The Mathematics of Arbitrage

9.7 Simple Integrands 191 particular this theorem shows that in the Main Theorem 9.1.1, the hypothesis that the price process is ...

192 9 Fundamental Theorem of Asset Pricing By Lemma 9.8.1 there aregn∈conv{fn,fn+1...}such thatgn→ga.e. whereg:Ω→[0,∞]. AlsoP[g& ...

9.7 Simple Integrands 193 where 0≤T 0 ≤...≤TNn+1<∞are stopping times and the functions fknareFTkn-measurable functions, bound ...

194 9 Fundamental Theorem of Asset Pricing Since|K ̃n|≤1 we still have E [( (K ̃n·Mn)TNnn+1 ) 2 ] ≤c+5. On the other hand (K ̃n· ...

9.7 Simple Integrands 195 to be independent. The countable set of rationals in the interval ]0,1[ is enu- merated as (qn)n≥ 1. B ...

196 9 Fundamental Theorem of Asset Pricing Theorem 9.7.6. (a)IfS :[0,1]×Ω→ Ris a continuous semi-martingale having the no- arbit ...

9.7 Simple Integrands 197 Example 9.7.7.We take a standard Wiener processWwith its natural filtra- tion (Gt) 0 ≤t≤ 1. Before we ...

198 9 Fundamental Theorem of Asset Pricing We now give some more examples motivating the introduction of gen- eral integrands. A ...

9.7 Simple Integrands 199 P[Tm<1]≤P [ sup 0 ≤t≤ 1 |Wtm|>m ] + ∑ m≥ 1 2 −(n+m) ≤ √ 2 π 1 m e− m 22 +2−m and hence ∑ P[Tm< ...

200 9 Fundamental Theorem of Asset Pricing Lemma 9.7.10.If(αm)m≥ 1 is a sequence in]0,1]such thatαm→ 0 fast enough, thenSsatisfi ...

9.7 Simple Integrands 201 min((Lj·S)m,1)−min((Lj·S)m− 1 ,1)≤(Lj·S)m−(Lj·S)m− 1. The processLjis bounded in intervals [0,m] and b ...

202 9 Fundamental Theorem of Asset Pricing EP [ lim inf j→∞ ( (^1) Ajmin(f 0 ,1)) ] ≤EP [ lim inf j→∞ (^1) Ajmin((Lj·S)∞,1) ] ≤l ...

9.8 Appendix: Some Measure Theoretical Lemmas 203 with the compact (metrisable) space [0,∞]. A sequence (xn)n≥ 1 of elements of ...

204 9 Fundamental Theorem of Asset Pricing Proof.We first take convex combinations of{fn−;n≥ 1 }that converge almost surely. Sin ...

9.8 Appendix: Some Measure Theoretical Lemmas 205 Lemma 9.8.6.Let(gk) 1 ≤k≤nbe non-negative functions defined on the prob- abili ...

10 A Simple Counter-Example to Several Problems in the Theory of Asset Pricing (1998) Abstract.We give an easy example of two st ...

208 10 Counter-Example applies to the more general situation of a locally bounded semi-martingaleS, but in the situation of cont ...

10.1 Introduction and Known Results 209 a strict local martingale. The terminology was introduced by Elworthy, Li, and Yor [ELY ...

210 10 Counter-Example Proof.TakeXas in the preceding theorem and define, through the stochas- tic logarithm, the processSasdS=d ...

10.2 Construction of the Example 211 natural filtration of the couple (B, W) and is supposed to satisfy the usual conditions. Th ...