164 CHAPTER 5. BROWNIAN MOTION
This is a statement of theMarkov propertyof the Wiener process.
Recall that the sum of independent random variables which are respec-
tively normally distributed with meanμ 1 andμ 2 and variancesσ^21 andσ^22 is a
normally distributed random variable with meanμ 1 +μ 2 and varianceσ^21 +σ^22 ,
see Moment Generating Functions Therefore for incrementsW(t 3 )−W(t 2 )
andW(t 2 )−W(t 1 ) the sumW(t 3 )−W(t 2 )+W(t 2 )−W(t 1 ) =W(t 3 )−W(t 1 ) is
normally distributed with mean 0 and variancet 3 −t 1 as we expect. Property
2 of the definition is consistent with properties of normal random variables.
Let
p(x,t) =
1
√
2 πtexp(−x^2 /(2t))denote the probability density for aN(0,t) random variable. Then to derive
the joint density of the event
W(t 1 ) =x 1 ,W(t 2 ) =x 2 ,...W(tn) =xnwitht 1 < t 2 < ... < tn, it is equivalent to know the joint probability density
of the equivalent event
W(t 1 )−W(0) =x 1 ,W(t 2 )−W(t 1 ) =x 2 −x 1 ,...,W(tn)−W(tn− 1 ) =xn−xn− 1.
Then by part 2, we immediately get the expression for the joint probability
density function:
f(x 1 ,t 1 ;x 2 ,t 2 ;...;xn,tn) =p(x 1 ,t)p(x 2 −x 1 ,t 2 −t 1 )...p(xn−xn− 1 ,tn−tn− 1 )
Comments on Modeling Security Prices with the Wiener Process
A plot of security prices over time and a plot of one-dimensional Brownian
motion versus time has least a superficial resemblance.
If we were to use Brownian motion to model security prices (ignoring
for the moment that some security prices are better modeled with the more
sophisticated geometric Brownian motion rather than simple Brownian mo-
tion) we would need to verify that security prices have the 4 definitional
5.6 Path Properties of Brownian Motion
- The assumption of normal distribution of stock price changes seems to
be a reasonable first assumption. Figure 5.2 illustrates this reasonable