The Mathematics of Arbitrage

272 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory The norm on the spaceGcan be calculated using Theorem ...

13.5 The Value of Maximal Admissible Contingent Claims on the SetMe 273 is lower semi-continuous for the weak topologyσ ( L^1 (P ...

274 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory For eachnthe set{Q|Q∈M;EQ[fn]=0}is a norm dense and (f ...

13.6 The SpaceGunder a Num ́eraire Change 275 is constructed with thed-dimensional processS,thespaceG (S V, 1 V ) is the space o ...

276 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory Remark 13.6.3.The previous theorem shows thatGis a num ...

13.7 The Closure ofG∞and Related Problems 277 implies that for alln, necessarily,Eμ[f∧n]≤0. The sequenceEμ[f∧n]is increasing and ...

278 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory Theorem 13.7.5.Suppose thatSis continuous and satisfie ...

14 The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes (1998) 14.1 Introduction The topic of the present ...

280 14 The FTAP for Unbounded Stochastic Processes through theS-integrable predictable processes satisfying a suitable admissi- ...

14.2 Sigma-martingales 281 Definition 14.2.1.AnRd-valued semi-martingale X =(Xt)t≥ 0 is called a sigma-martingaleif there exists ...

282 14 The FTAP for Unbounded Stochastic Processes Example 14.2.3. A sigma-martingaleSwhich does not admit an equivalent local m ...

14.2 Sigma-martingales 283 Proposition 14.2.5.For a semi-martingaleXthe following assertions are equivalent. (i) Xis a local mar ...

284 14 The FTAP for Unbounded Stochastic Processes Proof.For eachktakeφk,XTkintegrable such thatφk>0on[[0,Tk]] ,φk·XTk is a u ...

14.3 One-period Processes 285 below for a general version of this result; we refer to [S 94] for an account on the history of th ...

286 14 The FTAP for Unbounded Stochastic Processes Proof.ClearlyBis convex. Let us also remark that it is non-empty. To see this ...

14.3 One-period Processes 287 Indeed: EQ 1 [ exp ( η(x, S 1 )− ) (x, S 1 ) ] =−EQ 1 [ exp ( η(x, S 1 )− ) (x, S 1 )− ] +EQ 1 [ ( ...

288 14 The FTAP for Unbounded Stochastic Processes such thatSbecomes aQ-martingale. A glance at Example 14.2.3 above reveals tha ...

14.3 One-period Processes 289 follows. For an elementD∈E⊗B(Rd)oftheformD=A×B, we define λF(D)= ∫ AFη(B)π(dη). For eachη∈Ewe defi ...

290 14 The FTAP for Unbounded Stochastic Processes The Crucial Lemma 14.3.5.Let(E,E,π)be a probability measure space and let(Fη) ...

14.3 One-period Processes 291 In the above arguments we did suppose that (E,E,π) is complete. Now we drop this assumption. In th ...