The Mathematics of Arbitrage

178 9 Fundamental Theorem of Asset Pricing

Ynk=

∑Nk

j=0

λkjHk+j (^1) [[ 0,Tckn]]·M.
is, for eachn, converging in the space ofL^2 (Q)-martingales. An easy way to
prove this assertion, is via the following reasoning in Hilbert spaces.
LetM^2 be the Hilbert space ofL^2 (Q)-martingales and letH=

(∑

⊕M^2

)

^2
be its^2 -sum (see [D 75]). An element of this space is a sequenceX=(Xn)n
where eachXnis inM^2. This space is also a Hilbert space when equipped with
the norm‖X‖^2 =

∑

n≥ 1 ‖Xn‖

2

2. The sequenceX

k, defined by the co-ordinates

Xnk=

1

2 n

(

cn+3‖q‖L (^2) (Q)

)

(

Hk (^1) [[ 0,Tckn]]·M

)

is bounded in the Hilbert spaceHand hence there are convex combinations
Yk∈conv{Xk,Xk+1,...}that converge with respect to the norm ofH.It
follows that each “co-ordinate” converges inM^2. This implies the existence
of convex weightsλk 0 ,λk 1 ,...,λkNksuch that

Ynk=

∑Nk

j=0

λkjHk+j (^1) [[ 0,Tckn]]·M
is, for eachn, converging in the space ofL^2 (Q)-martingales.
The sequenceLk=
∑Nk
j=0λ
k
jH
k+j·Mis now a Cauchy sequence in the
space of semi-martingales. Indeed for givenε>0takeNsuch thatN^1 <ε.
We find that fork, l:
D

(

(Lk−Ll)·M

)

≤D(YNk−YNl)+D

⎛

⎝

∑n

j=1

λkjKckN+j·M

⎞

⎠+D

⎛

⎝

∑n

j=1

λljKclN+j·M

⎞

⎠

≤D(YNk−YNl)+2ε.

Forkandllarge enough this is smaller than 3ε.

Lemma 9.4.12.The sequence(Lk)k≥ 1 of Lemma 9.4.11 is such that(Lk·A)
converges in the semi-martingale topology.

Proof. We know that Lk·S ≥−1andthat(Lk·M) converges in the
semi-martingale topology. To show that (Lk·A) converges in the semi-
martingale topology we have to prove that for each∫ t≥0 the total variation
t
0

∣

∣d((Lk−Lm)·A)

∣

∣converges to 0 in probability askandmtend to∞.

We will show the stronger statement that

∫∞

0

∣

∣d((Lk−Lm)·A)

∣

∣tend to 0

in probability askandmtend to∞. If this were not the case then by the
Hahn decomposition, described in Sect. 9.2, we could findhkpredictable with