10.3 Incomplete Markets 213Proof.The proof is broken up in different lemmata. LetW′be the martin-
gale component in the Doob-Meyer decomposition ofW with respect to the
measureR. Clearly〈W′,W′〉=〈W, W〉.
Lemma 10.3.2.Under the hypothesis of the theorem, there is a real-valued
R-local martingaleU=0such that
(1)〈S, U〉=0
(2)there is a boundedRd-valued predictable processHthat isS-integrable and
such thatd〈U, U〉=H′d〈S, S〉Hso that the processN=H·Ssatisfies
〈N, N〉=〈U, U〉.
Proof of Lemma 10.3.2.Letλ=trace〈S, S〉.Sinced〈W, W〉is not singular
with respect todλthere is a predictable setAsuch that (^1) Ad〈W, W〉is not
identically zero and absolutely continuous with respect todλ. From the pre-
dictable Radon-Nikod ́ym theorem, see Chap. 12, it follows that there is a pre-
dictable processhsuch that (^1) Ad〈W, W〉=hdλ.Fornbig enough the process
h (^1) {‖h‖≤n} (^1) [[ 0,n]]isλ-integrable and is such that (^1) A (^1) {‖h‖≤n} (^1) [[ 0,n]]d〈W, W〉is
not zero a.s.. We takeU=
(
(^1) A (^1) {‖h‖≤n} (^1) [[ 0,n]]
)
·W′. To findHwe first con-struct a strategyK such that d〈K·S,Kdλ ·S〉 = 0 a.e.. This is easy. For each
coordinatei, we take an investmentPi=(0, 0 ,..., 0 , 1 , 0 ,...) in asset number
i. On the predictable setd〈P^1 ·S,Pdλ^1 ·S〉=0wetakeK=P 1 , on the predictable
set whered〈P^1 ·S,Pdλ^1 ·S〉=0andd〈P^2 ·S,Pdλ^2 ·S〉=0wetakeK=P 2 ,etc..Wenow
takeH=K (^1) A (^1) {‖h‖≤n} (^1) [[ 0,n]]h
12 ( dλ
d〈K·S,K·S〉
)
.
Remark 10.3.3.We define the stopping timeνuasνu=inf{t|〈N, N〉t>u},
whereN is defined as in Lemma 10.3.2 above. If we replace the process
(N, U), the filtrationFtand the probabilityRby, respectively, the process
(Nν 0 +t,Uν 0 +t)t≥ 0 ,(Fν 0 +t)t≥ 0 , and the conditional probabilityR[.|ν 0 <∞],
we may without loss of generality suppose thatR[ν 0 =0]=R[〈N, N〉∞>
0] = 1. In this case we have that limu→ 0 R[νu<∞] = 1 and limu→ 0 νu=0.
We will do so without further notice.
Remark 10.3.4.The idea of the subsequent construction is to see the strongly
orthogonal local martingalesUandNas time-transformed independent Brow-
nian motions and to use the construction of Sect. 10.2. The first step is to
prove that there is a strict local martingale that is an exponential. The idea
is to use the exponentialE(B)whereBis a time transform of a Brownian
motion. However, the exponential only tends to zero on the set{〈B, B〉=∞}.
Lemma 10.3.5.There is a predictable processK such that the local R-
martingaleE(K·N)is not uniformlyR-integrable.
Proof of Lemma 10.3.5.Take a strictly decreasing sequence of strictly positive
real numbers (εn)n≥ 1 such that
∑
n≥ 1 εn^2
n<^1
8.