13.4 Some Results on the Topology ofG 267Proof.We first show that the statement of the theorem holds for a well-chosen
subsequence (nk)k≥ 1. Afterwards we will fill in the remaining gaps.
The subsequencenkis chosen so that for allN≥nkwe have‖f−fN‖≤
2 −k−^1. It follows that‖fnk+1−fnk‖≤ 2 −k, for allk. We take, according
to Theorem 13.3.13, contingent claims inKmax, denoted by (ψk,φk)k≥ 1 such
that
fn 1 =ψ 1 −φ 1
fnk+1−fnk=ψk−φk,and such thatψkandφkare 2−k-admissible fork≥2. Letgnk=
∑k
l=1ψl
andhnk=
∑k
l=1φl. By Corollary 13.2.17 and the reasoning in the proof of
Theorem 13.4.1, these sequences converge in the norm ofGto respectivelyg
andh.Furthermorefnk=gnk−hnkand hencef=g−h.
We now fill in the gaps ]nk,nk+1[. Fornk<n<nk+1we choose maximal
2 −k-admissible contingent claimsρnandσnsuch thatfn−fnk=ρn−σn.To
complete the proof we just have to check the obvious fact thatgn=gnk+ρn
andhn=hnk+σnsatisfy the requirements of the theorem.
We will now discuss an example that serves as an illustration of what can
go wrong in an incomplete market.
Example 13.4.4.The example is a slight modification of the example presented
in Chap. 10, see also [S 93]. We start with a two-dimensional standard Brown-
ian motion (B, W), with its natural filtration (Ft)t≥ 0. For the price processS
we take a stochastic volatility process defined asdSt= (2 + arctan (Wt))dBt.
It is clear that the natural filtration ofSis precisely (Ft)t≥ 0 .Furthermore
it is easy to see that the set of stochastic integrals with respect toSis the
same as the set of stochastic integrals with respect toB. We will use this
fact without further notice. We defineL=E(B)andZ=E(W), whereE
denotes the stochastic exponential. The stopping timesτ andσare defined
asτ=inf{t|Lt≤^12 }andσ=inf{t|Zt≥ 2 }. The processXis defined as
X=Lτ∧σ. The measureQis nothing else butdQ=Zτ∧σdP. For the process
Xand the measureQ, the following hold:
(1) The processXis continuous, strictly positive, alsoX∞>0a.s.andX 0 =
1, it is a local martingale forP, i.e.P∈Me,
(2) underP, the processXis a strict local martingale, i.e.EP[X∞]<1,
(3) for eacht<∞the stopped processXtis aP-uniformly integrable mar-
tingale,
(4) there is an equivalent probability measureQ∈Mefor whichXbecomes
aQ-uniformly integrable martingale.
Let us now verify some additional features.
Proposition 13.4.5.In the setting of the above example, the spaceG∞is not
dense inG. In fact even the closure ofL∞for the norm‖g‖=^12 sup{‖g‖L (^1) (Q)|
Q∈Me}, does not containGas a subset.