Question

This problem is about "Modeling with Itô Stochastic Differential Equations - E. Allen"

1.1. Consider the random experiment of rolling one die. (a) Find the sample space N (b) Carefully determine the o-algebra, A, of sets generated by A1 and A2 2, 3} (c) Define a probability {1,2} measure P on the sample space 2.

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Answer 1

When we do the random experiment of rolling one die, we have possible outcomes, namely, 1, 2,3, 4, 5, or 6 a) Hence the sample space is {x:x = 1,2,3,4, 5, or 6 Or simply 1,2,3,4,5,6} b) To determine the o -algebra, A, of sets generated by 1,2, A2 2,3 A1 _ By definition of a -algebra, , E A, further A1, A2 E A implies A4, A E A {3,4,5,6}, Ag 1,4,5,6} As Ag _ Also their intersections and finite union belong to A. Thus A1 n A2 1,2,3 2}, A1 U A2 (A1 U A2)c [4,5, 6} (A1 n A2) U (A1 U A2)¢ = {2,4,5,6 A U A1,3,4, 5, 6} A U A2 2,3,4,5,6} A1 U AS1,2,4,5,6} _ A U A2 (3) _

A1 n Ag1 etc. Thus the a -algebra, wwwwww ,1},2}, 3, (1,2}, (1, 3}, [2,3}, (1,2,3}, {4,5,6},) {1,4,5,6},{2,4,5,6}, (3,4,5,6}, {1,2,4,5,6}, (1,3,4,5,6}, [2,3, 4,5,6},{1,2,3,4,5,6} A sample space n, is given by c) A probability measureP on PA [0, 1] (i) P(A) 0, for all A E A. (ii) P(n) 1. (iii) Countable additivity: If A1, A2, ... E A are pairwise disjoint, that is, A, n Ak Vj k, then P(Up A)P)