• Home
  • Textbooks
  • Introductory to Probability Models
  • Brownian Motion and Stationary Processes

Introductory to Probability Models

Sheldon M. Ross

Chapter 10

Brownian Motion and Stationary Processes - all with Video Answers

Educators

Chapter Questions

02:12

Problem 1

What is the distribution of $B(s)+B(t), s \leqslant t ?$

Willis James
Willis James
Numerade Educator
01:56

Problem 2

Compute the conditional distribution of $B(s)$ given that $B\left(t_{1}\right)=A$ and $B\left(t_{2}\right)=B$, where $0<t_{1}<s<t_{2}$.

Amany Waheeb
Amany Waheeb
Numerade Educator
03:20

Problem 3

Compute $E\left[B\left(t_{1}\right) B\left(t_{2}\right) B\left(t_{3}\right)\right]$ for $t_{1}<t_{2}<t_{3}$

Corinne Costa
Corinne Costa
Numerade Educator
07:46

Problem 4

Show that
$$
\begin{aligned}
&P\left\{T_{a}<\infty\right\}=1 \\
&E\left[T_{a}\right]=\infty, \quad a \neq 0
\end{aligned}
$$

Shafiq Rehman
Shafiq Rehman
Numerade Educator
01:25

Problem 5

What is $P\left\{T_{1}<T_{-1}<T_{2}\right\} ?$

Yiming Zhang
Yiming Zhang
Numerade Educator
01:37

Problem 6

Suppose you own one share of a stock whose price changes according to a standard Brownian motion process. Suppose that you purchased the stock at a price $b+c$, $c \geq 0$, and the present price is $b$. You have decided to sell the stock either when it reaches the price $b+c$ or when an additional time $t$ goes by (whichever occurs first). What is the probability that you do not recover your purchase price?

Lucas Finney
Lucas Finney
Numerade Educator
03:51

Problem 7

Compute an expression for
$$
P\left\{\max _{t_{1} \leqslant s \leqslant t_{2}} B(s)>x\right\}
$$

Ryan Williams
Ryan Williams
Numerade Educator
09:55

Problem 8

Consider the random walk that in each $\Delta t$ time unit either goes up or down the amount $\sqrt{\Delta t}$ with respective probabilities $p$ and $1-p$, where $p=\frac{1}{2}(1+\mu \sqrt{\Delta t})$.
(a) Argue that as $\Delta t \rightarrow 0$ the resulting limiting process is a Brownian motion process with drift rate $\mu$.
(b) Using part (a) and the results of the gambler's ruin problem (Section 4.5.1), compute the probability that a Brownian motion process with drift rate $\mu$ goes up $A$ before going down $B, A>0, B>0$

Gennady Notowidigdo
Gennady Notowidigdo
Numerade Educator
03:12

Problem 9

Let $\{X(t), t \geqslant 0\}$ be a Brownian motion process with drift coefficient $\mu$ and variance parameter $\sigma^{2}$. What is the joint density function of $X(s)$ and $X(t), s<t ?$

Nick Johnson
Nick Johnson
Numerade Educator
01:53

Problem 10

Let $\{X(t), t \geqslant 0\}$ be a Brownian motion process with drift coefficient $\mu$ and variance parameter $\sigma^{2}$. What is the conditional distribution of $X(t)$ given that $X(s)=c$ when
(a) $s<t$ ?
(b) $t<s$ ?

Victor Salazar
Victor Salazar
Numerade Educator
01:34

Problem 11

Consider a process whose value changes every $h$ time units; its new value being its old value multiplied either by the factor $e^{\sigma \sqrt{h}}$ with probability $p=\frac{1}{2}\left(1+\frac{\mu}{\sigma} \sqrt{h}\right)$ or by the factor $e^{-\sigma \sqrt{h}}$ with probability $1-p .$ As $h$ goes to zero, show that this process converges to geometric Brownian motion with drift coefficient $\mu$ and variance parameter $\sigma^{2}$.

Nick Johnson
Nick Johnson
Numerade Educator
06:32

Problem 12

A stock is presently selling at a price of $$\$ 50$$ per share. After one time period, its selling price will (in present value dollars) be either $$\$ 150$$ or $$\$ 25 .$$ An option to purchase $y$ units of the stock at time 1 can be purchased at cost $c y$.
(a) What should $c$ be in order for there to be no sure win?
(b) If $c=4$, explain how you could guarantee a sure win.
(c) If $c=10$, explain how you could guarantee a sure win.
(d) Use the arbitrage theorem to verify your answer to part (a).

R M
R M
Numerade Educator
02:10

Problem 13

Verify the statement made in the remark following Example $10.2$.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:28

Problem 14

The present price of a stock is 100 . The price at time 1 will be either 50,100 , or
200. An option to purchase $y$ shares of the stock at time 1 for the (present value) price $k y$ costs $c y$.
(a) If $k=120$, show that an arbitrage opportunity occurs if and only if $c>80 / 3$.
(b) If $k=80$, show that there is not an arbitrage opportunity if and only if $20 \leqslant$ $c \leqslant 40$.

Saad Ali Khan
Saad Ali Khan
Numerade Educator
04:11

Problem 15

The current price of a stock is 100 . Suppose that the logarithm of the price of the stock changes according to a Brownian motion process with drift coefficient $\mu=2$ and variance parameter $\sigma^{2}=1 .$ Give the Black-Scholes cost of an option to buy the stock at time 10 for a cost of (a) 100 per unit.
(b) 120 per unit.
(c) 80 per unit. Assume that the continuously compounded interest rate is 5 percent.
A stochastic process $\{Y(t), t \geqslant 0\}$ is said to be a Martingale process if, for $s<t$,
$$
E[Y(t) \mid Y(u), 0 \leqslant u \leqslant s]=Y(s)
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
01:10

Problem 16

If $\{Y(t), t \geqslant 0\}$ is a Martingale, show that
$$
E[Y(t)]=E[Y(0)]
$$

Manik Pulyani
Manik Pulyani
Numerade Educator
02:33

Problem 17

Show that standard Brownian motion is a Martingale.

Amany Waheeb
Amany Waheeb
Numerade Educator
03:23

Problem 18

Show that $\{Y(t), t \geqslant 0\}$ is a Martingale when
$$
Y(t)=B^{2}(t)-t
$$
What is $E[Y(t)] ?$ Hint: First compute $E[Y(t) \mid B(u), 0 \leqslant u \leqslant s]$.

SS
Sagar Singh
Numerade Educator
05:07

Problem 19

Show that $\{Y(t), t \geqslant 0\}$ is a Martingale when
$$
Y(t)=\exp \left\{c B(t)-c^{2} t / 2\right\}
$$
where $c$ is an arbitrary constant. What is $E[Y(t)] ?$
An important property of a Martingale is that if you continually observe the process and then stop at some time $T$, then, subject to some technical conditions (which will hold in the problems to be considered),
$$
E[Y(T)]=E[Y(0)]
$$
The time $T$ usually depends on the values of the process and is known as a stopping time for the Martingale. This result, that the expected value of the stopped Martingale is equal to its fixed time expectation, is known as the Martingale stopping theorem.

SS
Sagar Singh
Numerade Educator
03:12

Problem 20

Let
$$
T=\operatorname{Min}\{t: B(t)=2-4 t\}
$$
That is, $T$ is the first time that standard Brownian motion hits the line $2-4 t$. Use the Martingale stopping theorem to find $E[T]$.

Nick Johnson
Nick Johnson
Numerade Educator
01:45

Problem 21

Let $\{X(t), t \geqslant 0\}$ be Brownian motion with drift coefficient $\mu$ and variance parameter $\sigma^{2}$. That is,
$$
X(t)=\sigma B(t)+\mu t
$$
Let $\mu>0$, and for a positive constant $x$ let
$$
\begin{aligned}
T &=\operatorname{Min}\{t: X(t)=x\} \\
&=\operatorname{Min}\left\{t: B(t)=\frac{x-\mu t}{\sigma}\right\}
\end{aligned}
$$
That is, $T$ is the first time the process $\{X(t), t \geqslant 0\}$ hits $x .$ Use the Martingale stopping theorem to show that
$$
E[T]=x / \mu
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
27:31

Problem 22

Let $X(t)=\sigma B(t)+\mu t$, and for given positive constants $A$ and $B$, let $p$ denote the probability that $\{X(t), t \geqslant 0\}$ hits $A$ before it hits $-B$.
(a) Define the stopping time $T$ to be the first time the process hits either $A$ or $-B$. Use this stopping time and the Martingale defined in Exercise 19 to show that
$$
E\left[\exp \left\{c(X(T)-\mu T) / \sigma-c^{2} T / 2\right\}\right]=1
$$
(b) Let $c=-2 \mu / \sigma$, and show that
$$
E[\exp \{-2 \mu X(T) / \sigma\}]=1
$$
(c) Use part (b) and the definition of $T$ to find $p$. Hint: What are the possible values of $\exp \left\{-2 \mu X(T) / \sigma^{2}\right\} ?$

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
01:17

Problem 23

Let $X(t)=\sigma B(t)+\mu t$, and define $T$ to be the first time the process $\{X(t), t \geqslant 0\}$ hits either $A$ or $-B$, where $A$ and $B$ are given positive numbers. Use the Martingale stopping theorem and part (c) of Exercise 22 to find $E[T]$.

Manik Pulyani
Manik Pulyani
Numerade Educator
07:10

Problem 24

Let $\{X(t), t \geqslant 0\}$ be Brownian motion with drift coefficient $\mu$ and variance parameter $\sigma^{2}$. Suppose that $\mu>0$. Let $x>0$ and define the stopping time $T$ (as in Exercise 21 by
$$
T=\operatorname{Min}\{t: X(t)=x\}
$$
Use the Martingale defined in Exercise 18, along with the result of Exercise 21 , to show that
$$
\operatorname{Var}(T)=x \sigma^{2} / \mu^{3}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
04:53

Problem 25

Compute the mean and variance of
(a) $\int_{0}^{1} t d B(t)$
(b) $\int_{0}^{1} t^{2} d B(t)$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:58

Problem 26

Let $Y(t)=t B(1 / t), t>0$ and $Y(0)=0$
(a) What is the distribution of $Y(t)$ ?
(b) Compare $\operatorname{Cov}(Y(s), Y(t))$.
(c) Argue that $\{Y(t), t \geqslant 0\}$ is a standard Brownian motion process.

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator
02:44

Problem 27

Let $Y(t)=B\left(a^{2} t\right) / a$ for $a>0$. Argue that $\{Y(t)\}$ is a standard Brownian motion process.Let $Y(t)=B\left(a^{2} t\right) / a$ for $a>0$. Argue that $\{Y(t)\}$ is a standard Brownian motion process.

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 28

For $s<t$, argue that $B(s)-\frac{s}{t} B(t)$ and $B(t)$ are independent.

Hoan Nguyen
Hoan Nguyen
Numerade Educator
02:44

Problem 29

Let $\{Z(t), t \geqslant 0\}$ denote a Brownian bridge process. Show that if
$$
Y(t)=(t+1) Z(t /(t+1))
$$
then $\{Y(t), t \geqslant 0\}$ is a standard Brownian motion process.

Nick Johnson
Nick Johnson
Numerade Educator
01:24

Problem 30

Let $X(t)=N(t+1)-N(t)$ where $\{N(t), t \geqslant 0\}$ is a Poisson process with rate $\lambda$. Compute
$$
\operatorname{Cov}[X(t), X(t+s)]
$$

Hunza Gilgit
Hunza Gilgit
Numerade Educator
06:16

Problem 31

Let $\{N(t), t \geqslant 0\}$ denote a Poisson process with rate $\lambda$ and define $Y(t)$ to be the time from $t$ until the next Poisson event.
(a) Argue that $\{Y(t), t \geqslant 0\}$ is a stationary process.
(b) Compute $\operatorname{Cov}[Y(t), Y(t+s)]$.

Robin Corrigan
Robin Corrigan
Numerade Educator
14:13

Problem 32

Let $\{X(t),-\infty<t<\infty\}$ be a weakly stationary process having covariance function $R_{X}(s)=\operatorname{Cov}[X(t), X(t+s)]$
(a) Show that
$$
\operatorname{Var}(X(t+s)-X(t))=2 R_{X}(0)-2 R_{X}(t)
$$
(b) If $Y(t)=X(t+1)-X(t)$ show that $\{Y(t),-\infty<t<\infty\}$ is also weakly stationary having a covariance function $R_{Y}(s)=\operatorname{Cov}[Y(t), Y(t+s)]$ that satisfies
$$
R_{Y}(s)=2 R_{X}(s)-R_{X}(s-1)-R_{X}(s+1)
$$

Gennady Notowidigdo
Gennady Notowidigdo
Numerade Educator
12:53

Problem 33

Let $Y_{1}$ and $Y_{2}$ be independent unit normal random variables and for some constant $w$ set
$$
X(t)=Y_{1} \cos w t+Y_{2} \sin w t, \quad-\infty<t<\infty
$$
(a) Show that $\{X(t)\}$ is a weakly stationary process.
(b) Argue that $\{X(t)\}$ is a stationary process.

Ruirui Liu
Ruirui Liu
Numerade Educator
01:16

Problem 34

Let $\{X(t),-\infty<t<\infty\}$ be weakly stationary with covariance function $R(s)=$ $\operatorname{Cov}(X(t), X(t+s))$ and let $\widetilde{R}(w)$ denote the power spectral density of the process.
(i) Show that $\widetilde{R}(w)=\widetilde{R}(-w)$. It can be shown that
$$
R(s)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \widetilde{R}(w) e^{i w s} d w
$$
(ii) Use the preceding to show that
$$
\int_{-\infty}^{\infty} \widetilde{R}(w) d w=2 \pi E\left[X^{2}(t)\right]
$$

James Kiss
James Kiss
Numerade Educator