9.4 Proof of the Main Theorem 165We remark that ifDis a cone thenDisFatou closedif for every sequence
(fn)n≥ 1 inDwithfn≥−1andfn→falmost surely, we havef∈D.
The next result is the technical version of the main theorem.
Theorem 9.4.2.IfSis a bounded semi-martingale satisfying (NFLVR), then
(1)C 0 is Fatou closed and hence
(2)C=C 0 ∩L∞isσ(L∞,L^1 )-closed.
Proof.We will not prove the first part of Theorem 9.4.2 immediately, its proof
is quite complicated and will fill the rest of this section.
The second assertion is proved using Theorem 9.2.1. IfC 0 is Fatou closed
then we have to prove thatC=C 0 ∩L∞is closed for the topologyσ(L∞,L^1 ).
Take a sequence (fn)n≥ 1 in C, uniformly bounded in absolute value by 1 and
such thatfn→falmost surely. SinceC 0 is Fatou closed the elementfbelongs
toC 0 and hence alsof∈C.
We now show how Theorem 9.4.2 implies the main theorem of the paper.
For convenience of the reader we restate the main Theorem 9.1.1.
Theorem 9.1.1 (Main Theorem).LetSbe a bounded real-valued semi-
martingale. There is an equivalent martingale measureQforSif and only if
Ssatisfies (NFLVR).
Proof.We proceed on a well-known path ([D 92, MB 91, S 92, Str 90, L 92,
S 94]). SinceSsatisfies(NA)we haveC∩L∞+ ={ 0 }. BecauseC is weak-
star-closed inL∞we know that there is an equivalent probability measure
Qsuch thatEQ[f]≤0foreachfinC. This is precisely the Kreps-Yan
separation theorem, for a proof of which we refer to [S 94, Theorem 3.1]. For
eachs<t,B∈Fs,α∈Rwe haveα(St−Ss) (^1) B ∈C(S is bounded!).
ThereforeEQ[(St−Ss) (^1) B]=0andQis a martingale measure forS.
The condition(NFLVR)is not altered if we replace the original probability
measure by an equivalent one. In the proof that condition(NFLVR)is also
necessary, we may therefore suppose thatPis already a martingale measure
for the bounded semi-martingaleS.IfH is an admissible integrand then
by Theorem 9.2.9 we know that the process (H·S) is a super-martingale.
ThereforeE[(H·S)∞]≤E[(H·S) 0 ] = 0. Every functionf inCtherefore
satisfiesE[f]≤0. The same applies for elements in the norm closureCofC.
ThereforeC∩L∞+={ 0 }.
We now show how the main theorem implies Corollary 9.1.2 pertaining to
the locally bounded case. We refer to [DS 94a] for examples that show that
we can only obtain an equivalent local martingale measure for the processS.
The proof of Corollary 9.1.2 is similar to [S 94, Theorem 5.1].
Corollary 9.1.2.LetS be a locally bounded real-valued semi-martingale.
There is an equivalent local martingale measureQfor Sif and only ifS
satisfies (NFLVR).