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Unformatted text preview: e is a strictly positive probability vector so that all the stock prices are martingale. Proof. One direction is easy. If (i) is true, then for any strictly positive probaPm Pn bility vector i=1 j =1 xi ai,j pj > 0, so (ii) is false. Pm Suppose now that (i) is false. The linear combinations i=1 xi ai,j when viewed as vectors indexed by j form a linear subspace of n-dimensional Euclidean space. Call it L. If (i) is false, this subspace intersects the positive orthant O = {y : yj 0 for all j } only at the origin. By linear algebra we know that L can be extended to an n 1 dimensional subspace H that only intersects O at the origin. (Repeatedly find a line not in the subspace that only intersects O at the origin and add it to the subspace.) p ⇥⇥ HP PP P ⇥ ⇥ P⇥ PP PP O PP 181 6.1. TWO SIMPLE EXAMPLES Pn Since H has dimension n 1, it can be written as H = {y : j =1 yj pj = 0} where p is a vector with at least one positive component. Since for each fixed i the vector ai,j is in L ⇢ H, (ii) holds. To see that all the pj > 0 we leave it to the reader to che...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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