The Mathematics of Arbitrage

252 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory translated by the property thatH·Sis bounded below by ...

13.1 Introduction 253 use finitely additive measures in order to describe the closure of the space of bounded workable contingen ...

254 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory Definition 13.1.3.The locally bounded semi-martingaleS ...

13.2 Maximal Admissible Contingent Claims 255 We suppose from now on that the processSis a fixedd-dimensional locally bounded se ...

256 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory The(NA)property can be rephrased as the property that ...

13.2 Maximal Admissible Contingent Claims 257 Remark 13.2.7.The corollary also shows that iff=(H·S)∞is maximal admissible, then ...

258 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory Proof.Suppose thatK·S≥−(a+L·S)whereLis admissible and ...

13.2 Maximal Admissible Contingent Claims 259 Proof.This is a rephrasing of the Theorem 13.2.14 since by Theorem 13.2.5, the con ...

260 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory Corollary 13.2.18.IfS is a locally bounded semi-martin ...

13.3 The Banach Space Generated by Maximal Contingent Claims 261 byMaximalContingentClaims............................ 13.3 The ...

262 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory Proof.PutL=(H^1 −H^2 ), whereH^1 andH^2 are both maxim ...

13.3 The Banach Space Generated by Maximal Contingent Claims 263 The following lemma is an easy exercise in integration theory a ...

264 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory Proof.Take a sequence of real numbers such thatan↘‖g‖. ...

13.3 The Banach Space Generated by Maximal Contingent Claims 265 Proof.As in the previous result, for a contingent claimg=(H^1 · ...

266 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory 13.4 Some Results on the Topology ofG We now show that ...

13.4 Some Results on the Topology ofG 267 Proof.We first show that the statement of the theorem holds for a well-chosen subseque ...

268 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory Proof.For eacht≤∞we clearly have thatft=Xt− 1 ∈G. Supp ...

13.4 Some Results on the Topology ofG 269 Theorem 13.2.5, the random variableu(x)isinG. For arbitraryxwe split into the positive ...

270 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory EQn[(f−h)+]≥ ( 1 − 1 n ) 1 n (n−‖h‖∞) ≥ ( 1 − 1 n )( 1 ...

13.4 Some Results on the Topology ofG 271 with the usual Wiener measurePand where we take the uniform distribution mon [− 1 ,+1] ...