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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 ) n1 ... p # H (n1 ...N ) q #T (n1 ...N ) X (1...nn 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!!