18.445 Problem Set 2
Exercise 6 In class we showed that, in the nearest neighbor random walk on Z,
cfw_ Xn n1 , the time T0 of rst return to 0 has the following probability distribution:
P[ T0 = n] =
2
n 1 n/2 n/2
pq.
n 1 n/2
Prove, by a direct computatio
C HAPTER 1
I ntroduction
N otes f or t he I nstructor
T his c hapter i ntroduces t he m arkets for futures, forward, a nd o ptions c ontracts a nd
e xplains t he a ctivities of hedgers, speculators, a nd a rbitrageurs. Issues concerning futures
c ontracts
Study notes of Bodie, Kane & Marcus
By Zhipeng Yan
Investment
Zvi Bodie, Alex Kane and Alan J. Marcus Chapter One: The Investment Environment . 2 Chapter Two: Financial Instruments. 4 Chapter Three: How Securities Are Traded. 8 Chapter Six: Risk and risk
Notes on Stochastic Processes
Kiyoshi Igusa December 17, 2006
ii
These are lecture notes from a an undergraduate course given at Brandeis University in Fall 2006 using the second edition of Gregory Lawlers book Introduction to Stochastic Processes.
Conten
Markov Processes
Many of this issues with Markov processes are repeats from
Markov chains, e.g. distributions, initial distributions, stationary
distributions, exit times, exit distributions, reversibilty.
As with the Markov chains, our Markov processes w
This course covers Markov Chains, Poisson and Markov Processes, Renewal
Theory, Martingales, Brownian Motion and Stochastic Integration. That is
too much. I intend to emphasize the parts that are most useful to people
studying mathematical nance. This mea
Properties of Conditional Expectation
November 5, 2011
Ill speak in terms of 2 random variables, with a joint density, but things
generalize to more that 2, and to discrete distributions as well. Ill indicate
some of the mutatis mutandis at the end.
Given
Final Exam
Stochastic Processes
December 12, 2006
You can use the result of any problem from Karlin and Taylor that was assigned. No
calculators.
1. Let 1 < ci < 2, i = 0, . . . , and c1 = 0. Let Xn be a markov chain on the
non-negative integers with pi,i