5.2. THE DEFINITION OF BROWNIAN MOTION AND THE WIENER PROCESS 169
(a) Find the probability that 0< W(1)<1.
(b) Find the probability that 0< W(1)<1 and 1< W(2)<3.
(c) Find the probability that 0< W(1)<1 and 1< W(2)<3 and
0 < W(3)< 1 /2.
(d) Explain why this problem is different from the previous problem,
and also explain how to numerically evaluate to the proabilities.
- LetW(t) be standard Brownian motion.
(a) Find the probability thatW(5)≤3 given thatW(1) = 1.
(b) Find the numbercsuch that Pr[W(9)> c|W(1) = 1] = 0.10.
- Suppose that the fluctuations of a share of stock of a certain company
are well described by a Standard Brownian Motion process. Suppose
that the company is bankrupt if ever the share price drops to zero. If
the starting share price isA(0) = 5, what is the probability that the
share price is above 10 att= 25?. What is the probability that the
company is bankrupt byt= 25? Explain why these are not the same.
- Suppose you own one share of stock whose price changes according to a
Standard Brownian Motion Process. Suppose you purchased the stock
at a priceb+c,c >0 and the present price isb. You have decided to sell
the stock either when it reaches the priceb+cor when an additional
timetgoes by, whichever comes first. What is the probability that you
do not recover your purchase price?
- LetZ be a normally distributed random variable, with mean 0 and
variance 1,Z∼N(0,1). Then consider the continuous time stochastic
processX(t) =
√
tZ. Show that the distribution of X(t) is normal
with mean 0 with variancet. IsX(t) a Brownian motion?
- LetW 1 (t) be a Brownian motion andW 2 (t) be anotherindependent
Brownian motion, andρis a constant between−1 and 1. Then consider
the processX(t) =ρW 1 (t) +
√
1 −ρ^2 W 2 (t). Is thisX(t) a Brownian
motion?
- What is the distribution of W(s) +W(t), for 0 ≤ s ≤ t? (Hint:
Note that W(s) andW(t) are not independent. But you can write
W(s) +W(t) as a sum of independent variables. Done properly, this
problem requires almost no calculation.)
