Suppose x e l 1 q b p and let g c b be a sub cr field

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Suppose X E L 1 (Q, B, P) and let g c B be a sub-cr-field. Then there exists a random variable E (Xi g), called the conditional expectation of X with respect to g, such that
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340 10. Martingales (i) E (Xi g) is g-measurable and integrable. (ii) For all G E g we have L XdP = L E(Xig)dP. To test that a random variable is the conditional expectation with respect to g, one has to check two conditions: (i) the measurability condition and (ii) the integral condition. There are (at least) two questions one can ask about this definition. (a) Why does this definition of conditional expectation make mathematical sense? (b) Why does this definition make intuitive sense? It is relatively easy to answer (a) given the development of Radon-Nikodym differentiation. Suppose initially that X 0. Define v(A) = i XdP, A E !3. Then v is finite and v < < P. So vig <<Pig. From the Radon-Nikodym theorem, the derivative exists and we set E(Xig) = dvig dPig which by Corollary 10.1.2 is g-measurable, and so for all G E g r dvig vig(G) = v(G) = }G dPig dPig = { dvig dP since P =Pig on g }G dPig = L E(Xig)dP which is (ii) of the definition of conditional expectation. If X E L 1 is not necessarily non-negative, then satisfies (i) and (ii) in the definition. 0
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10.2 Definition of Conditional Expectation 341 Notes. (1) Definition of conditional probability: Given (Q, 8, P), a probability space, with g a sub-a-field of 8, define P(Ai9) = E(lAI9) , A E 8. Thus P(AI9) is a random variable such that (a) P(AI9) is 9-measurable and integrable. (b) P(AI9) satisfies Ia P(Ai9)dP = P(A n G), 'IG e g. (2) Conditioning on random variables: Suppose {X 1, t E T} is a family of ran- dom variables defined on (Q, 8) and indexed by some index set T. Define g := a(X 1, t E T) to be the a-field generated by the process {X 1, t E T} . Then define E(XIX 1, t E T) = E(X19). Note (1) continues the duality of probability and expectation but seems to place expectation in a somewhat more basic position, since conditional probability is defined in terms of conditional expectation. Note (2) saves us from having to make separate definitions for E(XIXI), E(XIX1, Xz), etc. We now show the definition makes some intuitive sense and begin with an example. Example 10.2.1 (Countable partitions) Let {An, n 1} be a partition of Q so that A; n A j = 0, i # j' and Ln An = n. (See Exercise 26 of Chapter 1.) Define g = a(An , n 1) so that g =\?:A;: J C {1, 2, . .. }J. IE] For X E L 1 (P), define EA.(X) = J XP(dwiAn) = i. XdP/PAn, if P(An) > 0 and EA.(X) = 17 if P(An) = 0. We claim 00 (a) E(XIQ) a,4. LEA.(X)1An n=l
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342 10. Martingales and for any A e B 00 (b) P(AIQ) a,;4. L P(AIAn)lAn• n=l Proof of (a) and (b). We first check (a) . Begin by observing 00 L EAn(X)lAn e Q. n=l Now pick A e g and it suffices to show for our proposed form of E(XIQ) that Since A e Q, A has the form A = LieJ A; for some J C {1, 2, ... }. Now we see if our proposed form of E (XIQ) satisfies (10.9). We have = !; r;; i; E An (X)lAndP (form of A) = LLEAn(X)P(A;An) n:;:l ieJ = LEA;(X) · P(A;) ({An} are disjoint) ieJ jA_XdP = L ' · P(A;) (definition of EA(X)) ieJ P(A;) = L { XdP = { XdP ieJ J A; }LieJ A; = i XdP.
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