MATH 56A SPRING 2008
STOCHASTIC PROCESSES
65
2.2.5. proof of extinction lemma. The proof of Lemma 2.3 is just like the proof of the lemma I did on Wednesday. It goes like this. Suppose that a is the s
MATH 56A SPRING 2008
STOCHASTIC PROCESSES
31
1.3. Invariant probability distribution.
Denition 1.4. A probability distribution is a function
: S [0, 1]
from the set of states S to the closed unit int
MATH 56A SPRING 2008
STOCHASTIC PROCESSES
45
1.6. The substo chastic matrix Q. On Monday, I used the MouseCat-Cheese problem to explain the use of the matrix Q. On Wednesday,
I explained this further
48
FINITE MARKOV CHAINS
1.6.2. expected time. Today I did a more thorough explanation of the expected time until we reach a recurrent state. I started by reviewing the basics of substochastic matrices
MATH 56A SPRING 2008
STOCHASTIC PROCESSES
51
1.6.3. Leontief model. The nal example is the Leontief economic model.
In this model, the numbers in a substochastic matrix are interpreted
as being the in
Math 56a: Introduction to Stochastic Processes and
Models
A stochastic process is a random process which evolves with time.
The basic model is the Markov chain. This is a set of states together
with t
MATH 56A: STOCHASTIC PROCESSES HOMEWORK From the syllabus: There will be weekly homework. The rst HW might have the problem: Using the SIR model, prove that the number of infected reaches its highest
MATH 56A: STOCHASTIC PROCESSES HOMEWORK From the syllabus: There will be weekly homework. Students are encouraged to work on their homework in groups and to access all forms of aid including expert ad
MATH 56A: STOCHASTIC PROCESSES HOMEWORK From the syllabus: There will be weekly homework. Students are encouraged to work on their homework in groups and to access all forms of aid including expert ad
MATH 56A: STOCHASTIC PROCESSES HOMEWORK From the syllabus: There will be weekly homework. Students are encouraged to work on their homework in groups and to access all forms of aid including expert ad
MATH 56A: STOCHASTIC PROCESSES HOMEWORK From the syllabus: There will be weekly homework. Students are encouraged to work on their homework in groups and to access all forms of aid including expert ad
MATH 56A SPRING 2008
STOCHASTIC PROCESSES
209
9.5. Black-Scholes Equation. On the last day of class I derived the Black-Scholes equation and solved it in the case when the drift and volatility of the
202
STOCHASTIC INTEGRATION
9.4. Extensions of Its formula. First I explained the idea of coo variation and the product rule. Then I used this to derive the second and third version of Its formula. o 9
102
OPTIMAL STOPPING TIME
4. Optimal Stopping Time
4.1. Denitions. On the rst day I explained the basic problem using
one example in the book. On the second day I explained how the
solution to the pro
108
OPTIMAL STOPPING TIME
4.4. Cost functions. The cost function g (x) gives the price you must
pay to continue from state x. If T is your stopping time then XT is
your stopping state and f (XT ) is y
114
OPTIMAL STOPPING TIME
4.5. discounted payo. Here we assume that the payo is losing
value at a xed rate so that after T steps it will only be worth T f (x)
where is the discount rate, say = .90. If
MATH 56A SPRING 2008
STOCHASTIC PROCESSES
157
8. Brownian Motion We will spend 5 days on this chapter and cover the following topics: 8.1. introduction 8.2. strong Markov property and the reection pri
162
BROWNIAN MOTION
8.2. strong Markov prop erty and reection principle. These are concepts that you can use to compute probabilities for Brownian motion. 8.2.1. strong Markov property. a) Brownian mo
168
BROWNIAN MOTION
8.3. fractal dimension of zero set. Today we calculated the fractal
dimension of the zero set.
First, I went over the calculation from last time. Suppose that Xt is
Brownian motion
MATH 56A SPRING 2008
STOCHASTIC PROCESSES
179
8.5. Recurrence and transience. The question is: Does Brownian
motion make particle go o to ?
(1)
(2)
(3)
(4)
Set up the probabilistic equation
Convert to
MATH 56A SPRING 2008 STOCHASTIC PROCESSES
KIYOSHI IGUSA
Contents 9. Stochastic integration 9.0. Concept, Its formula, quadratic variation o 9.1. Discrete stochastic integral 185 185 189
Date : April 1
MATH 56A SPRING 2008
STOCHASTIC PROCESSES
197
9.3. Its formula. First I stated the theorem. Then I did a simple o example to make sure we understand what it says. Then I proved it. The key point is Lv
202
STOCHASTIC INTEGRATION
9.4. Extensions of Its formula. First I explained the idea of coo
variation and the product rule. Then I used this to derive the second
and third version of Its formula.
o
9
MATH 56A: STOCHASTIC PROCESSES HOMEWORK
Homework 5 Martingales These problems are due Wednesday, March 26. Answers will be posted the following week. First problem: Suppose that X1 , X2 , are i.i.d. r
MATH 56A: STOCHASTIC PROCESSES HOMEWORK
Homework 6 Renewal These problems are due Thursday, April 3. Answers will be posted the following week. First problem: Suppose that we have a renewal process wi
MATH 56A: FALL 2006 HOMEWORK AND ANSWERS Math 56a: Homework 4 4. Homework 4 (Chap 2) p. 59 #2.7, 8, 18 2.7. Are these positive recurrent, null recurrent or transient? (a) This process is null recurren
MATH 56A: FALL 2006
HOMEWORK AND ANSWERS
Math 56a: Homework 5
1. Homework 5 (Chap 3)
p. 84 #3.5, 8, 11, 12
3.5. Let Xt be a Markov chain with state space S = cfw_1, 2 and rates (1, 2) = 1, (2, 1) = 4.
6. Homework 6 (Chap 4) p. 98 #4.1, 2, 3, 4 4.1. We have a simple random walk with absorbing walls on cfw_0, 1, 2, , 10 with payo function: x : 0 1 2 3 4 5 6 7 8 9 10 f (x) : 0 2 4 3 10 0 6 4 3 3 0 Fin
5. Homework 7 (Chap 5) p. 125 #5.2,5,7,15 5.2. If Xt is a Poisson process with = 1 then nd E(X1 | X2 ), E(X2 | X1 ). By the denition of a Poisson process, E(X2 X1 ) = E(X1 ) = = 1. Also, X2 X1 is inde
MATH 56A: STOCHASTIC PROCESSES
ANSWERS TO HOMEWORK 2006
Math 56a: Homework 8, Chap 6: Renewal
8. Time is money (Chap 6)
M/G/1 queueing is explained on p. 148.
8.1. The insurance company. We have an in