John Campbell Professor McPartland English 2 March 13, 2007 Prison Can Be A School Being that the mind is so complex it is understandable that when in control the minds capabilities of free thought are suddenly hindered. As a result there can only be
MATHEMATICAL METHODS
AUTUMN 2009
LECTURE 9: THE ITO STOCHASTIC INTEGRAL
(LECTURE NOTES, CHAPTERS 11 (AND 10)
1. Introduction
We will assume from now that our Brownian motion (Wt )t0 is defined
on some probability space (, F, P) (for which one may take BM
STOCHASTIC INTEGRALS
3
With this concept of a Brownian filtration, the formal definition of
adapted process then becomes:
Adapted process - rigorous definition: A process (Hy )t0 is adapted
to BM (Wt )t0 if for all t 0 and all a < b, the event cfw_a Ht b
Mathematical Methods I, Autumn 2009
113
We dene the integral of the random variable X over smaller sets F F, by
simply multiplying the integrand by the indicator function of F ; that is, we set
X IF dP :=
X IF dP
(9.5)
F
X() IF () dP().
=
We can also writ
Chapter 9
Expectations and integrals
If X : R is a random variable in the newly dened sense, i.e., any realvalued measurable function on the probability space , then we need to know
how to compute its expectation. The strategy is rst to dene the expectati
4.9 A south-facing collector at latitude 40" i5 tipped up at an angle equal to its latitude.
Compute the followinginsolations for January 15t at 3 olar noon:
3. The direct beam insolation normal to the sun's rays
1:. Beam insolation on the collector
c. Di
Question 2
Name 2 advantages and 2 disadvantages of
using smaller cells .
Solution 2
Name 2 advantages and 2 disadvantages of
using smaller cells .
Advantages:
Smaller cells increases channel reuse and improves
network capacity.
Smaller cells reduce tran
Question 3
What is the average load (in Erlangs) that 12
trunks can carry with blocking probability of
1%? (Assume one channel is 1 trunk.) What is
the average load that 36 trunks can support
with blocking probability of 1%? Which one is
more efficient?
S
Traders could have taken advantage of the fact that the fund is forced to
liquidate positions as it receives margin calls.
Against:
Bailing out funds creates moral hazard problems. That is, it motivates
other funds to take more risks in the future becau
Question 1
Why do we use hexagon-shaped cells?
Solution 1
Why do we use hexagon-shaped cells?
For a given value of radius, a hexagonshaped shall provide maximum coverage
area. Therefore, we can use the fewest
number of cells to cover a given geographic
re
Poem Explication 1) Dramatic Situation A. The Conflict that the poet himself is facing is the jealousy of life yet how his love takes all the pain away B. The speaker for this poem is himself C. The speaker is speaking directly to his love through th
STOCHASTIC INTEGRALS
5
Theorem (cf. Lecture Notes, Thm 11.6) If (Ht )t0 is an adapted
process such that
Z T
E(Ht2 ) dt < ,
0
then its stochastic integral
Z
T
Ht dWt
IT :=
0
is well-defined. It has mean 0:
E(IT ) = 0,
and variance
E(IT2 )
Z
=
Finally, IT c