Introduction to Financial Mathematics :: essays research papers

Introduction to Financial MathematicsTable of Contents1. Finite Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Elements of Continuous Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Lecture Notes MAP 5601 map5601LecNotes.tex i 8/27/20031. Finite Probability SpacesThe 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 warning space. For a coin wemight bring up this ideal space by H, T and for the die 1, 2, 3, 4, 5, 6. More generallyany convenient symbols may be used to represent outcomes. Along with the sample spacewe also specify a probability function, or step, of the likelihood of each outcome. Ifth e coin is a fair coin, then heads and tails are every bit likely. If we denote the probabilitymeasure by P, then we pull through P(H) = P(T) = 12 . Similarly, if each face of the die is equallylikely we may write P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 16 .Defninition 1.1. A finite probability space is a pair (, P) where is the sample space setand P is a probability measureIf = 1, 2, . . . , n, then(i) 0 P(i) 1 for all i = 1, . . . , n(ii)n Pi=1P(i) = 1.In general, given a set of A, we denote the power set of A by P(A). By definition thisis the set of all subsets of A. For example, if A = 1, 2, then P(A) = , 1, 2, 1, 2.