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A Mathematical Theory of Financial Bubbles
Philip Protter
November 3, 2012
Recurrent speculative insanity and the associated nancial deprivation
and larger devastation are, I am persuaded, inherent in the system. Perhaps it is better that this be recogniz
13-1-12
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Errata for
Network Flows: Theory, Algorithms, and Applications.
By
Ravindra K. Ahuja
Thomas L. Magnanti
and James B. Orlin
Errors listed in blue were corrected in the 4th printing of the textbook.
Errors listed in boldface black still remain as errors.
I
Department Listing: Mathematics Courses in the Spring 2013 Semester
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13-1-Depa r tment Lis ting: Ma thema tics @B a r na r d Co ur s es in the Spr ing 2013 Semes ter
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Department Listing: Mathematics @Barnard Courses in the Spring 2013 Semester
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13-1-6epartment Listing: Computer S c ienc e Courses in the S pring 2013 S emester
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Department Listing: Computer Science Courses in the Spring 2013 Semester
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CM E01 01 111
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Bayesian Sta+s+cs: STATW4640
Lecture Notes: Jan 25, 2013
Siddhartha Dalal
1
Introduc+on:
Bayesian inference has become remarkably widespread in the wake of the Monte
Carlo Markov Chain (MCMC) revoluMon of the 1990s.
Properties of Conditional Expectation
October 22, 2012
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
Stochastic Processes for Finance
May 21, 2012
Section 1: Martingales and Martingale transforms
1. Let Xi , i = 1, . . . be a sequence of random variables. The sequence is a
martingale if E cfw_Xn+1 |X1 , . . . , Xn = Xn . In principal this conditional
ex
References for Stochastic Processes for Finance
November 12, 2012
1
Texts
1. Random Walks and Electric Networks by Peter G. Doyle , J. Laurie
Snell is available freely on-line , very readable, and does techniques that
are equally applicable to higher dime
Stochastic Processes for Finance
November 12, 2012
Martingales and processes related to Brownian Motion are the two key elements of continuous time nance. Other processes, like markov chains, poisson
processes, continuous time markov chains, also nd lots
4606 PreReqs
M Hogan
December 22, 2011
1
Linear Algebra
1. For the matrix P =
1
3
2
4
Compute P 2 .
2/9 7/9
4/9 5/9
Compute (I.9P )1 .
2. For P =
.2
3. For the matrix P = .4
.5
for a vector satisng i
.3 .5
.4 .2 nd so that P = , = (1 , 2 , 3 )
.2 .3
0, i
Copyright c 2007 by Karl Sigman
1
1.1
Poisson processes, and Compound (batch) Poisson processes
Point Processes
Denition 1.1 A simple point process = cfw_tn : n 1 is a sequence of strictly increasing points 0 < t1 < t2 < , (1) with tn as n. With N (0) = 0
Copyright c 2007 by Karl Sigman
1
Simulating Markov chains
Many stochastic processes used for the modeling of nancial assets and other systems in engineering are Markovian, and this makes it relatively easy to simulate from them.
Here we present a brief i
Copyright c 2007 by Karl Sigman
1
Rare event simulation and importance sampling
Suppose we wish to use Monte Carlo simulation to estimate a probability p = P (A) when the event A is rare (e.g., when p is very small). An example would be p = P (Mk > b) wit
Copyright c 2007 by Karl Sigman
1
Estimating sensitivities
When estimating the Greeks, such as the , the general problem involves a random variable
Y = Y () (such as a discounted payo) that depends on a parameter of interest (such as initial
def
price S0
Copyright c 2007 by Karl Sigman
1
Simulating normal (Gaussian) rvs with applications to simulating Brownian motion and geometric Brownian motion in one and two dimensions
Fundamental to many applications in nancial engineering is the normal (Gaussian) dis
Copyright c 2007 by Karl Sigman
1
1.1
Introduction to reducing variance in Monte Carlo simulations
Review of condence intervals for estimating a mean
In statistics, we estimate an unknown mean = E (X ) of a distribution by collecting n iid samples from th
Copyright c 2007 by Karl Sigman
1
Acceptance-Rejection Method
As we already know, nding an explicit formula for F 1 (y ) for the cdf of a rv X we wish to generate, F (x) = P (X x), is not always possible. Moreover, even if it is, there may be alternative
Copyright c 2007 by Karl Sigman
1
Review of Probability
Random variables are denoted by X , Y , Z , etc. The cumulative distribution function (c.d.f.) of a random variable X is denoted by F (x) = P (X x), < x < , and if the random variable is continuous t
Copyright c 2005 by Karl Sigman
1
Portfolio mean and variance
Here we study the performance of a one-period investment X0 > 0 (dollars) shared among
several dierent assets. Our criterion for measuring performance will be the mean and variance
of its rate
Copyright c 2007 by Karl Sigman
1
1.1
IEOR 4700: Introduction to stochastic integration
Riemann-Stieltjes integration
b Recall from calculus how the Riemann integral a h(t)dt is dened for a continuous function h over the bounded interval [a, b]. We partit
1
Gamblers Ruin Problem
Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble
either wins $1 or loses $1 independent of the past with probabilities p and q = 1 p respectively.
Let Rn denote the total fortune after
Copyright c 2006 by Karl Sigman
1
Geometric Brownian motion
Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is not appropriate for modeling stock prices. Inste
Copyright c 2005 by Karl Sigman
1
Fund theorems
2 In the Markowitz problem, we assumed that all n assets are risky; i > 0, i cfw_1, 2, . . . , n. This lead to the efficient frontier as a curve starting from the minimum variance point. We learned that in t
1
Factor Models
The Markowitz mean-variance framework requires having access to many parameters: If there are n risky assets, with rates of return ri , i = 1, 2, . . . , n, then we must know 2 all the n means (ri ), n variances (i ) and n(n - 1)/2 covaria
Copyright c 2005 by Karl Sigman
1
Capital Asset Pricing Model (CAPM)
We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available to all. In addition we have a risk-free asset