Chuan Hsiang Han Oct 28 2009 A three period cointoss example Let a coin toss outcome The outcome of a coin tossed three times is denoted by Let is a algebra is ID: 1044653
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1. Information and ConditioningChuan-Hsiang HanOct. 28, 2009
2. A three period coin-toss exampleLet (a coin toss outcome).The outcome of a coin tossed three times is denoted by Let is a -algebra. is a measurable space.
3. A three period example: at period oneWhen the first coin toss is realized, the setsare resolved. Namely given the information of , for each , either or . and generate the -algebra , which contains the information learned from the first coin toss. and are atoms (indivisible sets) in .
4. A three period example: at period twoWhen the first two coin tosses are realized, a finer resolution is obtained:Complements, (countable) unions and intersections are also resolved. Atoms generate a -algebra at period 2 denoted by.Note:
5. A three period example: at period threeWhen the three coin tosses are all realized, each is resolved, so is every subset of . ThenNote:
6. A three period example revisedIf no information about the coin toss, the trivial -algebra is defined at time 0.Along this three time period, the collection of increasing -algebras is called a filtration. Each contains the information learned by observing the first coin-tosses. . contains more information than if .
7. FiltrationDenition Given a nonempty set and a fixed time , we assume there exists a -algebra for each and is increasing, i.e., for , . Then we call that is a filtration.A filtration represents an information flow. That is, at time , we know for each set in whether the true (or realized) outcome lies in that set. is also called the -algebra of events up to time .
8. An infinite Space . Suppose we observe but don't know its future value T. Then is resolved is not resolved.The sets resolved by time t are those can be described in terms of the path of up to time . is the path space that we can construct Brownian motion in Chapter 3.
9. Generate -algebra from r.v. (1) Denition Let be a r.v. defined on .The -algebra generated by is defined byEX: check example 2.1.4 in the text.
10. Generate -algebra from r.v. (2) Definition Let be a r.v. defined on .Let G be another -algebra in . If , we say that is -measurable.X is -measurable iff the information in is sufficient to determine the value of .Any Borel-measurable function of a -measurable r.v. is still -measurable. (If X and Y are -measurable and is a measurable function, then is -measurable.)
11. Stochastic ProcessesDefinition(1) Given a measurable space , the collection of random variables index by time , is called a stochastic process.(2) Given additional filtration , if for each , is -measurable, then we say that the stochastic process is adapted to the filtration.
12. Financial applicationsA portfolio position taken at time must be measurable. That is, must depend only on information available to the investor at time . An asset price at time must be measurable. That is, the asset is priced based on the information available to the trader at time . Asset prices, hedging positions, and wealth processes will be adapted to a filtration which contains the flow of public information.
13. Independence (1)When the value of the r.v. can be determined by the -algebra , we say that is -measurable.When the value of the r.v. is not affected by the -algebra , we say that is independent of .Let be a probability space. Two measurable sets A and B are independent if
14. Independence (2)Definition Let be a probability space.(1) Let and be sub--algebras. Two -algebras and are independent if(2) Let and are two random variables. and are independent if and are independent.EX: check example 2.2.2 in the text.Generalization to the multi-independence can be found in the Definition 2.2.3 in the text.
15. Independence (3)Theorem 2.2.5. Let and be independent random variables, and let f and g be Borel-measurable functions on . Then and are independent random variables.
16. Joint DistributionDefinition 2.2.6 The joint distribution measure of a pair of random variables is given by for all Borel sets The joint cumulative d. f. of isThe joint density satisfies
17. Marginal DistributionThe marginal distribution measures of and are , and , respectively.The marginal c.d.f.s of is The marginal density is that satisfies or, equivalently .
18. Jointly Normal Random VariablesSkip Theorem2.2.7.The joint density of isLet . denotes the mean vector and denotes covariance matrix (symmetric and positive definite matrix).
19. Conditioning (1)If and jointly possess a density and their marginals are and respectively. The conditional distribution of given is defined by the densityat any point where EX: check is a probability density of given .
20. Conditioning (2)If the expectation of this cond. dist. exists, we define the conditional expectation of given .Note: is a function of , say . Intuitively, by replacing by , we define a new r.v. , which is called the conditional expectation of given .
21. Conditioning (3) - General CaseLet -measurable r.v. satisfy or for any . We call the conditional expectation of given the -field and is denoted by .Replacing the generated -field by an arbitrary -field , one can generalize the definition to the conditional expectation of given .
22. N Coin TossesGiven , for any , , and let be a r.v. Then the conditional expectation of under , based on the information at time , is defined asWhen or , we define
23. Revisit 3 Period Binomial ModelGiven , we obtainRecall atoms in are , , ; so that , ,
24. We observe a partial averaging property:.Note in fact
25. Conditional ExpectationDef 2.3.1: Let be a probability space, let be a sub--algebra of , and let be a r.v. that is either nonnegative or integrable. The conditional expectation of given , denoted is any r.v. that satisfies(i) (Measurability) is -measurable.(ii) (Partial Averaging) for each ,If , then we write
26. Some Remarks(1) the r.v. is unique almost surely.(2) the r.v. exists based on the Radon-Nikodym theorem.(check Appendix B in the text.)(3) the r.v. is an estimate of based on the information in. (Check: Exercise 2.7 p.79 in the text.)
27. Properties of Conditional Expectation (1)Theorem 2.3.2: Let be a probability space and let be a sub--algebra of .(1) (2) If is -measurable, then(3) If is a sub--algebra of , then
28. Properties of Conditional Expectation (2)(4) If is independent of , then(5) If is a convex function, then
29. Computing Conditional expectations byindependenceLemma 2.3.4 Let be a probability space and let be a sub--algebra of .Suppose the r.v.s are -measurable and the r.v.s are independent of .Let,then.
30. Example 2.3.3Let be independent of and . Compute1 ( check )2
31. Martingale PropertyDefinition 2.3.5 Let be a probability space, let , and let be a filtration of sub--algebras of . Consider an adapted stochastic process (i) If for all , this process is martingale.(ii) If for all , this process is submartingale.(iii) If for all , this process is supermartingale.
32. Markov PropertyDefinition 2.3.6 Let be a probability space, let , and let be a filtration of sub--algebras of . Consider an adapted stochastic process Assuming that for all and for all nonnegative, Borel-measurable function there is another Borel-measurable function such thatThen we say that is a Markov process.