The Mathematics of Arbitrage

312 14 The FTAP for Unbounded Stochastic Processes (1)Vt= lims↘t;s∈Q+limn→∞(Kn·S)sexists a.s., for allt≥0, (2) (H^0 ·S)t−Vtis in ...

14.5 Duality Results and Maximal Elements 313 S)∗≤w. It follows that random variables of the form (^1) A ( φ·Sit−φ·Sis ) or − (^ ...

314 14 The FTAP for Unbounded Stochastic Processes Example 14.5.23.† There is a continuous process S,S 0 = 0 satisfying (NFLVR)a ...

14.5 Duality Results and Maximal Elements 315 Claim 1 can now be improved as follows: for−∞<β≤^18 we have E[exp(βν)] = 2 1 −√ ...

316 14 The FTAP for Unbounded Stochastic Processes Lemma 14.5.24.Ifg∈Kw, then there is a maximal elementh∈Kwsuch thath≥g. Proof. ...

14.5 Duality Results and Maximal Elements 317 Proof.Clearly (2) implies (1) by Theorem 14.5.12. For the reverse implication take ...

15 A Compactness Principle for Bounded Sequences of Martingales with Applications (1999) Abstract. ForH^1 -bounded sequences of ...

320 15 A Compactness Principle Note — and this is aLeitmotivof the present paper — that, for sequences (xn)n≥ 1 in a vector spac ...

15.1 Introduction 321 These authors have proved a remarkable decomposition theorem which es- sentially shows the following (see ...

322 15 A Compactness Principle Then there is anM-integrable predictable stochastic processH^0 such that H^0 ·Mis anL^2 -bounded ...

15.1 Introduction 323 tends to zero and such that the sequence of stopped processes ( (Mn)Tn ) n≥ 1 is relatively weakly compact ...

324 15 A Compactness Principle For general martingales, not necessarily of the formHn·Mfor a fixed local martingaleM, we can pro ...

15.1 Introduction 325 special assumptions, e.g., one-sided or two-sided bounds on the jumps of the processes (Hn·M), one may ded ...

326 15 A Compactness Principle 15.2 Notations and Preliminaries We fix a filtered probability space (Ω,F,(Ft)t∈R+,P), where the ...

15.2 Notations and Preliminaries 327 The Davis’ inequality forH^1 -martingales ([RY 91, Theorem IV.4.1], see also [M 76]) states ...

328 15 A Compactness Principle E[fnl∧βl(g+1)−fnl∧βl(g+1)∧K(g+1)] =E[fnl−fnl∧K(g+1)]−E[fnl−fnl∧βl(g+1)] ≤δ(K)− ( δ(∞)− δ(∞) 2 k(K ...

15.2 Notations and Preliminaries 329 As a first application of the Kadeˇc-Pelczy ́nski decomposition we prove the vector-valued ...

330 15 A Compactness Principle infinity. If (fk)k=1is a dense sequence in the unit ball ofC 0 , then for bounded sequences (μn)n ...

15.2 Notations and Preliminaries 331 This theorem immediately implies the following: Theorem 15.2.12.If(Nn)n≥ 1 is a relatively ...

332 15 A Compactness Principle Hence ∥ ∥ ∥[X, X] (^12) ∞ ∥ ∥ ∥ p ≤c (^12) ‖X‖ (^12) H^1 ‖X ∗ ∞‖ (^12) p 2 −p .
Corollary 15.2. ...