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Integration of trigonometric polynomials.
To integrate
sinm x cosn x dx
When one of them ( either m or n) is odd. Let n = 2k + 1 be odd Put u = sinx.
um (1 u2 )k du
Let both both be even. m = 2k and n = 2. Use sin2 x =
1
2+k
fn
dP
1. P and Q are absolutely continuous with respect to P + Q. gn = d(P +Q) Pn = fn +1 has
dP
a limit g = dQ as n . 0 g 1. g = 0 on X3 and g = 1 on X1 with 0 < g < 1 on
X2 . If gn =
fn
fn +1
then fn =
gn
1gn
tends to f =
g
1g .
g = 1 on X1 so f = .
2.
Chapter 1
Brownian Motion
1.1
Stochastic Process
A stochastic process can be thought of in one of many equivalent ways. We
can begin with an underlying probability space (, , P ) and a real valued
stochastic process can be dened as a collection of random
September 21, 2009.
Sequences, limits and series.
A sequence is a function dened on the set of positive integers. For n = 1, 2, 3, . . . , the
value an of the function has to be given.
Examples:
Ex 1. cfw_1, 1, 1, 1, 1, . . .. an = 1 for all n.
Ex 2. cfw_
September 23, 2009.
Sequences, limits and series (continued).
If f (x) is function dened on [a, ) i.e for all x a we can talk about
lim f (x) =
x
This means given any accuracy, i.e, a small number
| f ( x) |
for all suciently large x, i.e. for x A. One
cfw_Xn is a Markov Chain on the integers i = 0, 1, . . . with transition probabilities
p(i, j ) = ei
ij
j!
This is an example of a branching process, where each member of the current generation
has a random number of osprings distributed according to a P
Problemset 2. Due April 11.
The Gamma process is dened as a process with independent and stationary increments
whose distribution at time 1 is the exponential distribution with density
p(x) =
ex dx
0
if x 0
otherwise
1) What is its distribution at time t
Problemset 1. Due Feb 28.
1. Let P and Q be two probability measures on (X, ). Let Pn be an increasing family of
measurable nite partitions of X , i.e Pn = cfw_A1 , . . . , An with X = n Aj with disjoint
i=1
Aj s from and Pn+1 is a ner partition than Pn
Chapter 1
Poisson Processes
1.1
The Basic Poisson Process
The Poisson Process is basically a counting processs. A Poisson Process on
the interval [0 , ) counts the number of times some primitive event has
occurred during the time interval [0 , t]. The fol
September 16
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http : /math.nyu.edu/degree/undergrad/tutorschedule.html
Improper Integr
74
4.10
CHAPTER 4.
STOCHASTIC DIFFERENTIAL EQUATIONS.
Brownian Motion on the Haline.
It is not possible to construct the Brownian on the haline [0 , ). Sooner or
later it will hit 0 and then immeditely would turn negative as the following
lemmas show.
Lem
X (t) is the gamma process with Levy-Khintchine representation
itX (1)
E [e
itx
] = exp[
[e
0
ex
dx
1]
x
Let > 0, Write the Levy measure as the sum
ex
ex
ex
dx =
1x dx +
1x dx
x
x
x
with tow processes X (t) and X consisting entirely of jumps that are sma
Things to remember from Calculus 1.
Derivatives:
f ( x)
f ( x)
C
0
xp
p xp1
1
x
ex
cos x
log x
ex
sin x
cos x
sin x
sec2 x
tan x
sec x
cosec x
sec x tan x
cosec x cot x
cosec2 x
1
1 x2
1
1 x2
1
1 + x2
cot x
arcsin x
arccos x
arctan x
Product Rule:
f (