1
Temple University
Continuous Time Finance Conditional Expectation
2
Conditional Expectation
2.1 Information and algebras
3
Borel algebra of R
X 1 ( B ) = cfw_ : X ( w) B
B
The Borel algebra on the reals is the smallest algebra on R which contains al
1
Temple University
Continuous Time Finance Stochastic Calculus
2
Continuous Time Finance
Brownian Motion
3
Scaled Random Walk
= 123 .
1 if j = H Xj = 1 if j = T 1 nt n W (t ) = Xj n j =1
We speed up time (number of steps in any length of time) and scal
1
Temple University
Continuous Time Finance Probability Theory
2
Continuous Time Finance
Probability Theory
3
Probability Theory
Discrete probability theory deals with events that occur in countable sample spaces. a probability distribution is called dis
1
Temple University
Continuous Time Finance Basic Conditional Expectation
2
Conditional Expectation
See Section A.1 of the review on conditional expectations:
3
Conditional Expectation
Equivalently:
E[ X  Y = y ] = x
x
f X ,Y ( x, y ) fY ( y ) f X ,Y (
1
Temple University
Continuous Time Finance HW1
2
Shreve 1.6: Moment Generating Function
( x) = eux X Normal mean and stand dev
E (X) = e ux
 1 2
e
( x )2 2 2

1 2
e
1 ( x ) 2 2 2ux 2 2
(
) ) )
= e

1 2
e
1 x 2 2 x ( + 2u ) + 2 2 2
(
u + 1 u 2 2 2
1
Temple University
Continuous Time Finance Discretization of Ito Process
2
Discretization of Ito Process
Continuous Time Models
3
Formula for Ito Process
X(t) = X(0) + (s)dW(s) + (s)ds
0 0 t t
dX t = (t)dt + (t)dWt df (t , X t ) = f t (t , X t )dt + f x
1
Temple University
Continuous Time Finance Brownian Motion
2
Continuous Time Finance
Brownian Motion
3
Brownian Motion
One only has to place a drop of alcohol in the focal point of a microscope and introduce a little finely ground charcoal therein, and o