Stochastic Calculus, Spring 2010, 22 January, Lecture 1 Construction of the Brownian motion Reading for this lecture (for references see the end of the lecture): [1] pp. 72-108 Scaled Random Walks. Our main object of study in this course will be stochasti
Stochastic Calculus, Spring 2016,
Lecture 2
Properties of the Brownian motion
Reading for this lecture (for references see the end of the lecture):
[1] pp. 72-108
Let us summarize what we did during the last lecture. We proved that the symmetric random w
Stochastic Calculus, Spring 2016, February 18,
Lecture 4
Construction of the Ito Integral
Reading for
[1] pp.
[2] pp.
[3] pp.
[4] pp.
this lecture (for references see the end of the lecture):
125-137
33-67
128-148
21-42
Motivation. Consider an asset w
Stochastic Calculus, Spring 2016, March 3rd,
Lecture 5
Formula
Ito
Reading for this lecture (for references see the end of the lecture):
[1]
[2]
[3]
[4]
pp.
pp.
pp.
pp.
137-153
68-70
148-156
21-42
During the last lecture we defined Ito integral
Z t
f (s,
Measure Theory
1
Measurable Spaces
A measurable space is a set S, together with a nonempty collection, S, of
subsets of S, satisfying the following two conditions:
1. For any A, B in the collection S, the set1 A B is also in S.
2. For any A1 , A2 , S, Ai
STOCHASTIC CALCULUS, SPRING 2016, JANUARY 27,
LECTURE 1
CONSTRUCTION OF BROWNIAN MOTION
Contents
Reading for this lecture (for references see the end of the lecture):
[?] pp. 72-108
1. Definition of Brownian Motion
Our main object of study in this course
Stochastic Calculus, Spring 2016, April 7,
Lecture 7
Connection of the Stochastic Calculus and Partial Differential
Equation
Reading for this lecture:
(1) [1] pp. 125-175
(2) [2] pp. 239-280
(3) Professor R. Kohns lecture notes PDE for Finance, in particu
PDE for Finance Notes Section 1.
Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connection with the NYU course PDE for Finance, MATH-GA2706. Prepared in 2003, minor
updates made in 2011 and 2014.
Links between stochastic d
Stochastic Calculus, Spring 2016, April 14,
Lecture 9
Connection with PDEs - Multidimensional case
Boundary value problems and exit times
Reading for this lecture:
(1) [1] pp. 125-175
(2) [2] pp. 239-280
(3) Professor R. Kohns lecture notes PDE for Financ
Stochastic Calculus, Spring 2016, Lecture 3
Reading for
(1) [1] pp.
(2) [2] pp.
(3) [3] pp.
this lecture:
108-115
25-31
95-97
Background: Vanilla Barrier Options. In finance, a barrier option is a type
of contract where option to exercise at maturity depe
Stochastic Calculus, Fall 2002 (http:/www.math.nyu.edu/faculty/goodman/teaching/StochCalc/)
Practice Final Exam questions.
Given December 11, last revised December 12
Focus: Review
This should help you study for the final exam. There are more questions th
Stochastic Calculus, Spring 2016, Assignment # 4,
Due Date : Thursday, March 3, 2016
Integral
Ito
1. Let (t) be deterministic function of time, be constant and
define
X(T ) =
ZT
(t)et dBt .
0
Find the expectation and variance of X(T ). What is the distri
Stochastic Calculus, Spring 2016, Assignment # 2,
Due Date : Thursday, February 11, 2016
More on properties of Brownian Motion
1. Let Bt be a Brownian Motion starting from zero. Prove that for
any fixed constant > 0
Xt = eBt
2 t/2
,t 0
is a martingale wi
Stochastic Calculus, Assignment 6,
Due March 10th, 2016
Formula
Ito
1. Use Ito formula to prove that the following stochastic processes
are martingales:
t
(1): Xt = e 2 cos Bt ;
t
(2): Xt = e 2 sin Bt ;
t
(3): Xt = (Bt + t)eBt 2 .
2. Assume that f (x) is
Assignment 1, Spring 2016, Due Date February 4th, 2016
Symmetric Random Walk
1. M (n) is a symmetric random walk. Calculate:
h
i
a): E eM (n) using the fact that X1 , X2 , . . . , Xn are independent.
b): P [M (n) = k] for a given integer k [n, n].
2.
a):
Stochastic Calculus, Assignment 9, April 15, 2016
Due April 21, 2016
More on Stochastic Calculus and PDE, Exit Times
1. Let Xt satisfy the following SDE:
dXt = Xt dt + dBt
with X0 = x [a, b] and some fixed constant. Let = infcfw_t 0 :
Xt 6 [a, b]. Find EX
Stochastic Calculus, Spring 2016, Assignment # 2,
Due Date : Thursday, February 18, 2016
More on properties of Brownian Motion
Consider the simplest interest rate model
r(t) = r(0) + h(t) + B(t),
where r is an overnight interest rate, r(0) is its initial
Assignment 4, Spring 2016, Due Date February 25th, 2016
Running Maximum
Let Bt be a standard Brownian Motion starting from zero and Mt be its
running maximum and Ta be its hitting time.
1. In class we proved that EeTa = e 2a for all , a > 0.
a. Use the re
Stochastic Calculus, Assignment 8, April 9, 2016
Due April 12, 2016
Connection with PDE.
1. Let Bt be a standard Brownian Motion. Let be a constant and
T > 0 be a fixed maturity. Using Feynman-Kac theorem compute
EB0 =x e
RT
Bs2 ds
0
.
2. Let Bt be a stan