Homework #5 EN 553.633/433 Solved

Description

Three problems:
2.21 (You do not have to code or implement the inverse transform or A-R solution.)
A. Consider a Markov chain with state space {1, 2, 3} and with a stationary probability
distribution of π = 335
11 11 11 [ ] , , . Do the following:
(a) Determine the values of the entries in the transition matrix P under the constraint
that the values in the upper-left 2 × 2 block are equal to each other (i.e., p11 = p12
= p21 = p22) and that the sum of the entries in the first column is 13/15.
(b) Using the P in part (a), carry out a simulation of 5000 independent replicates of
the Markov chain for two steps of the process. In particular, compute 5000
independent values of X2 and compare the sample proportions of X2 equaling each
of {1, 2, 3} with the stationary probability values. Initialize each replicate at X0 =
1. It is not necessary to carry out a statistical test to do the comparison.
(c) Repeat the analysis of part (b) with the exception of considering 20 steps of the
Markov chain (producing values of X20). Comment on whether the results are
materially different from those in part (b) and, if so, comment on how the results
differ.
B. (a) If 𝑋𝑋 ∼ 𝑁𝑁(µ,σ2 ), derive 𝐸𝐸[𝑒𝑒𝑋𝑋].
(b) If 𝑊𝑊(𝑡𝑡) is the standard Wiener process and 𝑈𝑈(𝑊𝑊(𝑡𝑡)) = exp(𝑡𝑡 + 𝑊𝑊(𝑡𝑡)
2 ) , derive
𝐸𝐸[𝑈𝑈(𝑊𝑊(𝑡𝑡))].