182 CHAPTER 5. BROWNIAN MOTION
probability that the random walk goes up to valueabefore going down to
valuebwhen the step size is ∆xis
P[ toabefore−b] =
b∆x
(a+b)∆x
=
b
a+b
Thus, the probability ofa >0, before hitting−b <0 does not depend on
the step size, and also does not depend on the time interval. Therefore in
passing to the limit the probabilities should remain the same. Here is a place
where it is easier to derive the result from the coin-flipping game and pass to
the limit than to derive the result directly from Wiener process principles.
The Distribution of the Maximum
Lettbe a given time, leta >0 be a given value, then
P
[
max
0 ≤u≤t
W(u)≥a
]
=P[Ta≤t]
=
2
√
2 π
∫∞
a/
√
t
exp(−y^2 /2)dy
Sources
This section is adapted from: Probability Models, by S. Ross, andA First
Course in Stochastic ProcessesSecond Edition by S. Karlin, and H. Taylor,
Academic Press, 1975.
Problems to Work for Understanding
- Differentiate the c.d.f. ofTato obtain the expression for the p.d.f ofTa.
- Show thatE[Ta] =∞fora >0.
- Suppose that the fluctuations of a share of stock of a certain company
are well described by a Wiener 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 company is bankrupt
byt= 25? What is the probability that the share price is above 10 at
t= 25?.