answer all questions with explanation

Problem 1 Consider the economy in Example 1, Lecture 1 which lasts for three quarters. Similar to Example 5 of Lecture 1, consider a security that pays de = $1 if the economy state in quarter t is G and de = $0 if the economy state in quarter t is B. Define X1 = di X2 = d, + dz X3 = d1 + d2 + ds Note that both (dj, d2, da) and (X1, X2, X3) are stochastic processes. 1. Suppose that the event w = GBG is realized at the end of quarter 3. What are the values for hi, d2, d3 and X1, X2, X3? 2. Describe X2 as a map from the sample space ? (which you found in PS1) into the real line R. Find the smallest sigma algebra that makes X2 a random variable. 3. Suppose that d1, d2, d3 are random maps on a common probability space ($, F, P). Find F