182 9 Fundamental Theorem of Asset Pricing
The setW^0 is a subspace ofL∞. There is no problem in this notation
since ifH·Sis bounded, thenHas well as (−H) is admissible and therefore
f=(H·S)∞exists and is a bounded random variable. From Proposition 9.3.6
it follows thatW^0 =K∩(−K). The same notation for a space related toW^0
is already used in [D 92]. The setWis simply{α+f|α∈Randf∈W^0 }.
BecauseSis supposed to be locally bounded these vector spaces are quite
big. The following lemma seems to be obvious but, because unboundedS-
integrable processes are used, it is not so trivial as one might suspect. The
proof we give uses rather heavy material but it saves place.
Lemma 9.5.1.IfHisS-integrable andH·Sis bounded, thenH·Sis aQ-
martingale for allQ∈M(P).
Proof.TakeQ∈M(P). ClearlySis a special semi-martingale under the
measureQ. Since it is a local martingale it decomposes asS=S+0. The
stochastic integralH·Sis bounded and hence is a special martingale under
Q. Its decomposition is, according to Theorem 9.2.2,H·S=H·S+H·0, i.e.
H·Sis aQ-local martingale. Being bounded it is a martingale underQ.
It follows from the martingale property that ifH andGare two S-
integrable processes such thatH·SandG·Sare bounded and such that
(H·S)∞=(G·S)∞then necessarily (H·S)=(G·S). (This also follows from
arbitrage considerations.)
The following theorem is due to [AS 94] and [J 92] (see also Chap. 11).
Earlier versions can be found in [KLSX 91]. The theorem is particularly im-
portant in the setting of incomplete markets (e.g. semi-martingales with more
than one equivalent martingale measure). It shows exactly what elements can
be constructed or hedged, using admissible strategies.
Theorem 9.5.2.Iff∈L^0 (Ω,F,P)withf−∈L∞(Ω,F,P)then the follow-
ing are equivalent
(i) there isH predictable,S-integrable,Q∈Me(P)andα∈Rsuch that
H·Sis aQ-uniformly integrable martingale withf=α+(H·S)∞
(ii) there isQ∈Me(P)such thatER[f]≤EQ[f]for allR∈Me(P).
Forfbounded these two properties are also equivalent to
(iii)ER[f]is constant as a function ofR∈M(P).
Proof.We refer to [AS 94, Theorem 3.2]. For (iii) we remark thatMe(P)is
L^1 (P)-dense inM(P) and henceER[f] is constant onMe(P) if and only if
it is constant onM(P).
Corollary 9.5.3.Wisσ(L∞,L^1 )-closed inL∞.
Proof.This follows immediately from (iii) of the theorem.Wis the subspace
of these elements inL∞that are constant on a subset ofL^1.