The Mathematics of Arbitrage

15.3 An Example 333 Proof.Fix a collection ((εn,k)^2 n− 1 k=1)n≥^1 of independent random variables, εn,k= { − 2 −n with probabil ...

334 15 A Compactness Principle inH^1 and such that the pointwise limit equalsZt=− 2 t. Of course, the process Zis far from being ...

15.4 A Substitute of Compactness for Bounded Subsets ofH^1335 inequality this implies the existence of a constantc<∞such that ...

336 15 A Compactness Principle tends to zero in probability), we easily see thatMTnn tends to zero inL^1. Doob’s maximum inequal ...

15.4 A Substitute of Compactness for Bounded Subsets ofH^1337 is uniformly integrable. In this case the set {Mσn|n≥ 1 ,σa stoppi ...

338 15 A Compactness Principle or which is the same: Nn=(Mn)Tn∧σn−(∆ (Mn))Tn∧σn (^1) [[Tn∧σn,∞[[+(Cn)σn−. The maximal functions ...

15.4 A Substitute of Compactness for Bounded Subsets ofH^1339 15.4.3Proof of Theorem 15.B.The case where all martingales are of ...

340 15 A Compactness Principle show that these supports form a sequence of sets that tends to the empty set. This requires some ...

15.4 A Substitute of Compactness for Bounded Subsets ofH^1341 For eachnletT ̃nbe defined as T ̃n= { τn if|∆(Hn·M)τn|>γn ( (tr ...

342 15 A Compactness Principle Let us defineLn= ∑ kα k nH k. Clearly‖Ln·M‖ H^1 ≤1foreachn.From Theorem 15.1.3, it follows that t ...

15.4 A Substitute of Compactness for Bounded Subsets ofH^1343 We consider the martingales (∑ kλ k nV k)·M.ForeachnletDnbe the co ...

344 15 A Compactness Principle tend to a process of finite variation. The third term has a maximal function that tends to zero s ...

15.4 A Substitute of Compactness for Bounded Subsets ofH^1345 above linear system would only admit the solutionαk=0forallk≤d+1. ...

346 15 A Compactness Principle P [ sup t |(Hn·M)t−Nt|≥ 2 −n ] ≤ 2 −n. The Borel-Cantelli lemma then implies that sup t sup n |(H ...

15.4 A Substitute of Compactness for Bounded Subsets ofH^1347 Exactly as in Chap. 9 we introduce the cone C 1 ,w={f|there is aw- ...

348 15 A Compactness Principle as well as the negative parts of the super-martingales ((Hn·S)t)t∈R+areP- uniformly integrable. I ...

15.4 A Substitute of Compactness for Bounded Subsets ofH^1349 Vt= lims↘t ,s∈Q+ V̂s,t∈R+ is an a.s. well-defined cadlag super-mar ...

350 15 A Compactness Principle DefineL̂kas L̂k=Lnk (^1) [[ 0,T k]]+L mk 1 ]]Tk,∞[[ so thatL̂kis aw-admissible predictable integr ...

15.4 A Substitute of Compactness for Bounded Subsets ofH^1351 M)∗Un)n≥ 1 , showing that ((Ln·M)Un)n≥ 1 is a uniformly bounded se ...

352 15 A Compactness Principle lim n→∞ ‖(rLn,j−H^0 ,j)·M‖H (^1) (P)=0 (Zj)t= limq↘t q∈Q+ lim n→∞ (sLn,j·M)q whereZjis a well-def ...