212 10 Counter-Example
∫
{σ<∞}Lσ∧τ=∫∞
0P[σ∈dt]E[Lτ∧t]=P[σ<∞].The first line follows from the independence ofσand the processL. Putting
together both terms yieldsE[Lτ∧σ]=^12 P[σ=∞]+P[σ<∞]=^34 <1.
On the other hand the productXYis a uniformly integrable martingale.
To see this, it is sufficient to show thatE[X∞Y∞] = 1. The calculation is
similar to the preceding calculation and uses the same arguments.
E[X∞Y∞]=E[Lτ∧σMτ∧σ]
=E[Lτ∧σMσ] becauseMσis a uniformly integrable martingale=2E[Lτ∧σ (^1) {σ<∞}]
=2P[σ<∞]=1.
10.3 Incomplete Markets
All processes will be defined on a filtered probability space (Ω,(Ft)t≥ 0 ,P).
For the sake of generality the time set is supposed to be the setR+of all
non-negative real numbers. The filtration is supposed to satisfy the usual
hypothesis, i.e. it is right continuous and theσ-algebraF 0 is saturated with
all the negligible sets ofF∞=
∨
t≥ 0 Ft.ThesymbolSdenotes ad-dimensional
semi-martingaleS:Ω×R+→Rd. For vector stochastic integration we refer
to [J 79]. If needed, we denote byx′the transpose of a vectorx.
We assume thatShas the(NFLVR)property and the set of(ELMM)
is denoted byMe. The market is supposed to be incomplete in the following
sense. We assume that there is a real-valued non-zero continuous local martin-
galeWsuch that the bracket〈W, S〉= 0 but such that the measured〈W, W〉
(defined on the predictableσ-algebra of Ω×R+) is not singular with respect
to the measuredλwhereλ=trace〈S, S〉.
Let us first try to give some economic interpretation to this hypothesis.
The existence of a local martingaleW such that〈S, W〉= 0 implies that
the processSis not sufficient to hedge all the contingent claims. The extra
assumption that the measured〈W, W〉is not singular todtrace〈S, S〉then
means that at least part of the local martingaleWmoves at the same time as
the processS. The incompleteness of the market, therefore, is not only due to
the fact thatSandWare varying in disjoint time sets but the incompleteness
is also due to the fact that locally the processSdoes not span all of the random
movements that are possible.
Theorem 10.3.1.IfSis a continuousd-dimensional semi-martingale with
the (NFLVR) property, if there is a continuous local martingaleWsuch that
〈W, S〉=0butd〈W, W〉is not singular todtrace〈S, S〉, then for eachRin
Me, there is a maximal elementfsuch thatER[f]< 0.