The Mathematics of Arbitrage

232 12 Absolutely Continuous Local Martingale Measures models preference relations are supposed to be strictly monotone and henc ...

12.1 Introduction 233 A strategy is a predictable process that is integrable with respect to the semi-martingaleSand that satisf ...

234 12 Absolutely Continuous Local Martingale Measures (b) (i)Ssatisfies the property (NA) and(ii)K 1 is bounded inL^0. (c) (i)S ...

12.2 The Predictable Radon-Nikod ́ym Derivative 235 complicated analytic proof had already been presented by the present au- tho ...

236 12 Absolutely Continuous Local Martingale Measures [DM 80]). One of these results says that there is a sequence of predictab ...

12.2 The Predictable Radon-Nikod ́ym Derivative 237 This shows thatCis continuous. Next we putφ=C (^1) [[ 0,Tn∧t]]and we find th ...

238 12 Absolutely Continuous Local Martingale Measures First note that, for each strictly positive operatorb 0 ,themapb→b−^1 is ...

12.3 The No-Arbitrage Property and Immediate Arbitrage 239 and only if for each predictableRd-processf, such that‖f(t, ω)‖is eit ...

240 12 Absolutely Continuous Local Martingale Measures T 2 =inf{t|t>T 1 ,(H·S)t≥−ε} is finite on the set{T 1 <∞}. We now p ...

12.3 The No-Arbitrage Property and Immediate Arbitrage 241 We now give some more motivation why such a form of arbitrage is call ...

242 12 Absolutely Continuous Local Martingale Measures sign functionφis a predictable process equal to +1 or−1. The predictable ...

12.3 The No-Arbitrage Property and Immediate Arbitrage 243 The proof of the theorem is based on the following lemma: Lemma 12.3. ...

244 12 Absolutely Continuous Local Martingale Measures P [ (H (^1) [[ 0,T 2 ]]·S)ε≥ 1 ] ≥P[{(H·A)ε≥1+a}∩{(H·M)∗<a}] ≥P[Λ]−P[( ...

12.4 The Existence of an Absolutely Continuous Local Martingale Measure 245 continuous local martingale, then the Girsanov-Maruy ...

246 12 Absolutely Continuous Local Martingale Measures F={LT> 0 }. Note that the no-arbitrage condition implies thatP[F]>0 ...

12.4 The Existence of an Absolutely Continuous Local Martingale Measure 247 This settles the problem of the usual hypotheses. Ea ...

248 12 Absolutely Continuous Local Martingale Measures stopping timeμ=∞onGcand equal to inf{t|Lt≤^12 Lσ}on the setG.The outcome ...

12.4 The Existence of an Absolutely Continuous Local Martingale Measure 249 It follows that L ̃∞> 0 PF-a.s.. It is chosen in ...

250 12 Absolutely Continuous Local Martingale Measures H ̃=H (^1) {σ 1 <T}, H ̃n=H ̃ (^1) [[ 0,τ n]]. From the preceding cons ...

13 The Banach Space of Workable Contingent Claims in Arbitrage Theory (1997) Abstract.For a locally bounded local martingaleS, w ...