2.3 General 2.3 General Conditional Conditional Expectations Expectations 報報報 報報報 : 報報報 報報報 :
2.3 General 2.3 General Conditional Conditional
ExpectationsExpectations
報告人:李振綱報告人:李振綱
ReviewReview• Def 2.1.1 (P.51)
Let be a nonempty set. Let T be a fixed positive number, and assume that for each there is a . Assume further that if , then every set in is also in .Then we call ,, a filtration.
• Def 2.1.5 (P.53)Let X be a r.v. defined on a nonempty sample space . Let be aIf every set in is also in , we say that X is .
[0, ]t T - algebra F(t)s t F(s)
the collection of - algebra F(t) 0 t T
- algebra of subsets of (X)
- measurable
F(t)
ReviewReview• Def 2.1.6 (P.53)
Let be a nonempty sample space equipped with a filtration , .Let be a collection of r.v.’s is an adapted stochastic process if, for each t, the r.v. is .
F(t) 0 t T
( )X t( )X t
F(t) measurable
IntroductionIntroduction• and a
If X is the information in is sufficient to determine the value of X.
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.
( , , )F
measurable
sub - - algebra of F
Toss coins Toss coins Let be the set of all possible outcomes ofN coin tosses, p : probability for head
q=(1-p) : probability for tail
Special cases n=0 and n=N,
1 1
1
1
# ( ... ) # ( ... )1 1
...
[ ]( ...... )
( ... ... ).n N n N
n N
n n
H Tn n N
E X
p q X
0 0
0
# ( ... ) # ( ...0 0
)
...
[ ] ( ... ) [ ]N N
N
H TNE X p q X E X
0 0[ ]( ... ) = X( ... )N N NE X
Example (discrete Example (discrete continous) continous)
• Consider the three-period model.(P.66~68)
0S1 0S ( )H uS
1 0( )S T dS
22 0( ) S HH u S
2 2 0( ) ( ) S HT S TH udS
22 0( ) S TT d S
33 0( ) S HHH u S
33 0( ) S TTT d S
3 3 3
20
( ) ( ) ( )
S HTT S THT S TTH
ud S
3 3 3
20
( ) ( ) ( )
S HHT S HTH S THH
u dS
2 3 3 3E [S ](HH) = pS (HHH) +qS (HHT) ....(2.3.4)
HH
2 3 HH 3A
E [S ](HH) P(A ) = S ( )P( ) ....(2.3.8)
2 3 3[ ]( ) dP( ) = ( ) dP( )HH HHA A
E S S (Lebesgue integral)
(連續 )
(間斷 )
HH P(A )
General Conditional ExpectationsGeneral Conditional Expectations
• Def 2.3.1.let be a probability space, let be a , and let X be a r.v. that is either nonnegative or integrable. The conditional expectation of X given , denoted , is any r.v. that satisfies
(i) (Measurability) is (ii) (Partial averaging)
( , , )F
[ | ]E X
measurable
P( ) =[ | ]( ) X( dP( ) for ) all AA A
E X d
sub - - algebra of F
[ | ]E X
unique ?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, their difference Y-Z is as well, and thus the set A={Y-Z>0} is in . So we have
and thus
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.
[ | ]E X
( ( ) ( )) ( ) 0A
Y Z dP
( ) ( ) ( ) ( ) ( ) ( )A A AY dP X dP Z dP
measurable
General Conditional Expectations General Conditional Expectations PropertiesProperties• Theorem 2.3.2
let be a probability space and let be a .
(i) (Linearity of conditional expectation) If X and Y are integrable r.v.’s and and are constants, then
(ii) (Taking out what is known) If X and Y are integrable r.v.’s, Y and XY are integrable, and X is
( , , )F sub - - algebra of F
1 2 1 2E[c X+c Y| ] = c E[X| ] + c E[Y| ]
E[XY| ] = XE[Y| ]
1c 2c
measurable
General Conditional Expectations ProGeneral Conditional Expectations Properties(conti.)perties(conti.)
(iii) (Iterated condition)If H is a and X is an integrable r.v., then
(iv) (Independence)If X is integrable and independent of , then
(v) (Conditional Jensen’s inequality)If is a convex function of
a dummy variable x and X is integrable, then
p.f(Volume1 P.30)
E[X| ] = E[X]
E[ (X)| ] (E[X| ])
sub - - algebra of
E[E[X| ]| ] = E[X| ]H H
(X)
Example 2.3.3. (P.73)Example 2.3.3. (P.73)• X and Y be a pair of jointly normal random variables. Defi
ne so that X and W are independent, we know W is normal with mean and variance . Let us take the conditioning to be .We estimate Y, based on X. so,
(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.
1
2
-W Y X
1 11 2
2 2
E[Y|X] = +EW = (X- )+X
Y-E[Y|X] = W-E[W]
1
2
Y X W
= (X) 2 1
3 21
= -
2 2 23 2 = (1- )
Lemma 2.3.4.(Independence)Lemma 2.3.4.(Independence)• let be a probability space, and let be a
. Suppose the r.v.’s are and the r.v.’sare independent of . Let be a function of the dummy variables and define
Then
( , , )F sub - - algebra of F
1.... KX X measurable 1.... LY Y 1, , , 1, ,( ... ... )K Lf x x y y
1, ,... Kx x1, ,... Ly y
1, , 1, , , 1, ,( ... ) ( ... ... )K K Lg x x Ef x x y y
1, , , 1, , 1, , ( ... ... | ) ( ... )K L KEf X X Y Y g X X
Example 2.3.3.(conti.)Example 2.3.3.(conti.) (P.73)(P.73)• Estimate some function of the r.v.’s X and Y base
d on knowledge of X.
By Lemma 2.3.4
Our final answer is random but .
( , )f x y
1
2
( ) ( , )g x Ef x x W
[ ( , ) | ] ( ) E f X Y X g X( ) -X measurable
MartingaleMartingale
• Def 2.3.5. let be a probability space, let T be a fixed positiv
e number, and let , , be a filtration of . Consider an adapted stochastic process M(t), .
(i) If we say this process is a martingale. It has no tendency to rise or
fall. (ii) If we say this process is a submartingale. It has no tendency to fal
l; it may have a tendency to rise. (iii) If we say this process is a supermartingale. It has no tendency to
rise; it may have a tendency to fall.
( , , )F ( )F t 0 t T
sub - - algebras of F
0 t T E[M(t)|F(s)] = M(s) for all 0 s t T,
E f[M or(t)| allF(s)] M( 0 s T) t ,s
E f[M or(t)| allF(s)] M( 0 s T) t ,s
Markov processMarkov process• Def 2.3.6.
Continued Def 2.3.5. Consider an adapted stochastic process , .Assume that for all and for every nonnegative, Borel-measurable function f, there is another Borel-measurable function g such that
Then we say that the X is a Markov process.
[ ( ( )) | ( )] ( ( )).E f X t F s g X s
0 t T ( )X t0 s t T
[ ( , ( )) | ( )] ( , ( )), 0 .E f t X t F s f s X s s t T
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