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Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve

Chapter 1

General Probability Theory - all with Video Answers

Educators

Chapter Questions

02:30

Problem 1

. Using the properties of Definition 1.1.2 for a probability measure $\mathbb{P}$, show the following.
(i) If $A \in \mathcal{F}, B \in \mathcal{F}$, and $A \subset B$, then $\mathbb{P}(A) \leq \mathbb{P}(B)$.
(ii) If $A \in \mathcal{F}$ and $\left\{A_n\right\}_{n=1}^{\infty}$ is a sequence of sets in $\mathcal{F}$ with $\lim _{n \rightarrow \infty} \mathbb{P}\left(A_n\right)=0$ and $A \subset A_n$ for every $n$, then $\mathbb{P}(A)=0$. (This property was used implicitly in Example 1.1.4 when we argued that the sequence of all heads, and indeed any particular sequence, must have probability zero.)

Mengchun Cai
Mengchun Cai
Numerade Educator

Problem 2

he infinite coin-toss space $\Omega_{\infty}$ of Example 1.1.4 is uncountably infinite. In other words, we cannot list all its elements in a sequence.
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1 General Probability Theory
To see that this is impossible, suppose there were such a sequential list of all elements of $\Omega_{\infty}$ :
$$
\begin{aligned}
& \omega^{(1)}=\omega_1^{(1)} \omega_2^{(1)} \omega_3^{(1)} \omega_4^{(1)} \ldots, \\
& \omega^{(2)}=\omega_1^{(2)} \omega_2^{(2)} \omega_3^{(2)} \omega_4^{(2)} \ldots \\
& \omega^{(3)}=\omega_1^{(3)} \omega_2^{(3)} \omega_3^{(3)} \omega_4^{(3)} \ldots,
\end{aligned}
$$
$\vdots$
An element that does not appear in this list is the sequence whose first component is $H$ if $\omega_1^{(1)}$ is $T$ and is $T$ if $\omega_1^{(1)}$ is $H$, whose second component is $H$ if $\omega_2^{(2)}$ is $T$ and is $T$ if $\omega_2^{(2)}$ is $H$, whose third component is $H$ if $\omega_3^{(3)}$ is $T$ and is $T$ if $\omega_3^{(3)}$ is $H$, etc. Thus, the list does not include every element of $\Omega_{\infty}$.

Now consider the set of sequences of coin tosses in which the outcome on each even-numbered toss matches the outcome of the toss preceding it, i.e.,
$$
A=\left\{\omega=\omega_1 \omega_2 \omega_3 \omega_4 \omega_5 \ldots ; \omega_1=\omega_2, \omega_3=\omega_4, \ldots\right\} .
$$
(i) Show that $A$ is uncountably infinite.
(ii) Show that, when $0<p<1$, we have $\mathbb{P}(A)=0$.

Uncountably infinite sets can have any probability between zero and one, including zero and one. The uncountability of the set does not help determine its probability.

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09:01

Problem 3

Consider the set function $\mathbb{P}$ defined for every subset of $[0,1]$ by the formula that $\mathbb{P}(A)=0$ if $A$ is a finite set and $\mathbb{P}(A)=\infty$ if $A$ is an infinite set. Show that $\mathbb{P}$ satisfies (1.1.3)-(1.1.5), but $\mathbb{P}$ does not have the countable additivity property (1.1.2). We see then that the finite additivity property (1.1.5) does not imply the countable additivity property (1.1.2).

Mengchun Cai
Mengchun Cai
Numerade Educator
02:39

Problem 4

. (i) Construct a standard normal random variable $Z$ on the probability space $\left(\Omega_{\infty}, \mathcal{F}_{\infty}, \mathbb{P}\right)$ of Example 1.1.4 under the assumption that the probability for head is $p=\frac{1}{2}$. (Hint: Consider Examples 1.2.5 and 1.2.6.)
(ii) Define a sequence of random variables $\left\{Z_n\right\}_{n=1}^{\infty}$ on $\Omega_{\infty}$ such that
$$
\lim _{n \rightarrow \infty} Z_n(\omega)=Z(\omega) \text { for every } \omega \in \Omega_{\infty}
$$
and, for each $n, Z_n$ depends only on the first $n$ coin tosses. (This gives us a procedure for approximating a standard normal random variable by random variables generated by a finite number of coin tosses, a useful algorithm for Monte Carlo simulation.)

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
01:43

Problem 5

When dealing with double Lebesgue integrals, just as with double Riemann integrals, the order of integration can be reversed. The only
1.9 Exercises
43
assumption required is that the function being integrated be either nonnegative or integrable. Here is an application of this fact.

Let $X$ be a nonnegative random variable with cumulative distribution function $F(x)=\mathbb{P}\{X \leq x\}$. Show that
$$
\mathbb{E} X=\int_0^{\infty}(1-F(x)) d x
$$
by showing that
$$
\int_{\Omega} \int_0^{\infty} \mathbb{I}_{[0, X(\omega))}(x) d x d \mathbb{P}(\omega)
$$
is equal to both $\mathbf{E} X$ and $\int_0^{\infty}(1-F(x)) d x$.

Yiming Zhang
Yiming Zhang
Numerade Educator

Problem 6

Exer Let $u$ be a fixed number in $\mathbb{R}$, and define the convex function $\varphi(x)=e^{u x}$ for all $x \in \mathbb{R}$. Let $X$ be a normal random variable with mean $\mu=\mathbb{E} X$ and standard deviation $\sigma=\left[\mathbb{E}(X-\mu)^2\right]^{\frac{1}{2}}$, i.e., with density
$$
f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}} .
$$
(i) Verify that
$$
\mathrm{E} e^{u X}=e^{u \mu+\frac{1}{2} u^2 \sigma^2} .
$$
(ii) Verify that Jensen's inequality holds (as it must):
$$
\mathbb{E} \varphi(X) \geq \varphi(\mathbb{E} X) .
$$

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02:56

Problem 7

For each positive integer $n$, define $f_n$ to be the normal density with mean zero and variance $n$, i.e.,
$$
f_n(x)=\frac{1}{\sqrt{2 n \pi}} e^{-\frac{x^2}{2 n}} .
$$
(i) What is the function $f(x)=\lim _{n \rightarrow \infty} f_n(x)$ ?
(ii) What is $\lim _{n \rightarrow \infty} \int_{-\infty}^{\infty} f_n(x) d x$ ?
(iii) Note that
$$
\lim _{n \rightarrow \infty} \int_{-\infty}^{\infty} f_n(x) d x \neq \int_{-\infty}^{\infty} f(x) d x .
$$

Explain why this does not violate the Monotone Convergence Theorem, Theorem 1.4.5.

Tyler Gaona
Tyler Gaona
Numerade Educator
10:40

Problem 8

(Moment-generating function). Let $X$ be a nonnegative random variable, and assume that
$$
\varphi(t)=\mathbb{E} e^{t X}
$$
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1 General Probability Theory
is finite for every $t \in \mathbb{R}$. Assume further that $\mathbb{E}\left[X e^{t X}\right]<\infty$ for every $t \in$ $\mathbb{R}$. The purpose of this exercise is to show that $\varphi^{\prime}(t)=\mathbb{E}\left[X e^{t X}\right]$ and, in particular, $\varphi^{\prime}(0)=\mathbb{E} X$.
We recall the definition of derivative:
$$
\varphi^{\prime}(t)=\lim _{s \rightarrow t} \frac{\varphi(t)-\varphi(s)}{t-s}=\lim _{s \rightarrow t} \frac{\mathbf{E} e^{t X}-\mathbf{E} e^{s X}}{t-s}=\lim _{s \rightarrow t} \mathbb{E}\left[\frac{e^{t X}-e^{s X}}{t-s}\right] .
$$

The limit above is taken over a continuous variable $s$, but we can choose a sequence of numbers $\left\{s_n\right\}_{n=1}^{\infty}$ converging to $t$ and compute
$$
\lim _{s_n \rightarrow t} \mathbb{E}\left[\frac{e^{t X}-e^{s_n X}}{t-s_n}\right],
$$

Chris Trentman
Chris Trentman
Numerade Educator

Problem 9

Suppose $X$ is a random variable on some probability space $(\Omega, \mathcal{F}, \mathbb{P}), A$ is a set in $\mathcal{F}$, and for every Borel subset $B$ of $\mathbb{R}$, we have
$$
\int_A \mathbb{I}_B(X(\omega)) d \mathbb{P}(\omega)=\mathbb{P}(A) \cdot \mathbb{P}\{X \in B\} .
$$
Then we say that $X$ is independent of the event $A$.
Show that if $X$ is independent of an event $A$, then
$$
\int_A g(X(\omega)) d \mathbb{P}(\omega)=\mathbb{P}(A) \cdot \mathbb{E} g(X)
$$
for every nonnegative, Borel-measurable function $g$.

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Problem 10

Exer Let $\mathbb{P}$ be the uniform (Lebesgue) measure on $\Omega=[0,1]$. Define
$$
Z(\omega)=\left\{\begin{array}{l}
0 \text { if } 0 \leq \omega<\frac{1}{2}, \\
2 \text { if } \frac{1}{2} \leq \omega \leq 1 .
\end{array}\right.
$$

For $A \in \mathcal{B}[0,1]$, define
$$
\widetilde{\mathbb{P}}(A)=\int_A Z(\omega) d \mathbb{P}(\omega) .
$$
(i) Show that $\widetilde{\mathbb{P}}$ is a probability measure.
(ii) Show that if $\mathbb{P}(A)=0$, then $\widetilde{\mathbb{P}}(A)=0$. We say that $\widetilde{\mathbb{P}}$ is absolutely continuous with respect to $\mathbb{P}$.
(iii) Show that there is a set $A$ for which $\widetilde{\mathbb{P}}(A)=0$ but $\mathbb{P}(A)>0$. In other words, $\widetilde{\mathbb{P}}$ and $\mathbb{P}$ are not equivalent.

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02:12

Problem 11

. In Example 1.6.6, we began with a standard normal random variable $X$ under a measure $\boldsymbol{P}$. According to Exercise 1.6, this random variable has the moment-generating function
$$
\mathbb{E} e^{u X}=e^{\frac{1}{2} u^2} \text { for all } u \in \mathbb{R} .
$$

The moment-generating function of a random variable determines its distribution. In particular, any random variable that has moment-generating function $e^{\frac{1}{2} u^2}$ must be standard normal.

In Example 1.6.6, we also defined $Y=X+\theta$, where $\theta$ is a constant, we set $Z=e^{-\theta X-\frac{1}{2} \theta^2}$, and we defined $\widetilde{\mathbb{P}}$ by the formula (1.6.9):
$$
\widetilde{\mathbb{P}}(A)=\int_A Z(\omega) d \mathbb{P}(\omega) \text { for all } A \in \mathcal{F} .
$$

We showed by considering its cumulative distribution function that $Y$ is a standard normal random variable under $\widetilde{\mathbb{P}}$. Give another proof that $Y$ is standard normal under $\widetilde{\mathbb{P}}$ by verifying the moment-generating function formula
$$
\widetilde{\mathbb{E}} e^{u Y}=e^{\frac{1}{2} u^2} \text { for all } u \in \mathbb{R} \text {. }
$$

Amany Waheeb
Amany Waheeb
Numerade Educator

Problem 12

n Example 1.6.6, we began with a standard normal random variable $X$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and defined the random variable $Y=X+\theta$, where $\theta$ is a constant. We also defined $Z=e^{-\theta X-\frac{1}{2} \theta^2}$ and used $Z$ as the Radon-Nikodým derivative to construct the probability measure $\widetilde{\mathbb{P}}$ by the formula (1.6.9):
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1 General Probability Theory
$$
\tilde{\mathbb{P}}(A)=\int_A Z(\omega) d \mathbb{P}(\omega) \text { for all } A \in \mathcal{F} .
$$

Under $\widetilde{\mathbb{P}}$, the random variable $Y$ was shown to be standard normal.
We now have a standard normal random variable $Y$ on the probability space $(\Omega, \mathcal{F}, \widetilde{\mathbb{P}})$, and $X$ is related to $Y$ by $X=Y-\theta$. By what we have just stated, with $X$ replaced by $Y$ and $\theta$ replaced by $-\theta$, we could define $\widehat{Z}=e^{\theta Y-\frac{1}{2} \theta^2}$ and then use $\widehat{Z}$ as a Radon-Nikodým derivative to construct a probability measure $\widehat{\mathbb{P}}$ by the formula
$$
\widehat{\mathbb{P}}(A)=\int_A \widehat{Z}(\omega) d \tilde{\mathbb{P}}(\omega) \text { for all } A \in \mathcal{F},
$$
so that, under $\widehat{\mathbb{P}}$, the random variable $X$ is standard normal. Show that $\widehat{Z}=\frac{1}{Z}$ and $\widehat{\mathbb{P}}=\mathbb{P}$.

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Problem 13

Change of measure for a normal random variable). A nonrigorous but informative derivation of the formula for the Radon-Nikodým derivative $Z(\omega)$ in Example 1.6.6 is provided by this exercise. As in that example, let $X$ be a standard normal random variable on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and let $Y=X+\theta$. Our goal is to define a strictly positive random variable $Z(\omega)$ so that when we set
$$
\widetilde{\mathbb{P}}(A)=\int_A Z(\omega) d \mathbb{P}(\omega) \text { for all } A \in \mathcal{F},
$$
the random variable $Y$ under $\widetilde{\mathbb{P}}$ is standard normal. If we fix $\bar{\omega} \in \Omega$ and choose a set $A$ that contains $\bar{\omega}$ and is "small," then (1.9.4) gives
$$
\tilde{\mathbb{P}}(A) \approx Z(\bar{\omega}) \mathbb{P}(A),
$$
where the symbol $\approx$ means "is approximately equal to." Dividing by $\mathbb{P}(A)$, we see that
$$
\frac{\widetilde{\mathbb{P}}(A)}{\mathbb{P}(A)} \approx Z(\bar{\omega})
$$
for "small" sets $A$ containing $\bar{\omega}$. We use this observation to identify $Z(\bar{\omega})$.
With $\bar{\omega}$ fixed, let $x=X(\bar{\omega})$. For $\epsilon>0$, we define $B(x, \epsilon)=\left[x-\frac{\epsilon}{2}, x+\frac{\epsilon}{2}\right]$ to be the closed interval centered at $x$ and having length $\epsilon$. Let $y=x+\theta$ and $B(y, \epsilon)=\left[y-\frac{\epsilon}{2}, y+\frac{\epsilon}{2}\right]$.
(i) Show that
$$
\frac{1}{\epsilon} \mathbb{P}\{X \in B(x, \epsilon)\} \approx \frac{1}{\sqrt{2 \pi}} \exp \left\{-\frac{X^2(\bar{\omega})}{2}\right\} .
$$
(ii) In order for $Y$ to be a standard normal random variable under $\widetilde{\mathbb{P}}$, show that we must have
$$
\frac{1}{\epsilon} \widetilde{\mathbb{P}}\{Y \in B(y, \epsilon)\} \approx \frac{1}{\sqrt{2 \pi}} \exp \left\{-\frac{Y^2(\bar{\omega})}{2}\right\} .
$$
1.9 Exercises
47
(iii) Show that $\{X \in B(x, \epsilon)\}$ and $\{Y \in B(y, \epsilon)\}$ are the same set, which we call $A(\bar{\omega}, \epsilon)$. This set contains $\bar{\omega}$ and is "small" when $\epsilon>0$ is small.
(iv) Show that
$$
\frac{\tilde{\mathbb{P}}(A)}{\mathbb{P}(A)} \approx \exp \left\{-\theta X(\bar{\omega})-\frac{1}{2} \theta^2\right\} .
$$

The right-hand side is the value we obtained for $Z(\bar{\omega})$ in Example 1.6.6.

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05:41

Problem 14

(Change of measure for an exponential random variable). Let $X$ be a nonnegative random variable defined on a probability space $(\Omega, \mathcal{F}, P)$ with the exponential distribution, which is
$$
\mathbb{P}\{X \leq a\}=1-e^{-\lambda a}, a \geq 0,
$$
where $\lambda$ is a positive constant. Let $\tilde{\lambda}$ be another positive constant, and define
$$
Z=\frac{\tilde{\lambda}}{\lambda} e^{-(\bar{\lambda}-\lambda) X} .
$$

Define $\widetilde{\mathbb{P}}$ by
$$
\tilde{\mathbb{P}}(A)=\int_A Z d \mathbb{P} \quad \text { for all } A \in \mathcal{F} .
$$
(i) Show that $\tilde{\mathbb{P}}(\Omega)=1$.
(ii) Compute the cumulative distribution function
$$
\widetilde{\mathbb{P}}\{X \leq a\} \text { for } a \geq 0
$$
for the random variable $X$ under the probability measure $\widetilde{\mathbb{P}}$.
Exercise 1.15 (Provided by Alexander Ng). Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and assume $X$ has a density function $f(x)$ that is positive for every $x \in \mathbb{R}$. Let $g$ be a strictly increasing, differentiable function satisfying
$$
\lim _{y \rightarrow-\infty} g(y)=-\infty, \quad \lim _{y \rightarrow \infty} g(y)=\infty,
$$
and define the random variable $Y=g(X)$.
Let $h(y)$ be an arbitrary nonnegative function satisfying $\int_{-\infty}^{\infty} h(y) d y=1$. We want to change the probability measure so that $h(y)$ is the density function for the random variable $Y$. To do this, we define
$$
Z=\frac{h(g(X)) g^{\prime}(X)}{f(X)} .
$$
(i) Show that $Z$ is nonnegative and $\mathbb{E} Z=1$.

Now define $\tilde{\mathbb{P}}$ by
$$
\tilde{\mathbb{P}}(A)=\int_A Z d \mathbb{P} \text { for all } A \in \mathcal{F} .
$$
(ii) Show that $Y$ has density $h$ under $\tilde{\mathbb{P}}$.

Heena Haldankar
Heena Haldankar
Numerade Educator

Problem 15

(Provided by Alexander $\mathrm{Ng}$ ). Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and assume $X$ has a density function $f(x)$ that is positive for every $x \in \mathbb{R}$. Let $g$ be a strictly increasing, differentiable function satisfying
$$
\lim _{y \rightarrow-\infty} g(y)=-\infty, \quad \lim _{y \rightarrow \infty} g(y)=\infty,
$$
and define the random variable $Y=g(X)$.
Let $h(y)$ be an arbitrary nonnegative function satisfying $\int_{-\infty}^{\infty} h(y) d y=1$. We want to change the probability measure so that $h(y)$ is the density function for the random variable $Y$. To do this, we define
$$
Z=\frac{h(g(X)) g^{\prime}(X)}{f(X)} .
$$
(i) Show that $Z$ is nonnegative and $\mathbb{E} Z=1$.

Now define $\tilde{\mathbb{P}}$ by
$$
\widetilde{\mathbb{P}}(A)=\int_A Z d \mathbb{P} \quad \text { for all } A \in \mathcal{F} .
$$
(ii) Show that $Y$ has density $h$ under $\tilde{\mathbb{P}}$.

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