Problem 1 Consider the random walk approximation Bi of the Brownian motion in LNl with 2 time steps, that is 30:0 Bl=X1 E2=X1+X2 Denote an upward move of the random walk by + and a downward move by , each of whcih 1 happens with probability 5. Here X1 and X2 have identical and independent distribution as follows: Xi = A for an + move7 and Xi = A for an move7 i = 1.2. Similar to Example 1 of LN1, consider the following sample space n = {++,+, +.} 1. Describe 31-. i = 0. 1,2 as maps from 9 into the set of real numbers R. 2. Find the aalgebra generated by B. for each i = 07 17 2. 3. What is the smallest Ualgebra F that makes Bi, 1' = 0, 17 2 a random map? 4. What is the probability measure [P dened on .F that tells us how likely an event in .F happens? Describe P as a map from .7: into [0, 1]