You know I will downvote if you are not answering the question.
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 dt = $1 if the economy state in quarter t is G and dt $0 if the economy state in quarter t is B. Define X1 = di X2 = d + d2 X3 = d + d2 + d3 Note that both (di, d2, dz) 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 d, d2, d3 and X1, X2, X3? 2. Describe X2 as a map from the sample space 12 (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, dz are random maps on a common probability space (12, F,P). Find F. Example 1 Suppose the state of the economy in each quarter is either good (G) with probability q and bad (B) with probability 1 9. Suppose further that these probabilities are identical and independent over time. Consider the following probability space of the state of the economy in the next two quarters: (1) 12 = {GG, GB, BG, BB} (2) Even in this simple example, there are several o-algebras. Let's consider the following class of o-algebras called a filtration which will play an important role later: Example 5 In Example 1, consider a security that pays dt = $1 if the economy state in quarter t is G and dt = $0 if the economy state in quarter t is B. Then X = di + d2 is a random variable with 0(X) ={0, 12, {BB}, {GG}, {GB, BG}, {BB, GG}, {GB, BG, BB}, {GB, BG, GG}} C 22 Note that o(X) represents information known by observing X. 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 dt = $1 if the economy state in quarter t is G and dt $0 if the economy state in quarter t is B. Define X1 = di X2 = d + d2 X3 = d + d2 + d3 Note that both (di, d2, dz) 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 d, d2, d3 and X1, X2, X3? 2. Describe X2 as a map from the sample space 12 (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, dz are random maps on a common probability space (12, F,P). Find F. Example 1 Suppose the state of the economy in each quarter is either good (G) with probability q and bad (B) with probability 1 9. Suppose further that these probabilities are identical and independent over time. Consider the following probability space of the state of the economy in the next two quarters: (1) 12 = {GG, GB, BG, BB} (2) Even in this simple example, there are several o-algebras. Let's consider the following class of o-algebras called a filtration which will play an important role later: Example 5 In Example 1, consider a security that pays dt = $1 if the economy state in quarter t is G and dt = $0 if the economy state in quarter t is B. Then X = di + d2 is a random variable with 0(X) ={0, 12, {BB}, {GG}, {GB, BG}, {BB, GG}, {GB, BG, BB}, {GB, BG, GG}} C 22 Note that o(X) represents information known by observing X
