Stat433/833 Lecture Notes
Stochastic Processes
Jiahua Chen
Department of Statistics and Actuarial Science University of Waterloo c Jiahua Chen
Key Words: -eld, Brownian motion, diusion process, ergordic, nite dimensional distribution, Gaussian process, Ko
STAT 433/833 PROBLEM SET 3
1. Exercise 3.7, page 84 of the textbook.
2. Exercise 3.12, page 85 of the textbook.
In addition, add in the following part: Suppose now that = 0, X(0) = 1, and that at (deterministic) time T the process stops growing and is rep
STAT 433/833 PROBLEM SET 1
1. Exercise 1.7, page 36 of the textbook.
2. Exercise 1.14, page 39 of the textbook.
3. Exercise 2.2, page 58 of the textbook.
4. Exercise 2.5, page 58 of the textbook.
5. Exercise 2.7, page 59 of the textbook.
6. (a) Let cfw_Xn
Xing Shuo Zhai
ID: 20332540
STAT 433 Problem Set 2 Solution
1. Rearrange the order of states to cfw_1, 2, 0, then
"
#
0.25 0.25 0.5
0.25 0.25
P = 0.25 0.25 0.5 and Q =
.
0.25 0.25
0
0
1
" #
2
(a) v = (I Q)1 e =
.
2
Thus, the expected number of years until
Xing Shuo Zhai
ID: 20332540
STAT 833 Final Assignment Solution
1.
2. (a)
i. N 0 (0) = N (f (0) = N (0) = 0
ii. For 0 t1 < t2 t3 < t4 , since f (t) is strictly increasing, we have that 0 f (t1 ) <
f (t2 ) f (t3 ) < f (t4 ). Also since N (t) is a Poisson pr
STAT 433/833 PROBLEM SET 2
1. (Resnick, pp. 151-152) The business of a restaurant fluctuates in successive years between three
states: 0(bankruptcy), 1(verge of bankruptcy) and 2(solvency). The transition matrix giving the
probabilities of evolving from s
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ID: 20332540
STAT 433 Problem Set 1 Solution
1. Exercise 1.7, Page 36
(a)
P (Xm 6= j, m = 0, , n|X0 = i)
= P (Xm does not reach j for m = 0, , n|X0 = i)
= 1 P (Xm ever reaches j for m = 0, , n|X0 = i)
n
X
(m)
=1
fij .
m=1
Since the state sp
Xing Shuo Zhai
ID: 20332540
STAT 433 Problem Set 3 Solution
1. Since the CTMC is irreducible, we have P (Xt = j|X0 = i) > 0 for some t > 0 and there
(n)
must exist some n Z+ such that Pij > 0. Let the states through which such path passes
t
be cfw_i0 , i1