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Introduction to Financial Mathematics 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}}.