270 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory
EQn[(f−h)+]≥(
1 −
1
n)
1
n(n−‖h‖∞)≥
(
1 −
1
n)(
1 −
‖h‖∞
n)
.
From Theorem 13.3.17 we can now deduce that the distance offtoG∞is
precisely equal to 1.
This completes the discussion of the Example 13.4.4.Example 13.4.8.This is an example showing that the spaceGcan be one-
dimensional, whereas the setMeremains very big. For this we take a finite
time set [0,1], and we take Ω = [0,1] with the Lebesgue measure. Fort<1,
we putFtequal to theσ-algebra generated by the zero sets with respect to
Lebesgue-measure. Fort= 1 we putFtequal to theσ-algebra of all Lebesgue-
measurable sets. The price process is defined asSt=0fort<1andS 1 (ω)=
ω−^12 .OfcourseG= span(S 1 ). The setMeis the set{f|f> 0 ,
∫ 1
0 (t−
1
2 )f(t)dt=0}. This set is big in the sense that it is not relatively weakly
compact inL^1 [0,1].
Example 13.4.9.This example shows that the spaceGcan actually be iso-
morphic to anL∞-space. The example is constructed is the same spirit as the
previous one. We take [0,2] as the time set and Ω = [− 1 ,1]×[− 1 ,1] with the
two-dimensional Lebesgue measure. Letg 1 , respectivelyg 2 , be the first and
second coordinate projection defined on Ω. Fort<1theσ-algebraFtis the
σ-algebra generated by the zero sets, for 1≤t<2wehaveFt=σ(F 0 ,g 1 )and
F 2 =σ(F 1 ,g 2 ), which is also theσ-algebra of Lebesgue-measurable subsets
of Ω. The processSis defined asSt=0fort<1,St=g 1 for 1≤t<2and
S 2 =g 1 +g 2. We remark that the filtration is generated by the processS.
Clearly (H·S) 2 ∈Gif and only if it is of the form (H·S) 2 =αg 1 +hg 2 ,
wherehisF 1 -measurable and bounded. This implies thatGcan be identified
withR×L∞(Ω,F 1 ,P). We will not calculate the norm of the spaceG, but
instead we will use the closed graph theorem to see that this norm is equivalent
to the norm defined as‖(α, h)‖=|α|+‖h‖∞. It follows thatGis isomorphic
to anL∞-space.
Example 13.4.10.The following example is in the same style as the processS
has exactly one jump. But this time the behaviour of the processSbefore the
jump is such that the spaceGis not ofL∞-type.
We start with the one-dimensional Brownian motionW, starting at zero
and with its natural filtration (Ht) 0 ≤t≤ 1 .Attimet=1weaddajumpguni-
formly distributed over the interval [− 1 ,1] and independent of the Brownian
motionW. So the price process becomesSt=Wtfort<1andS 1 =W 1 +g.
The filtration becomes, up to null sets,Ft=Htfort<1andF 1 =σ(H 1 ,g).
For simplicity we assume that this process is defined on the probability space
Ω×[− 1 ,+1] where Ω is the trajectory space of Brownian motion, equipped