284 14 The FTAP for Unbounded Stochastic Processes
Proof.For eachktakeφk,XTkintegrable such thatφk>0on[[0,Tk]] ,φk·XTk
is a uniformly integrable martingale and‖φk·XTk‖H^1 < 2 −k.PutT 0 =0and
φ^0 =φ^11 [[ 0 ]]. It is now obvious thatφ=φ^0 +
∑
k≥ 1 φk 1
]]Tk− 1 ,Tk]]is strictly
positive, isX-integrable and is such thatφ·Xis anH^1 -martingale.
14.3 One-period Processes
In this section we shall present the basic idea of the proof of the main theorem
in the easy context of a process consisting only of one jump. LetS 0 ≡0and
S 1 ∈L^0 (Ω,F,P;Rd) be given and consider the stochastic processS=(St)^1 t=0;
as filtration we choose (Ft)^1 t=0whereF 1 =FandF 0 is some sub-σ-algebra
ofF. At a first stage we shall in addition make the simplifying assumption
thatF 0 is trivial, i.e., consists only of null-sets and their complements. In this
setting the definition of theno-arbitrage condition(NA)(see [DMW 90] or
Chap. 9) for the processSboils down to the requirement that, forx∈Rd,
the condition (x, S)≥0 a.s. implies that (x, S) = 0 a.s., where (., .) denotes
the inner product inRd.
From the theorem of Dalang-Morton-Willinger [DMW 90] we deduce that
the no-arbitrage condition(NA)implies the existence of an equivalent martin-
gale measure forS, i.e., a measureQon (Ω,F),Q∼P, such thatEQ[S 1 ]=0.
By now there are several alternative proofs of the Dalang-Morton-Wil-
linger theorem known in the literature ([S 92], [KK 94], [R 94]) and we shall
present yet another proof of this theorem in the subsequent lines. While some
of the known proofs are very elegant (e.g., [R 94]) our subsequent proof is
rather clumsy and heavy. But it is this method which will be extensible to
the general setting of anRd-valued (not necessarily locally bounded) semi-
martingale and will allow us to prove the main theorem in full generality.
Let us fix some notation: by Adm we denote the convex cone ofadmissible
elementsofRdwhich consists of thosex∈Rdsuch that the random variable
(x, S 1 ) is (almost surely) uniformly bounded from below.
ByKwe denote the convex cone inL^0 (Ω,F,P) formed by the admissible
stochastic integrals on the processS, i.e.,
K={(x, S 1 )|x∈Adm}and we denote byCthe convex cone inL∞(Ω,F,P) formed by the uniformly
bounded random variables dominated by some element ofK, i.e.,
C=(K−L^0 +(Ω,F,P))∩L∞(Ω,F,P)
={f∈L∞(Ω,F,P)| there isg∈K, f≤g}.Under the assumption thatSsatisfies(NA), i.e.K∩L^0 +={ 0 },wewantto
find an equivalent martingale measureQfor the processS. The first argu-
ment is well-known in the present context (compare [S 92] and Theorem 14.4.1