The Mathematics of Arbitrage

292 14 The FTAP for Unbounded Stochastic Processes we obtain anF 1 -measurable density of a probability measure. Assertion (i) o ...

14.3 One-period Processes 293 Defining dQ dP = ∏T t=1 Zt, we obtain a probability measureQ,Q∼Psuch that (St)Tt=0is a martingale ...

294 14 The FTAP for Unbounded Stochastic Processes 14.4 The GeneralRd-valued Case In this sectionS =(St)t∈R+ denotes a general R ...

14.4 The GeneralRd-valued Case 295 Theorem 14.4.1.Under the assumption (NFLVR) the coneCis weak-star- closed inL∞(Ω,F,P). Hence ...

296 14 The FTAP for Unbounded Stochastic Processes Then there is, forε> 0 , a probability measureQ∼P,‖Q−Q 1 ‖<ε, such that ...

14.4 The GeneralRd-valued Case 297 B=b·A ν(ω, dt, dx)=Fω,t(dx)dAt(ω) whereb=(bi)di=1is a predictable process andFω,t(dx) a trans ...

298 14 The FTAP for Unbounded Stochastic Processes The boundedness property translates to the fact thatH=(Ht(ω))t∈R+is anadmissi ...

14.4 The GeneralRd-valued Case 299 Using the identities EQ 1 [ (H (^1) P·(Xˇ+B))∞ ] = ∫ Ω×R+ (Hω,t∗Fω,t+(Hω,t,bω,t)) (^1) PdA(ω, ...

300 14 The FTAP for Unbounded Stochastic Processes E[Y(ω, T(ω),∆ST(ω)) (^1) T<∞] =E [∫ R+×Rd Y(ω, t, y)μ(ω, dt, dy) ] =E [∫ R ...

14.4 The GeneralRd-valued Case 301 Next observe thatQ 2 |FT−=Q 1 |FT−: indeed, we have to show thatQ 1 andQ 2 coincide on the se ...

302 14 The FTAP for Unbounded Stochastic Processes (ii) Qk+1|FTk−=Qk|FTk−anddQdQk+1k isFTk-measurable. (iii)S(k)is a sigma-marti ...

14.4 The GeneralRd-valued Case 303 Denote byDthe predictable set D= ⋃ k≥ 1 [[Tk]]⊆Ω×R+ and splitSintoS=Sa+Si,where Sa= (^1) D·S ...

304 14 The FTAP for Unbounded Stochastic Processes ‖Q ̃j−Q ̃j+1‖<εj,j=0,...,k− 1. In addition we assume thatQ ̃jandQ ̃j− 1 ag ...

14.5 Duality Results and Maximal Elements 305 The proof of the main theorem is complete now.
For later use, let us resume in t ...

306 14 The FTAP for Unbounded Stochastic Processes In the general case, i.e. when the processS is not necessarily locally bounde ...

14.5 Duality Results and Maximal Elements 307 Proof.LetZsbe a cadlag version of the density processZs=EQ 0 [ dQ dQ 0 |Fs ] . Now ...

308 14 The FTAP for Unbounded Stochastic Processes If no confusion can arise to which process the feasibility condition refers, ...

14.5 Duality Results and Maximal Elements 309 by the inequalityH·S≥−EQ[w|Ft]whereEQ[w]<∞.Comparethe formulation of Theorems 1 ...

310 14 The FTAP for Unbounded Stochastic Processes this result frequently. We also remark that ifHisw-admissible and if (H· S)∞≥ ...

14.5 Duality Results and Maximal Elements 311 Lemma 14.5.18.Ifw≥ 1 ,ifforsomeQ 0 ∈Meσwe haveEQ 0 [w]<∞,if g wis bounded and i ...