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Introduction To Financial Mathematics

Introduction to Financial Mathematics
Lecture Notes — MAP 5601
Department of Mathematics
Florida State University
Fall 2003
Table of Contents
1. Finite Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Elements of Continuous Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Lecture Notes — MAP 5601 map5601LecNotes.tex i 8/27/2003
1. Finite Probability Spaces
The toss of a coin or the roll of a die results in a finite number of possible outcomes.
We represent these outcomes by a set of outcomes called a sample space. For a coin we
might denote this sample space by {H, T} and for the die {1, 2, 3, 4, 5, 6}. More generally
any convenient symbols may be used to represent outcomes. Along with the sample space
we also specify a probability function, or measure, of the likelihood of each outcome. If
the coin is a fair coin, then heads and tails are equally likely. If we denote the probability
measure by P, then we write P(H) = P(T) = 1
2 . Similarly, if each face of the die is equally
likely we may write P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1
6 .
Defninition 1.1. A finite probability space is a pair (
, P) where
is the sample space set
and P is a probability measure:
If
= {!1, !2, . . . , !n}, then
(i) 0 < P(!i)  1 for all i = 1, . . . , n
(ii)
n Pi=1
P(!i) = 1.
In general, given a set of A, we denote the power set of A by P(A). By definition this
is the set of all subsets of A. For example, if A = {1, 2}, then P(A) = {;, {1}, {2}, {1, 2}}.
Here, as always, ; is the empty set. By additivity, a probability measure on
extends to
P(
) if we set P(;) = 0.
Example. For the toss of a fair die, P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1
6 ,...