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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