.11

Let $\xi (t)$ be a martingale with respect to the filtration $\{ \mathcal{A}_{t}\}$ . Prove that the process $\xi (t)$ has uncorrelated increments, i.e. $\operatorname {Cov}\left[ \xi (s)-\xi (t), \xi (v)-\xi (u) \right] = 0$ for $t<s<u<v$ .

Problem 30.12

Let $\xi (t)$ be a homogeneous Poisson process with parameter $\lambda$ . Prove that $\zeta (t) = \mathrm{exp}\left(\xi (t)-at\right)$ is a submartingale for $a \leq \lambda (e-1)$ and a supermartingale for $a \geq \lambda (e-1)$ .

Problem 30.13

Prove that a Wiener process started at zero is a continuous-time martingale.

Problem 30.14

Let $\{ \xi_{n}\}$ be a martingale with $\mathbb {E}\left[(\xi_{n}-\xi_{n-1})^{2}\right] \leq c$ for some $c<\infty$ . Prove that $\xi_{n}/n \to \mathbb {E}\left[\xi_{1}\right]$ in probability.

Problem 30.15

Let $\xi_{0}, \xi_{1}, \ldots$ be a branching process, $\xi_{0}=1$ and $\mathbb {E}\left[\xi_{1}\right]=m$ . Set $\eta_{n} = \xi_{n}/m^{n}$ . Prove that $\{ \eta_{n}\}$ forms a martingale.

Problem 30.16

(a)(b)

Let $w(t)$ be a standard Wiener process. Determine which of the following processes are martingales:

(a)

$e^{cw(t)}$ ;

(b)

$tw(t) - \displaystyle \int_{0}^{t} w(s)\, ds$ .