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 innitesimal mean, or drift, which is a(x, t). The process i
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 calculus
that the fundamental theorem of calculus plays in
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 limit t
0 and the Ito integral. This is one of the most t
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 all t in some range, such as t [0, T ]. The process take
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
state space and continuous sample paths. This course is la
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. We introduce three mathematical ideas: a algebra to rep
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 before doing any
work on the assignment.
Assignment 7,
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 the
rst time Xt hits S.
= min cfw_t | Xt S .
(1)
This
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 discuss
ways to formulate the solution of a PDE in terms
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 correct answer with no explanation
may receive no credit.
Y
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 on academics, then
on Derivative Securities) before do
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 Monday, December 17 from 7:10 to 9pm
Explain all answers, p
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 allowed one piece of standard size paper with whatever
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 density function u (x, t) of the OrnsteinUhlenbeck proc
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 before doing any
work on the assignment.
Assignment 4,
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 before doing any
work on the assignment.
Assignment 5,
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 before doing any
work on the assignment.
Assignment 6,
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
#
Daniel Schwabe, [email protected]
#
Set parameters
T
= 5.
#
# 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 motion
#
Daniel Schwabe, [email protected]
#
Set parameters
T
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]
#
Do simulations
m
T
L
x0
x0
p
h
ia
r
=
=
=
=
=
=
=
=