Springer Finance

108 3 Brownian Motion

Lemma, Lemma 2.3.4. In order to compute the expectation on the right-hand

side of (3.5.2), we replace W(s) by a dummy variable x to hold it constant

and then take the unconditional expectation of the remaining random variable

(i.e., we define g(x) = lEf(W(t)- W(s) + x)). But W(t)- W(s) is normally

distributed with mean zero and variance t -s. Therefore,

1 100 ,.,^2

g(x) = f(w + x)e-^2 C•-•> dw.

J2rr (t -s) -oo

(3.5.3)

The Independence Lemma states that if we now take the function g ( x) defined

by (3.5.3) and replace the dummy variable x by the random variable W(s),

then equation (3.5.1) holds. 0

We may make the change of variable T = t-s and y = w + x in (3.5.3) to
obtain
1 100 (y-:c)^2
g(x) = � f(y)e- 2 T dy.
y21l"T -oo
We define the transition density p(r, x,y) for Brownian motion to be

1 (y-:c)^2

p(r, x,y) =. �e-^2 T ,

y21l"T

so that we may further rewrite (3.5.3) as

g(x) =I: f(y)p(r, x,y) dy

and (3.5.1) as

IE[f(W(t))j.F(s)] =I: f(y)p(r, W(s),y)dy.

(3.5.4)

(3.5.5)

This equation has the following interpretation. Conditioned on the informa­
tion in .F(s) (which contains all the information obtained by observing the
Brownian motion up to and including times), the conditional density of W(t)
is p(r, W(s), y). This is a density in the variable y. This density is normal with
mean W(s) and variance T = t-s. In particular, the only information from
F(s) that is relevant is the value of W(s). The fact that only W(s) is relevant
is the essence of the Markov property.

3.6 First Passage Time Distribution

In Chapter 5 of Volume I, we studied the first passage time for a random walk,
first using the optional sampling theorem for martingales to obtain the distri­
bution in Section 5.2 and then rederiving the distribution using the reflection