Chapter 3. Brownian Motion
Denition 3.3.1. Let (, F , P ) be a probability space. For each w , suppose there
is a continuous function W (t) of t 0 that
satises W (0) = 0 and that depends on w.
Then W (t), t 0, is a Brownian motion if for
all 0 = t0 < t1 <
Chapter 5. Risk-Neutral Pricing
We have a probability space (, F , P ) and a
ltration F (t), dened for 0 t T . where T
is a xed nal time. Suppose further that Z is
an almost surely positive random variable with
EZ = 1, and we dene
P ( A) =
A
Z (w) dP (w)
Chapter 1. General Probability Theory
Denition 1.1.1. Let be a nonempty set,
and let F be a collection of subsets of . We
say F is a -algebra (or eld) provided that
(i) the empty set belongs to F ;
(ii) whenever a set A belongs to F , its complement Ac al
Chapter 2. Information and Conditioning
Denition 2.1.1. Let be a nonempty set.
Let T be a xed positive number, and assume
that for each t [0, T ] there is a -eld F (t).
Assume further that if s t, then every set in
F (s) is also in F (t). Then we call the