Chapter 2, Information and Conditioning Video Solutions, Stochastic Calculus for Finance II : Continuous-Time Models

Problem 1

. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a general probability space, and suppose a random variable $X$ on this space is measurable with respect to the trivial $\sigma$-algebra $\mathcal{F}_0=\{\emptyset, \Omega\}$. Show that $X$ is not random (i.e., there is a constant $c$ such that $X(\omega)=c$ for all $\omega \in \Omega)$. Such a random variable is called degenerate.

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01:59

Problem 2

Independence of random variables can be affected by changes of measure. To illustrate this point, consider the space of two coin tosses $\Omega_2=\{H H, H T, T H, T T\}$, and let stock prices be given by
$$
\begin{aligned}
& S_0=4, S_1(H)=8, S_1(T)=2, \\
& S_2(H H)=16, S_2(H T)=S_2(T H)=4, S_2(T T)=1 .
\end{aligned}
$$

Consider two probability measures given by
$$
\begin{aligned}
& \tilde{\mathbb{P}}(H H)=\frac{1}{4}, \tilde{\mathbb{P}}(H T)=\frac{1}{4}, \tilde{\mathbb{P}}(T H)=\frac{1}{4}, \tilde{\mathbb{P}}(T T)=\frac{1}{4}, \\
& \mathbb{P}(H H)=\frac{1}{9}, \mathbb{P}(H T)=\frac{2}{9}, \mathbb{P}(T H)=\frac{2}{9}, \mathbb{P}(T T)=\frac{1}{9} .
\end{aligned}
$$

Define the random variable
$$
X=\left\{\begin{array}{l}
1 \text { if } S_2=4, \\
0 \text { if } S_2 \neq 4 .
\end{array}\right.
$$
78
2 Information and Conditioning
(i) List all the sets in $\sigma(X)$.
(ii) List all the sets in $\sigma\left(S_1\right)$.
(iii) Show that $\sigma(X)$ and $\sigma\left(S_1\right)$ are independent under the probability measure $\tilde{\mathbb{P}}$.
(iv) Show that $\sigma(X)$ and $\sigma\left(S_1\right)$ are not independent under the probability measure $\mathbb{P}$.
(v) Under $\mathbb{P}$, we have $\mathbb{P}\left\{S_1=8\right\}=\frac{2}{3}$ and $\mathbb{P}\left\{S_1=2\right\}=\frac{1}{3}$. Explain intuitively why, if you are told that $X=1$, you would want to revise your estimate of the distribution of $S_1$.

Manik Pulyani

Numerade Educator

06:19

Problem 3

(Rotating the axes). Let $X$ and $Y$ be independent standard normal random variables. Let $\theta$ be a constant, and define random variables
$$
V=X \cos \theta+Y \sin \theta \text { and } W=-X \sin \theta+Y \cos \theta .
$$

Show that $V$ and $W$ are independent standard normal random variables.

Shu-Ting Huang

Numerade Educator

03:41

Problem 4

In Example 2.2.8, $X$ is a standard normal random variable and $Z$ is an independent random variable satisfying
$$
\mathbb{P}\{Z=1\}=\mathbb{P}\{Z=-1\}=\frac{1}{2} .
$$

We defined $Y=X Z$ and showed that $Y$ is standard normal. We established that although $X$ and $Y$ are uncorrelated, they are not independent. In this exercise, we use moment-generating functions to show that $Y$ is standard normal and $X$ and $Y$ are not independent.
(i) Establish the joint moment-generating function formula
$$
\mathbb{E} e^{u X+v Y}=e^{\frac{1}{2}\left(u^2+v^2\right)} \cdot \frac{e^{u v}+e^{-u v}}{2} .
$$
(ii) Use the formula above to show that $\mathrm{E} e^{v Y}=e^{\frac{1}{2} v^2}$. This is the momentgenerating function for a standard normal random variable, and thus $Y$ must be a standard normal random variable.
(iii) Use the formula in (i) and Theorem 2.2.7(iv) to show that $X$ and $Y$ are not independent.

Aimal Hassan

Numerade Educator

03:54

Problem 5

Let $(X, Y)$ be a pair of random variables with joint density function
$$
f_{X, Y}(x, y)= \begin{cases}\frac{2|x|+y}{\sqrt{2 \pi}} \exp \left\{-\frac{(2|x|+y)^2}{2}\right\} & \text { if } y \geq-|x|, \\ 0 & \text { if } y<-|x| .\end{cases}
$$

Show that $X$ and $Y$ are standard normal random variables and that they are uncorrelated but not independent.

Christopher Stanley

Numerade Educator

Problem 6

Consider a probability space $\Omega$ with four elements, which we call $a, b, c$, and $d$ (i.e., $\Omega=\{a, b, c, d\}$ ). The $\sigma$-algebra $\mathcal{F}$ is the collection of all subsets of $\Omega$; i.e., the sets in $\mathcal{F}$ are
$$
\begin{aligned}
& \Omega,\{a, b, c\},\{a, b, d\},\{a, c, d\},\{b, c, d\}, \\
& \{a, b\},\{a, c\},\{a, d\},\{b, c\},\{b, d\},\{c, d\}, \\
& \{a\},\{b\},\{c\},\{d\}, \emptyset .
\end{aligned}
$$

We define a probability measure $\mathbb{P}$ by specifying that
$$
\mathbb{P}\{a\}=\frac{1}{6}, \mathbb{P}\{b\}=\frac{1}{3}, \mathbb{P}\{c\}=\frac{1}{4}, \mathbb{P}\{d\}=\frac{1}{4},
$$
and, as usual, the probability of every other set in $\mathcal{F}$ is the sum of the probabilities of the elements in the set, e.g., $\mathbb{P}\{a, b, c\}=\mathbb{P}\{a\}+\mathbb{P}\{b\}+\mathbb{P}\{c\}=\frac{3}{4}$.
We next define two random variables, $X$ and $Y$, by the formulas
$$
\begin{aligned}
& X(a)=1, X(b)=1, X(c)=-1, X(d)=-1, \\
& Y(a)=1, Y(b)=-1, Y(c)=1, Y(d)=-1 .
\end{aligned}
$$

We then define $Z=X+Y$.
(i) List the sets in $\sigma(X)$.
(ii) Determine $\mathbb{E}[Y \mid X]$ (i.e., specify the values of this random variable for $a$, $b, c$, and $d$ ). Verify that the partial-averaging property is satisfied.
(iii) Determine $\mathbb{E}[Z \mid X]$. Again, verify the partial-averaging property.
(iv) Compute $\mathbb{E}[Z \mid X]-\mathbb{E}[Y \mid X]$. Citing the appropriate properties of conditional expectation from Theorem 2.3.2, explain why you get $X$.

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05:01

Problem 7

et $Y$ be an integrable random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and let $\mathcal{G}$ be a sub- $\sigma$-algebra of $\mathcal{F}$. Based on the information in $\mathcal{G}$, we can form the estimate $\mathbb{E}[Y \mid \mathcal{G}]$ of $Y$ and define the error of the estimation $\mathrm{Err}_{\mathrm{rr}}=Y-\mathbb{E}[Y \mid \mathcal{G}]$. This is a random variable with expectation zero and some variance $\operatorname{Var}(\operatorname{Err})$. Let $X$ be some other $\mathcal{G}$-measurable random variable, which we can regard as another estimate of $Y$. Show that
$$
\operatorname{Var}(\operatorname{Err}) \leq \operatorname{Var}(Y-X) .
$$

In other words, the estimate $\mathbb{E}[Y \mid \mathcal{G}]$ minimizes the variance of the error among all estimates based on the information in $\mathcal{G}$. (Hint: Let $\mu=\mathbb{E}(Y-X)$. Compute the variance of $Y-X$ as
$$
\mathbf{E}\left[(Y-X-\mu)^2\right]=\mathbb{E}\left[((Y-\mathbb{E}[Y \mid \mathcal{G}])+(\mathbb{E}[Y \mid \mathcal{G}]-X-\mu))^2\right] .
$$

Multiply out the right-hand side and use iterated conditioning to show the cross-term is zero.)

Amany Waheeb

Numerade Educator

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Problem 8

Let $X$ and $Y$ be integrable random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Then $Y=Y_1+Y_2$, where $Y_1=\mathbb{E}[Y \mid X]$ is $\sigma(X)$-measurable and $Y_2=Y-E[Y \mid X]$. Show that $Y_2$ and $X$ are uncorrelated. More generally, show that $Y_2$ is uncorrelated with every $\sigma(X)$-measurable random variable.

Rashmi Sinha

Numerade Educator

04:56

Problem 9

. Let $X$ be a random variable.
(i) Give an example of a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a random variable $X$ defined on this probability space, and a function $f$ so that the $\sigma$-algebra generated by $f(X)$ is not the trivial $\sigma$-algebra $\{\emptyset, \Omega\}$ but is strictly smaller than the $\sigma$-algebra generated by $X$.
(ii) Can the $\sigma$-algebra generated by $f(X)$ ever be strictly larger than the $\sigma$-algebra generated by $X$ ?

Barsha Rana

Numerade Educator

Problem 10

Let $X$ and $Y$ be random variables (on some unspecified probability space $(\Omega, \mathcal{F}, \mathbb{P})$ ), assume they have a joint density $f_{X, Y}(x, y)$, and assume $\mathbb{E}|Y|<\infty$. In particular, for every Borel subset $C$ of $\mathbb{R}^2$, we have
$$
\mathbb{P}\{(X, Y) \in C\}=\int_C f_{X, Y}(x, y) d x d y .
$$

In elementary probability, one learns to compute $\mathbb{E}[Y \mid X=x]$, which is a nonrandom function of the dummy variable $x$, by the formula
$$
\mathbb{E}[Y \mid X=x]=\int_{-\infty}^{\infty} y f_{Y \mid X}(y \mid x) d y,
$$
where $f_{Y \mid X}(y \mid x)$ is the conditional density defined by
$$
f_{Y \mid X}(y \mid x)=\frac{f_{X, Y}(x, y)}{f_X(x)} .
$$

The denominator in this expression, $f_X(x)=\int_{-\infty}^{\infty} f_{X, Y}(x, \eta) d \eta$, is the marginal density of $X$, and we must assume it is strictly positive for every $x$. We introduce the symbol $g(x)$ for the function $\mathbb{E}[Y \mid X=x]$ defined by (2.6.1); i.e.,
$$
g(x)=\int_{-\infty}^{\infty} y f_{Y \mid X}(y \mid x) d y=\int_{-\infty}^{\infty} \frac{y f_{X, Y}(x, y)}{f_X(x)} d y .
$$

In measure-theoretic probability, conditional expectation is a random variable $\mathbb{E}[Y \mid X]$. This exercise is to show that when there is a joint density for $(X, Y)$, this random variable can be obtained by substituting the random variable $X$ in place of the dummy variable $x$ in the function $g(x)$. In other words, this exercise is to show that
$$
\mathbf{E}[Y \mid X]=g(X) .
$$
(We introduced the symbol $g(x)$ in order to avoid the mathematically confusing expression $E[Y \mid X=X]$.)

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Problem 11

(i) Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and let $W$ be a nonnegative $\sigma(X)$-measurable random variable. Show there exists a function $g$ such that $W=g(X)$. (Hint: Recall that every set in $\sigma(X)$ is of the form $\{X \in B\}$ for some Borel set $B \subset \mathbb{R}$. Suppose first that $W$ is the indicator of such a set, and then use the standard machine.)
(ii) Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and let $Y$ be a nonnegative random variable on this space. We do not assume that $X$ and $Y$ have a joint density. Nonetheless, show there is a function $g$ such that $\mathbb{E}[Y \mid X]=g(X)$.

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