Download 2.3 General Conditional Expectations 報告人：李振綱

2.3 General
Conditional
Expectations
報告人：李振綱
Review
• Def 2.1.1 (P.51)
Let  be a nonempty set. Let T be a fixed
positive number, and assume that for each
t  [0, T ] there is a  - algebra F(t) . Assume
further that if s  t , then every set in F(s) is
also in F(t) .
Then we call the collection of  - algebra F(t) , 0  t  T
, a filtration.
• Def 2.1.5 (P.53)
Let X be a r.v. defined on a nonempty sample
space  . Let  be a  - algebra of subsets of 
If every set in  (X) is also in  , we say that X is
 - measurable .
Review
• Def 2.1.6 (P.53)
Let  be a nonempty sample space equipped
with a filtration F(t) , 0  t  T .
Let X (t ) be a collection of r.v.’s is an adapted
stochastic process if, for each t, the r.v. X (t ) is
F(t)  measurable .
Introduction
• 

(, F , ) and a sub -  - algebra  of F
If X is   measurable the information in
sufficient to determine the value of X.
 is
If X is independent of  , then the information in
 provides no help in determining the value of
X.

In the intermediate case, we can use the
information in  to estimate but not precisely
evaluate X.
Toss coins
Let  be the set of all possible outcomes of
N coin tosses,
p : probability for head
q=(1-p) : probability for tail
En [ X ](1......n )




n1 ...
p # H (n1 ...N ) q #T (n1 ...N ) X (1...nn 1... N ).
N
Special cases n=0 and n=N,
E0 [ X ] 

 
0 ... N
p # H (0 ...N ) q #T (0 ...N ) X (0 ...N )  E[ X ]
EN [ X ](0 ...N ) = X(0 ...N )
Example (discrete  continous)
S3 ( HHH )  u 3 S0
S 2 ( HH )  u 2 S0
S1 ( H )  uS0
S0
S1 (T )  dS0
S3 ( HHT )  S3 ( HTH )  S3 (THH )
S2 ( HT )  S2 (TH )  udS0
S 2 (TT )  d 2 S0
 u 2 dS0
S3 ( HTT )  S3 (THT )  S3 (TTH )
 ud 2 S0
S3 (TTT )  d 3 S0
• Consider the three-period model.(P.66~68)
E 2 [S3 ](HH) = pS3 (HHH) +qS3 (HHT) ....(2.3.4)
(間斷)
E2 [S3 ](HH) P(AHH ) =
(連續)

AHH
S ()P()


AHH
E2 [ S3 ]( ) dP( ) =
3

AHH
 P(A HH )
....(2.3.8)
S3 ( ) dP( )
(Lebesgue
integral)
General Conditional Expectations
• Def 2.3.1.
let (, F , ) be a probability space, let  be a
sub -  - algebra of F , and let X be a r.v. that is either
nonnegative or integrable. The conditional
expectation of X given  , denoted E[ X |  ] , is
any r.v. that satisfies
(i) (Measurability)
E[ X |  ] is
  measurable
(ii) (Partial averaging)

A
E[ X |  ]() dP() =  X() dP() for all A 
A
E[ X |  ] unique ?
• (See P.69)
Suppose Y and Z both satisfy condition(i) ans (ii)
of Def 2.3.1. Suppose both Y and Z are   measurable
, their difference Y-Z is as well, and thus the set
A={Y-Z>0} is in  . So we have
 Y ()dP() 
A
and thus
A
X ()dP()  Z ()dP()
A
 (Y ()  Z ())dP()  0
A
The integrand is strictly positive on the set A, so
the only way this equation can hold is for A to
have probability zero(i.e. Y  Z almost surely).
We can reverse the roles of Y and Z in this
argument and conclude that Y  Z almost surely .
Hence Y=Z almost surely.
General Conditional Expectations
Properties
• Theorem 2.3.2
let (, F , ) be a probability space and let
a sub -  - algebra of F .

be
(i) (Linearity of conditional expectation) If X and Y are
integrable r.v.’s and
c1and c2 are constants, then
E[c1X+c2 Y| ] = c1E[X| ] + c2E[Y| ]
(ii) (Taking out what is known) If X and Y are integrable
r.v.’s, Y and XY are integrable, and X is
E[XY| ] = XE[Y| ]
  measurable
General Conditional Expectations
Properties(conti.)
(iii) (Iterated condition)If H is a
is an integrable r.v., then
sub -  - algebra of 
and X
E[E[X| ]| H ] = E[X|H ]
(iv) (Independence)If X is integrable and independent of
, then

E[X| ] = E[X]
(v) (Conditional Jensen’s inequality)If  (X) is a convex
function of a dummy variable x and X is integrable, then
E[ (X)| ]   (E[X| ])
p.f(Volume1 P.30)
Example 2.3.3. (P.73)
• X and Y be a pair of jointly
normal random

variables. Define W  Y -  X so that X and W are
independent, we know W is normal with mean
 
2
2
2
 = 
=
(1
)

and
variance
3
2 . Let us take

the conditioning to be  = (X) .We estimate Y,
based on X.
 1
so,
Y
X W
1
2
2
3
1
2
1


2
 1
 1
E[Y|X] =
X +EW =
(X-1 )+2
2
2
Y-E[Y|X] = W-E[W]
(The error is random, with expected value zero, and is
independent of the estimate E[Y|X].)
• In general, the error and the conditioning r.v. are
uncorrelated, but not necessarily independent.
Lemma 2.3.4.(Independence)
• let (, F , ) be a probability space, and let  be
a sub -  - algebra  of F . Suppose the r.v.’s
X1.... X K are   measurable and the r.v.’s Y1....YL
are independent of  . Let f ( x1, ..., xK , y1, ..., yL ) be
a function of the dummy variables x1, ..., xK and
y1, ..., y L define
g ( x1, ..., xK )  Ef ( x1, ..., xK , y1, ..., yL )
Then
Ef ( X 1, ..., X K ,Y1, ..., YL |  )  g ( X 1, ..., X K )
Example 2.3.3.(conti.) (P.73)
• Estimate some function f ( x, y ) of the r.v.’s X and Y
based on knowledge of X.
By Lemma 2.3.4
 1
g ( x)  Ef ( x,
x W )
2
 E[ f ( X , Y ) | X ]  g ( X )
Our final answer is random but
 ( X ) - measurable.
Martingale
• Def 2.3.5.
let (, F , ) be a probability space, let T be a
fixed positive number, and let F (t ) , 0  t  T ,
be a filtration of sub -  - algebras of F.
Consider an adapted stochastic process
M(t), 0  t  T .
(i) If E[M(t)|F(s)] = M(s) for all 0  s  t  T,
we say this process is a martingale. It has no tendency
to rise or fall.
(ii) If E[M(t)|F(s)]  M(s) for all 0  s  t  T,
we say this process is a submartingale. It has no
tendency to fall; it may have a tendency to rise.
(iii) If E[M(t)|F(s)]  M(s) for all 0  s  t  T,
we say this process is a supermartingale. It has no
tendency to rise; it may have a tendency to fall.
Markov process
• Def 2.3.6.
Continued Def 2.3.5. Consider an adapted
stochastic process X (t ) , 0  t  T.
Assume that for all 0  s  t  T and for every
nonnegative, Borel-measurable function f, there
is another Borel-measurable function g such that
E[ f ( X (t )) | F ( s)]  g ( X ( s )).
Then we say that the X is a Markov process.
E[ f (t , X (t )) | F ( s)]  f ( s, X ( s)), 0  s  t  T .
Thank you for your listening!!