5. SIGMA-ALGEBRA {-40 POINTS, EQUALL'Y DISTRIBUTED) Let the price of one share of a. stock be represented by a stochastic process X = {Xf : t E {0.1. 2, 3. 4}}. Consider the following possible realizations for the stochastic process X0 (w) X105) X2 (w) X3010 X4 (w) [P (w) ml 11 13 15 16 16 0, 06 mg 11 13 15 16 14 0, 065 mg 11 13 15 15 16 0, 06 M4 11 13 15 15 14 0, 065 0:15 11 13 15 14 16 0, 06 we 11 13 15 14 14 0, 065 [U7 11 13 12 13 14 0, 0625 we 11 13 12 13 18 0, 08 mg 11 13 12 13 12 0, 04 win 11 13 12 13 11 0,0675 w\" 11 11 12 13 14 0, 0625 [4.113 11 11 12 13 12 0,0625 (#13 11 11 12 12 12 0,0625 cam 11 11 11 14 20 0,0625 L015 11 11 11 14 12 0,0625 0.116 11 11 11 12 12 0,0625 FIGURE 1. Realizations of a stochastic process (1) What a-algebras [F0,}'1_..F2) are generated by random variables X0. X1. and X2? (2) 1\"that is the conditional distribution of X4 if you know X2? (3] Calculate the conditional expectation of X4 if you know X2. (4) This exercise explains why)r ltration f0 C .71 C .772 is called information. Assume t = 1. List some events from F2 occurence of which you current I}: cannot determine