A Probability Path.pdf

# For such k we have from 1076 that vkij 0 since on vn

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For such k we have from (10.76) that Vk('I/J) 0 since on [Vn 0 (</>) < OJ the term Vn 0 (</>)S! 0 > < 0. This verifies the admissibility of 1/J. STEP (III). Now we finish the verification that 1/J is an arbitrage strategy. This follows directly from (10.76) with N substituted for k since VN('I/J) > 0 on [Vn 0 ( </>) < 0]. So an arbitrage strategy exists and the market is not viable. 0 We now return to the proof of the converse of Theorem 10.16.1. Suppose we have a viable market so no arbitrage strategies exist. We need to find an equivalent martingale measure. Begin by defining two sets of random variables r :={X : Q R : X 0, E (X) > 0} V :={VN(</>): Yo(</>)= 0, </>is self-financing and predictable}. Lemma 10.16.3 implies that r n V = 0. (Otherwise, there would be a strategy</> such that VN(</>) 0, E(VN(</>)) > 0 and Vo(</> = 0 and Lemma 10.16.3 would imply the existence of an arbitrage strategy in violation of our assumption that the market is viable.)

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10.16 Fundamental Theorems of Mathematical Finance 423 We now think of r and V as subsets of the Euclidean space !Rn, the set of all functions with domain Q and range JR. (For example, if Q has m elements WJ, .•. , Wm, we can identify !Rn with !Rm .) The set Vis a vector space. To see this, suppose tjl(1) and tjl(2) are two self-financing predictable strategies such that VN(tP(i)) e V fori= 1, 2. For real numbers a, b we have and 0 = aVo(<JI(1)) + bVo(<JI(2)) = Vo(atjl(1) + btjl(2)) and aVN(tP(1)) + bVN(tP(2)) is the value function corresponding to the self- financing predictable strategy a¢(1) + btjl(2); this value function is 0 at time 0. Thus aVN(tP(1)) + bVN(¢(2)) E V. Next define /(:={X E r: L X(w) = 1} wen so that JC c r. Observe that JC is closed in the Euclidean space !Rn and is compact and convex. (If X, Y e JC, then we have LwaX(w)+(1-a)Y(w) = a+1-a = 1 for 0 1.) Furthermore, since V n r = 0, we have V n JC = 0. Now we apply the separating hyperplane theorem. There exists a linear function A : !Rn f-+ lR such that (i) A(X) > 0, for X e JC, (ii) A(X) = 0, for X e V. We represent the linear functional A as the vector A= (A(w),w e Q) and rewrite (i) and (ii) as (i') Lwen A(w)X(w) > 0, for X E JC, (ii') Lwen VN(<JI)(w) = 0, for VN(tP) E V, so that <Pis self-financing and predictable. From (i') we claim A(w) > 0 for all we Q. The reason for the claim, is that if A(wo) = 0 for some wo, then X= 1(wol satisfies LwenX(w) = 1 so X E JC but L A(w)X(w) = A(wo) = 0 wen violates (i'). Define P* by P*(w)- A(w) - Lw'en A(w') .
424 10. Martingales Then P*(w) > 0 for all w so that P* = P. It remains to check that P* is that which we desire: an equivalent martingale measure. For any VN(¢) e V, (ii') gives L .X(w)VN(¢)(w) = 0, weQ so that, since Vo(¢) = 0, we get from (10.65) N E*(L(cPj• dj)) = 0, (10.77) j=l for any ¢which is predictable and self-financing. Now pick 1 ::: i ::: d and suppose 0 ::: n ::: N) is any predictable process. Using Lemma 10.16.2 with Vo = 0 we know there exists a predictable process ( 0 ::: n ::: N) such that ¢# := { 0, ... , 0, 0, ... , 0), 0 ::: n ::: N} is predictable and self-financing. So applying (10.77), we have N N d 0 =E*(L<¢ 1 , dj)) = E*(LL¢r>-aj>) j=l j=Il=I N =E*(E (10.78) j=I Since (10.78) holds for an arbitrary predictable process {¢ji), 0 ::: j ::: N}, we conclude from Lemma 10.5.1 that Bn), 0::: n::: N} is a P*-martingale.

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