The Mathematics of Arbitrage

9.4 Proof of the Main Theorem 171 (iv) The jumps ofLn·Sare bounded from below by−nn+1 2. Indeed the process (Kn·S)Tnis bounded a ...

172 9 Fundamental Theorem of Asset Pricing Tchebycheff’s inequality now yieldsQ[(Ln·M)∗ (^1) Bc ≥n]≤α^2 which impliesQ[Bc∩{(Ln·M ...

9.4 Proof of the Main Theorem 173 and we therefore obtain Q [ (Rn·A)Tn,i−(Rn·A)Tn,i− 1 ≥α− 2 λ n ] >β−n−^2 fori=1,...,knand a ...

174 9 Fundamental Theorem of Asset Pricing From Q [ (Rn·A)Tn,i−(Rn·A)Tn,i− 1 ≥α−^2 nλ ] β−n−^2 and from Corollary 9.8.7 we dedu ...

9.4 Proof of the Main Theorem 175 the technique. Let us introduce the following sequence of stopping times (cis supposed to be&g ...

176 9 Fundamental Theorem of Asset Pricing (K·S)t= ∑n i=1 λi (^1) {t>Tci} ( (Hi·S)min(t,τ,σ)−(Hi·S)min(Tci,τ,σ) ) ≥ ∑n i=1 λi ...

9.4 Proof of the Main Theorem 177 Proof.Letε>0andtakec 0 as in Lemma 9.4.9 i.e. Q [(∑ λiKci·M )∗ >ε ] <ε for all (λ 1 . ...

178 9 Fundamental Theorem of Asset Pricing Ynk= ∑Nk j=0 λkjHk+j (^1) [[ 0,Tckn]]·M. is, for eachn, converging in the space ofL^2 ...

9.4 Proof of the Main Theorem 179 values in{+1,− 1 },α >0 and two increasing sequences (ik,jk)k≥ 1 such that Q[φk>α]>αw ...

180 9 Fundamental Theorem of Asset Pricing (R ̃k·S)t=(Rk·S)t =(Rk·A)t+(Rk·M)t ≥max ( (Lik·A)t,(Ljk·A)t ) +(Rk·M)t ≥max ( (Lik·A) ...

9.5 The Set of Representing Measures 181 (L·S)∞= lim t→∞ (L·S)t= lim t→∞ lim n→∞ (Ln·S)t = lim n→∞ lim t→∞ (Ln·S)t= lim n→∞ (Ln· ...

182 9 Fundamental Theorem of Asset Pricing The setW^0 is a subspace ofL∞. There is no problem in this notation since ifH·Sis bou ...

9.5 The Set of Representing Measures 183 Remark 9.5.4.The corollary was known long before Theorem 9.5.2 was known. The earliest ...

184 9 Fundamental Theorem of Asset Pricing atom{ 3 n+1, 3 n+2,...}.S 0 =0andSn−Sn− 1 is defined as the variable gn(3(n−1) + 1) = ...

9.5 The Set of Representing Measures 185 set we will use is precisely the setCintroduced in Sections 9.2, 9.3 and 9.4. In Sect. ...

186 9 Fundamental Theorem of Asset Pricing Remark 9.5.10.Let us recall that the dual ofL∞isba(Ω,F,P), the space of all bounded, ...

9.6 No Free Lunch with Bounded Risk 187 From the definitions and the results of Sect. 9.3 it follows that(NFL) implies(NFLBR)imp ...

188 9 Fundamental Theorem of Asset Pricing L^2. Therefore there is a sequence of convex combinationsgn∈conv{hk;k≥n} that converg ...

9.6 No Free Lunch with Bounded Risk 189 The Proposition 9.6.3 allows us to obtain a sharpening of the main theorem of [S 94, The ...

190 9 Fundamental Theorem of Asset Pricing the sequence (gn)n≥ 1 tends to 0 inL^1 (Q). Therefore the sequence (gn)n≥ 1 tends to ...