Week 7
Diusion processes
Jonathan Goodman
October 29, 2012
1
Introduction to the material for the week
This week we discuss a random process Xt that is a diusion process. A diusion
process has an inni
Week 6
Itos lemma for Brownian motion
Jonathan Goodman
October 22, 2012
1
Introduction to the material for the week
sec:intro
Itos lemma is the big thing this week. It plays the role in stochastic cal
Week 5
Integrals with respect to Brownian motion
Jonathan Goodman
October 7, 2012
1
Introduction to the material for the week
This week starts the other calculus aspect of stochastic calculus, the lim
Week 3
Continuous time Gaussian processes
Jonathan Goodman
September 24, 2012
1
Introduction to the material for the week
This week we take the limit t 0. The limit is a process Xt that is dened
for a
Week 4
Brownian motion and the heat equation
Jonathan Goodman
October 1, 2012
1
Introduction to the material for the week
A diusion process is a Markov process in continuous time with a continuous
sta
Week 2
Discrete Markov chains
Jonathan Goodman
September 17, 2012
1
Introduction to the material for the week
This week we discuss Markov random processes in which there is a list of possible states.
Stochastic Calculus, Courant Institute, Fall 2012
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012/index.html
Always check the class message board on the blackboard site from home.nyu.edu
Week 8
Stopping times, martingales, strategies
Jonathan Goodman
November 12, 2012
1
Introduction to the material for the week
Suppose Xt is a stochastic process and S is some set. The hitting time is
Week 11
Backwards again, Feynman Kac, etc.
Jonathan Goodman
November 26, 2012
1
Introduction to the material for the week
sec:intro
This week has more about the relationship between SDE and PDE. We di
Sample questions for the nal exam
Partial Answer
Chen-Hung Wu
December 16, 2012
Instructions for the nal:
The nal is Monday, December 17 from 7:10 to 9pm
Explain all answers, possibly briey. A corre
Stochastic Calculus, Courant Institute, Fall 2011
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2011/index.html
Always check the class bboard on the blackboard site from home.nyu.edu (click
Stochastic Calculus, Courant Institute, Fall 2012
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012/index.html
Sample questions for the nal exam
Instructions for the nal:
The nal is Monda
Stochastic Calculus, Spring, 2007 (http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2007/)
Practice for the Final Exam.
The nal exam is Thursday, May 3, from 5:10 to 7 pm in room 1302. You are
Stochastic Calculus, Courant Institute, Fall 2012
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012/index.html
Problem Session
November 29, 2012
Corrections: (none yet)
1. The probability
Stochastic Calculus, Courant Institute, Fall 2012
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012/index.html
Always check the class message board on the blackboard site from home.nyu.edu
Stochastic Calculus, Courant Institute, Fall 2012
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012/index.html
Always check the class message board on the blackboard site from home.nyu.edu
Stochastic Calculus, Courant Institute, Fall 2012
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012/index.html
Always check the class message board on the blackboard site from home.nyu.edu
0.06
0.04
0.02
0.00
estimated density
0.08
0.10
Probability density for the final value of an OrnsteinUhlenbeck process
20
10
0
end value
200000 paths, time step = 0.010, T =
10
10.00
20
# Stochastic Calculus, Courant Institute, NYU, Fall 2012
#
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012
/
# Assignment 3, simulate the distribution of the max of a
Brownian motion
#
D
# Stochastic Calculus, Courant Institute, NYU, Fall 2012
#
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012
/
# Assignment 3, simulate the distribution of the end value of
a Brownian moti
0.06
0.04
0.02
0.00
estimated density
0.08
0.10
Probability density for the final value of an OrnsteinUhlenbeck process
20
10
0
end value
200000 paths, time step = 0.005, T =
10
5.00
20
0.04
0.03
0.02
0.01
0.00
estimated density
0.05
0.06
Probability density for the final value of a BM path
15
10
5
end value
200000 paths, time step = 0.020, T =
0
20.00
#
#
Stochastic Calculus, Courant Institute, NYU, Fall 2012
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012/
#
Assignment 2, simulations of an urn process
#
Daniel Schwabe, [email protected]