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Introduction to Financial Mathematics

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Introduction to Financial Mathematics

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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}}.
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