The Mathematics of Arbitrage

6.11 Interpretation of theL∞-Bound in the DMW Theorem 109 Proof.IfQ 0 ∈Ma∩Pkthen for allY∈W 1 we haveuk(Y)=inf{EQ[Y]| Q∈Pk}≤EQ 0 ...

7 A Primer in Stochastic Integration 7.1 The Set-up ............................................. In the previous chapters we ma ...

112 7 A Primer in Stochastic Integration Cash is convenient to transform money from one date to another. We assume that there ar ...

7.2 Introductory on Stochastic Processes 113 (ii)F 0 contains all null sets ofF∞. This means:A⊂B∈F∞andP[B]=0 implyA∈F 0 (F 0 is ...

114 7 A Primer in Stochastic Integration IfT:Ω→R+∪{∞}is a stopping time,FTis theσ-algebra of events prior toTi.e.FT={A|A∈F∞and f ...

7.2 Introductory on Stochastic Processes 115 SequencesTn↗∞, such that eachXTnsatisfies (P), are called localising sequences. In ...

116 7 A Primer in Stochastic Integration Example 7.2.9.IfLis a local martingale, it is tempting to use the following sequence of ...

7.3 Strategies, Semi-martingales and Stochastic Integration 117 7.3 Strategies, Semi-martingales and Stochastic Integration Inte ...

118 7 A Primer in Stochastic Integration quite a delicate task. The difficulties appear already in the case of Brownian motion; ...

7.3 Strategies, Semi-martingales and Stochastic Integration 119 uniformly as a sequence of continuous functions onR+. Let us den ...

120 7 A Primer in Stochastic Integration To develop the natural degree of generality also for processes with jumps we have to ex ...

7.3 Strategies, Semi-martingales and Stochastic Integration 121 (H·S)t:= (H·M)t+(H·A)t,t∈R+, (7.12) is well-defined. One can eve ...

122 7 A Primer in Stochastic Integration to be special forQ. For more details on how to define stochastic integrals for bounded ...

7.3 Strategies, Semi-martingales and Stochastic Integration 123 topology induced by (7.13) (or, equivalently, by (7.14)). The li ...

124 7 A Primer in Stochastic Integration E[|Xt|]= ∫t 0 ∣ ∣ ∣ ∣ B u ∣ ∣ ∣ ∣dP[T=u]= ∫t 0 1 u 2 e−^2 udu=∞. HenceX is not a martin ...

7.3 Strategies, Semi-martingales and Stochastic Integration 125 Clearly the Lebesgue-Stieltjes integrability ofHwith respect toA ...

126 7 A Primer in Stochastic Integration On the other hand we have on{T> 1 }and fort≥ 1 (H·A)t= ∫t∧T 0 du 1 −u =+∞. The next ...

7.3 Strategies, Semi-martingales and Stochastic Integration 127 (i) H·Vis a local martingale and (ii)H·Nis a local martingale. T ...

128 7 A Primer in Stochastic Integration Corollary 7.3.8.IfMis a local martingale, ifHisM-integrable and if(H· M)−is locally int ...

8 Arbitrage Theory in Continuous Time: an Overview 8.1 Notationand Preliminaries ............................... After all this ...