5.6. PATH PROPERTIES OF BROWNIAN MOTION 183
- Suppose you own one share of stock whose price changes according to a
Wiener 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?
Outside Readings and Links:
- Russell Gerrard, City University, London, Stochastic Modeling Notes
for the MSc in Actuarial Science, 2003-2004. Contributed by S. Dunbar
October 30, 2005. - Yuval Peres, University of California Berkeley, Department of Statis-
tics Notes on sample paths of Brownian Motion. Contributed by S.
Dunbar, October 30, 2005.
5.6 Path Properties of Brownian Motion
Rating
Mathematically Mature: may contain mathematics beyond calculus with
proofs.
Section Starter Question
Provide an example of a continuous function which is not differentiable at
some point. Why does the function fail to have a derivative at that point?
What are the possible reasons that a derivative could fail to exist at some
point?
Key Concepts
- With probability 1 a Brownian Motion path is continuous butnowhere
differentiable.