ONE-DIMENSIONAL RANDOM WALKS
1. SIMPLE RANDOM WALK
Definition 1. A random walk on the integers Z with step distribution F and initial state x Z is
a sequence Sn of random variables whose increments are independent, identically distributed
random variables
CONDITIONAL EXPECTATION AND MARTINGALES
1. I NTRODUCTION
Martingales play a role in stochastic processes roughly similar to that played by conserved
quantities in dynamical systems. Unlike a conserved quantity in dynamics, which remains
constant in time,
LECTURE 9: LVY PROCESSES
STEVEN P. LALLEY
1. DEFINITIONS AND EXAMPLES
Definition 1.1. A continuoustime process cfw_X t = X (t )t 0 with values in R (or, more
generally, in Rd ) is called a Lvy process if (1) its sample paths are right-continuous and
have
INTEGRAL
THE ITO
1. Introduction: Geometric Brownian motion
vys representation theorem, quoted at the beginning of the last lecture, every
According to Le
continuoustime martingale with continuous paths and finite quadratic variation is a timechanged
Bro
CHANGE OF MEASURE: GIRSANOVS THEOREM
1. Exponential Martingales
Girsanov Theorem is a far-reaching generalization of the CameronMartin theorem that allows the drift to be not only time-varying, but random. It provides a
description of the likelihood ratio
CONDITIONAL EXPECTATION AND MARTINGALES
1. I NTRODUCTION
Martingales play a role in stochastic processes roughly similar to that played by conserved
quantities in dynamical systems. Unlike a conserved quantity in dynamics, which remains
constant in time,
ONE-DIMENSIONAL RANDOM WALKS
1. SIMPLE RANDOM WALK
Definition 1. A random walk on the integers Z with step distribution F and initial state x Z is
a sequence Sn of random variables whose increments are independent, identically distributed
random variables
Expected Crowd
Average Concessions Expenditure
Fixed Cost
Ticket Price
3000 people
$15.00 per person
$10,000.00
$10.00 per person
REVENUE FROM TICKETS
CONCESSION SALES
$30,000.00
$45,000.00
PROFIT PERCENTAGE
80%
BAND PROFIT
$50,000.00
Number of Offers
Acc
LECTURE 1
ARBITRAGE PRICING: THE FUNDAMENTAL THEOREM
1. Introduction
The BlackScholes theory, which is the main subject of this course and its sequel, is based
on the Efficient Market Hypothesis, that arbitrages (the term will be defined shortly) do
not e