Introduction to Financial Mathematics :: essays research papers

Introduction to Financial MathematicsTable of Contents1. impermanent Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Elements of Continuous Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123. first derivative Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Lecture Notes MAP 5601 map5601LecNotes.tex i 8/27/20031. bounded Probability SpacesThe toss of a strickle or the roll of a leave results in a finite number of possible outcomes.We guard these outcomes by a determine of outcomes called a sample space. For a coin wemight denote this sample 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 speciate a hazard function, or footfall, of the likelihood of severally outcome. Ifthe coin is a fair coin, then heads and tails are equally likely. If we denote the probabilitymeasure by P, then we write 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 couplet (, 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.