The Mathematics of Arbitrage

9.1 Introduction 151 strictly positive with positive probability. The economic interpretation is that by betting on the processS ...

152 9 Fundamental Theorem of Asset Pricing (see [S 92, KK 94, R 94] for elementary proofs). For the case of discrete but infinit ...

9.1 Introduction 153 stricted in order to allow the definition of integralsH·Sfor more general trad- ing strategies.Shas to be a ...

154 9 Fundamental Theorem of Asset Pricing In [DS 94a] counter-examples are given which show that in the above corol- lary one c ...

9.2 Definitions and Preliminary Results 155 9.2 Definitions and Preliminary Results Throughout the paper we will work with rando ...

156 9 Fundamental Theorem of Asset Pricing to [P 90] for the details. The predictableσ-algebraPonR+×Ωistheσ- algebra generated b ...

9.2 Definitions and Preliminary Results 157 The theorem and more precisely its Corollary 9.2.4, will be used in Sect. 9.4. It al ...

158 9 Fundamental Theorem of Asset Pricing Remark 9.2.6.In Corollary 9.2.4 we cannot replace (∆X)∗by (∆X)T.The following example ...

9.2 Definitions and Preliminary Results 159 Throughout the paper, with the exception of Sect. 9.7,Swill be a fixed semi-martinga ...

160 9 Fundamental Theorem of Asset Pricing We close this section by quoting a result due to ́Emery and Ansel and Stricker. The r ...

9.3 No Free Lunch with Vanishing Risk 161 the existence ofα>0 such thatP[(Hn·S)∞≥n]>α>0. The sequencefn= min ( 1 n(H n· ...

162 9 Fundamental Theorem of Asset Pricing A={lim inft→∞(H·S)t<β<γ<lim supt→∞(H·S)t}. Since the Boolean algebra ⋃ 0 ≤tF ...

9.3 No Free Lunch with Vanishing Risk 163 Corollary 9.3.4.If the semi-martingaleSsatisfies (NFLVR) then the set {(H·S)∞|His 1 -a ...

164 9 Fundamental Theorem of Asset Pricing hn≥gn.Ifhn=(Ln·S)∞then‖h−n‖∞≤n^1 and henceLnisn^1 -admissible by Proposition 9.3.6 an ...

9.4 Proof of the Main Theorem 165 We remark that ifDis a cone thenDisFatou closedif for every sequence (fn)n≥ 1 inDwithfn≥−1andf ...

166 9 Fundamental Theorem of Asset Pricing Proof.SinceS is locally bounded, there is a sequenceαn→+∞and an increasing sequence o ...

9.4 Proof of the Main Theorem 167 Remark 9.4.5.Let us motivate why we introduced the maximal elementf 0 in the above lemma. As a ...

168 9 Fundamental Theorem of Asset Pricing by the larger outcome 0. Replacinggby a maximal element is in this sense a “best try” ...

9.4 Proof of the Main Theorem 169 Proof.Suppose to the contrary that there isα>0, sequences (nk,mk)k≥ 1 tending to∞and for ea ...

170 9 Fundamental Theorem of Asset Pricing From now onQwill be fixed. SinceS is bounded it is a special semi- martingale and its ...