FRE 6233 Stochastic Calculus and Option pricing
Week 2: The Ito integral and Itos lemma in
several dimensions
Agn`es Tourin
September 19, 2016
This Lecture draws heavily on the Textbook by S. Shreve entitled
Stochastic Calculus for Finance, Part 2, Spring
Stochastic Calculus and Option Pricing
Week 1: From the Random walk to the Wiener
process, passing to the continuous-time limit,
modeling the flow of information by using
filtrations, application to the modeling of asset
prices.
Agn`es Tourin
February 7,
Week 10
Change of measure, Girsanov
Jonathan Goodman
November 25, 2013
1
Reweighting
Suppose X is a random variable with probability density u(x). Then the expected value of f (X) is
Eu [ f (X)] =
f (x)u(x) dx .
(1)
Suppose v(x) is a dierent probability d
Week 9
Generators, duality, change of measure
Jonathan Goodman
November 18, 2013
1
Generators
This section describes a common abstract way to describe many of the dierential equations related to Markov processes. The forward or backward equation
of a nite
Week 8
Diusion processes, part 2
Jonathan Goodman
October 28, 2013
1
Integration and Itos lemma for dXt
sec:il
Outline of this section:
1. Ways to dene new processes using old ones:
(a) An Ito integral with respect to an old process, Yt =
t
0
as dXs
(b) A
Week 7
Diusion processes
Jonathan Goodman
October 28, 2013
1
Introduction to the material for the week
This week we discuss a random process Xt that is a diusion process. A diusion
is a continuous time Markov process with continuous sample paths. Brownian
Week 4
Brownian motion and the heat equation
Jonathan Goodman
September 30, 2013
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
Week 2
Discrete Markov chains
Jonathan Goodman
September 16, 2013
1
Introduction to the material for the week
This week we discuss discrete probability and discrete Markov random processes.
We introduce four mathematical ideas:
A algebra to represent a s
Week 6
Itos lemma for Brownian motion
Jonathan Goodman
October 21, 2013
1
Introduction to the material for the week
Itos lemma is the big thing this week. It plays the role in stochastic calculus
that the fundamental theorem of calculus plays in ordinary
Week 5
Integrals with respect to Brownian motion
Jonathan Goodman
October 7, 2013
1
Introduction to the material for the week
This week continues the calculus aspect of stochastic calculus, the limit t 0
and the Ito integral. This is one of the most techn
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
Stochastic Calculus, Courant Institute, Fall 2013
http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc2013/index.html
Always check the class message board before doing any work on the assignment.
Sample questions for the nal exam.
Instructions for th