9.5 The Set of Representing Measures 181(L·S)∞= lim
t→∞
(L·S)t= lim
t→∞
lim
n→∞
(Ln·S)t
= lim
n→∞
lim
t→∞
(Ln·S)t= lim
n→∞
(Ln·S)∞=f 0.The interchange of the limits is allowed because almost surely (Ln·S)t→
(L·S)tuniformly int, by Lemma 9.4.6. Indeed (Hn·S)t converge uni-
formly onR+and the convex combinations Lk∈conv{Hk,Hk+1,...}pre-
serve this uniform convergence. This shows thatf 0 ∈K 0 and as remarked
before Lemma 9.4.6 this implies Theorem 9.4.2.
Remark 9.4.13.The topology of semi-martingales was defined in Sect. 9.2. It
was defined using the open end interval [0,∞[. A similar but stronger topology
could have been defined using the time interval [0,∞]. This amounts to using
the distance function:
D(X)=sup{E[min(|(H·X)∞|,1)]|Hpredictable,|H|≤ 1 }.The difference between the two topologies is comparable to the difference
between uniform convergence on compact sets of [0,∞[ and uniform conver-
gence on [0,∞]. A careful inspection of the proofs, mainly devoted to checking
the existence of the limits at∞, shows that the semi-martingales (Ln·S)tend
to (L·S) in the semi-martingale topology on [0,∞] and not only on [0,∞[.
We preferred not to use this approach in order to keep the proofs easier.
9.5 The Set ofRepresenting Measures .........................
In this section we use the results obtained in Ansel and Stricker [AS 94] “Cou-
verture des actifs contingents” and we give a new criterion under which the
market is complete. Throughout this paragraph the processSis supposed to
be locally bounded and to be a local martingale under the measureP.This
will facilitate the notation. We will study the following sets of “represent-
ing measures” defined on theσ-algebraF(see e.g. [D 92] for an explanation
concerning the name “representing measures”):
M(P)={Q|Q P,Qisσ-additive andSis aQ-local martingale}
Me(P)={Q|Q∼P,Qisσ-additive andSis aQ-local martingale}.The spaceM(P) consists of all absolutely continuous local martingale
measures and it can happen that some of the elements will give a measure
zero to events that under the original measure are supposed to have a strictly
positive probability to occur. This phenomenon was studied in detail in [D 92].
We will show thatMe(P)=M(P) impliesM(P)={P}.
We will need the following set of attainable assets:
W^0 ={f|there is anS-integrableH,H·Sbounded and (H·S)∞=f}.