Diagramas de casos de uso

As announced in the introduction, we now discuss closure properties in L0 _{of the}

decomposition of aV-optionally representable acceptance setA.

By analogy to the definition in [Schachermayer, 2004], we define a trading cone as follows:

Definition 3.4.1. C ⊂ L0₍

Rd+1,Ft) is said to be a (time-t) trading cone if C

is closed in L0 _{and is closed under multiplication by non-negative, bounded, F} t-

measurable random variables.

We recall Lemma 4.6 of [Jacka et al., 2008] which we quote here (suitably rephrased) for ease of reference:

Theorem 3.4.2. Let C be a closed convex cone inL0_{(F), then}

C is stable under multiplication by (scalar) elements ofL∞₊(F) (3.9)

if and only if there is a random closed coneMC _{such that}

C={X∈ L0(F) : X ∈MC a.s.}. (3.10) We shall demonstrate that if Ais V-representable then (K_t0)0≤t≤T, the L0-

closures of the cones Kt, are trading cones, whose sum is closed, and equal to the

L0-closure of A(V), which is is arbitrage-free.

This is a version of the (First) Fundamental Theorem of Asset Pricing (FTAP). For the rest of this section closures inL0 or L0 _{will be denoted by a simple}

overline, whereas weak∗closure of a setSwill be denotedSwWe setA0_{(V) :=A(V),}

the closure ofA(V) in L0_{. Recall that A}0_{(V) is arbitrage-free whenever}

A0(V)∩ L0₊={0},

and define the trading cone

Ct={X ∈ L0t :cX ∈ A0(V) for all c∈L∞+(Ft)}.

Note that closure inL0 _{of C}

tfollows immediately from the closure ofA0(V).

Lemma 3.4.3. For each t, K_t0 is a trading cone and if X ∈ L0

t then X ∈ Kt0 iff

Proof. Now if X ∈ L∞

t then X ∈ Kt if and only if EQ[X.V|Ft] = X.EQ[V|Ft] ≤

0 for allQ ∈ Q. It follows that Kt0 ={X ∈ Lt0 : X.EQ[V|Ft] ≤ 0 for allQ ∈ Q}

and this is obviously a trading cone.

It follows from Theorem3.4.2and Lemma3.4.3that

Lemma 3.4.4. There are random closed conesM_tC and M_tK such that Ct={Y ∈ L0t :Y ∈MtC a.s.}

K_t0 ={Y ∈ L_t0 :Y ∈M_tK a.s.}

and the polar (in Rd+1) of MtK is cone({EQ[V|Ft] : Q ∈ Q}), the random closed

cone generated by{_EQ[V|Ft] : Q∈ Q}.

We now give the main theorem of this section:

Theorem 3.4.5. The setG :=⊕_tCt is closed inL0, arbitrage-free and equalsH:=

⊕tKt0(A,V). Moreover, ifA isV-representable, then their common value isA0(V)

and then A0_{, the closure in L}0 _{of A is given by}

A=A0(V).V=⊕tKt0(A,V).V. (3.11)

Proof. The proof is in three steps. We will show that:

1. G is closed inL0_.

2. Ct=Kt0(A,V) (andG is arbitrage-free) establishing equality ofG andH.

3. A0_{(V) =HifA isV-representable andA}0_=A0_(V).V.

Proof of 1. We recall Definition 2.6 and Lemma 2.7 (suitably rephrased) from [Jacka et al., 2008]

Definition 3.4.6. SupposeJ is a sum of convex cones inL0_:

J =M0+. . .+MT.

We call elements ofM0×. . .×MT whose components almost surely sum to 0, null-

strategies (with respect to the decompositionM0+. . .+MT) and denote the set of

them byN(M0×. . .×MT).

Lemma 3.4.7. (Lemma 2 in [Kabanov et al., 2003]) Suppose that

J =M0+. . .+MT

is a decomposition of J into trading cones; then J is closed if N(M0×. . .×MT)

is a vector space and each Mt is closed in L0.

Since we have already established that each Ct is a trading cone, applying

Lemma3.4.7to the decomposition ofG, we only need to prove that the null strategies N(C) form a vector space. The argument is standard: since G is a cone, we need only show thatξ= (ξ0, . . . , ξT)∈ N(C) implies that−ξ∈ N(C). To do this, given

ξ ∈ N(C), fix a t and a bounded non-negative c ∈ L0_t with a.s. bound b. Then, sinceξ is null,

−cξt=bξ0+. . . bξt−1+ (b−c)ξt+bξt+1+. . .+bξT,

and each term in the sum is clearly in the relevantCs and hence inA0. Sincecand

tare arbitrary,−ξt∈Ct for each tand so −ξ∈ N(C).

It is clear from (3.4) thatKt⊆Ctand hence, by closure ofCtthatKt0 ⊆Ct.

ThusH ⊆ G.

Proof of 2. Recall from [Jacka et al., 2008] that consistent price processes for H are those martingales valued in (M_tK)∗ at each time step. Since ΛQ

tVtQ is

such a martingale (for any Q∈ Q), the collection of consistent price processes for the sequence of trading cones K_t0(A,V) is non-empty and so, by Theorem 4.11 of [Jacka et al., 2008], His arbitrage-free.

The consistent price processes for ⊕tCt are those martingales valued in

(M_tC)∗ at each time step. We now claim that, for each t,

(M_tC)∗ = (M_tK)∗.

Once we establish this, equality follows on taking the random polar cones inRd+1. First, observe that Ct ⊇Kt0(A,V) implies (MtC)∗ ⊆(MtK)∗ almost surely.

So, assume that (M_tC)∗ is a strict subset of (M_tK)∗. Then there exists Q∈ Q such that

P(EQ[V|Ft]6∈M

C t )>0.

For thisQ, we form the consistent price processZt= ΛQtEQ[V|Ft]∈(MtK)∗. Form

the frictionless trading cones

and we have an arbitrage-free and closed coneAe=⊕_tC_t(Z) from the FTAP. Clearly e

Acontains A0_{(V), and soC}

t(Z) is contained in Ct, whence Zt ∈MtC a.s., contra-

dicting the assumption of strict inclusion.

Proof of 3. If Ais V-representable then A0(V) = (⊕tKt)

w

=⊕tKt0 =H

but, as we have already established,His closed. Finally, sinceA ⊃Kt.V it is clear

that A0 _{⊇ H.V. Conversely, since A}0 _{=A(V).V =⊕K} t

w

.V = ⊕Kt.V, it follows

thatA0 _{⊆ H.V}