0 these results are easily extended when we drop the

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0 These results are easily extended when we drop the assumption of positivity which was only assumed in order to be able to apply Dubins' inequality 10.8.4. Corollary 10.15.1 Suppose (Bn} is a decreasing family and X e L Then almost surely and in L (The result also holds if { Bn } is an increasing family. See Proposition 10 . 11.2.) Proof. Observe that if we define {Xn} by Xn := E(XIBn), then this sequence is a reversed martingale from smoothing. From the previous theorem, we know a.s. and in L 1. We must identify X 00 From L 1 -convergence we have that for all A E 13, L E(Xi13n)dP L X 00 dP. Thus for all A e Boo L E(XiBn)dP = L XdP (definition) = L E(Xi13oo)dP (definition) L X 00 dP So by the Integral Comparison Lemma 10.1.1 Xoo = E(XiBoo). (from (10.60)). (10.60) 0
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10.15 Reversed Martingales 415 Example 10.15.1 (Dubins and Freedman) Let {Xn} be some sequence of ran- dom elements of a metric space(§, S) defined on the probability space (Q, B, P) and define Bn = a(Xn, Xn+b · · · ). Define the tail a-field Proposition 10.15.2 Tis a.s. trivial (that is, A E T implies P(A) = 0 or 1) iff VA e B: sup IP(AB)- P(A)P(B)I 0. Bel3n Proof. . If T is a.s. trivial, then P(AIBn) P(AIBoo) = P(AIT) = P(AI{0, Q}) = P(A) (10.61) a.s. and in L Therefore, sup IP(AB)- P(A)P(B)I = sup IE(P(ABIBn))- P(A)£(1B)i Bel3n Bel3n = sup IE (1B{P(AIBn)- P(A)})I Bel3n sup E IP(AIBn) - P(A)I 0 Bel3n from (10.61). +-.If A E T, then A E Bn and therefore P(A n A) = P(A)P(A) which yields P(A) = (P(A)) 2 . 0 Call a sequence {X n} of random elements of (§, S) mixing if there exists a probability measure F on S such that for all A e S P[Xn e F(A) and P([Xn e ·] n A) F(·)P(A). So {Xn} possesses a form of asymptotic independence. Corollary 10.15.2 If the tail a-field T of {Xn} is a.s. trivial, and P[Xn e ·] F(·), then {Xn} is mixing.
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416 10. Martingales 10.16 Fundamental Theorems of Mathematical Finance This section briefly shows the influence and prominence of martingale theory in mathematical finance. It is based on the seminal papers by Harrison and Pliska (1981), Harrison and Krebs (1979) and an account in the book by Lamberton and Lapeyre (1996). 10.16.1 A Simple Market Model The probability setup is the following. We have a probability space (Q, B, P) where n is finite and B is the set of all subsets. We assume P({w}) > 0, Yw E Q. (10.62) We think of w as a state of nature and (10.62) corresponds to the idea that all investors agree on the possible states of nature but may not agree on probability forecasts. There is a finite time horizon 0, 1, ... , N and N is the terminal date for eco- nomic activity under consideration. There is a family of a-fields Bo C 81 C · · · c BN = B. Securities are traded at times 0, 1, ... , Nand we think of Bn as the information available to the investor at time n. We assume Bo = {Q, 0}. Investors traded+ 1 assets (d :::: 1) and the price of the ith asset at time n is fori = 0, 1, ... , d. Assets labelled 1, ... , dare risky and their prices change randomly. The asset labelled 0 is a riskless asset with price at time n given by and we assume as a normalization = 1. The riskless asset may be thought of as a money market or savings account or as a bond growing deterministically.
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