The Mathematics of Arbitrage

9.4 Proof of the Main Theorem 171

(iv) The jumps ofLn·Sare bounded from below by−nn+1 2. Indeed the process
(Kn·S)Tnis bounded above bynon [[0,Tn[[. Its value is always bigger
than−1 and hence jumps of (Kn·S)Tn are bounded from below by
−(n+1).
(v) ‖(Ln·M)∗‖L^2 (Q)≤n+‖∆(Ln·M)Tn‖L^2 (Q)≤n+^3 nλ 2. The last inequality

follows from Corollary 9.2.4 and the inequality‖(Ln·S)∗‖L (^2) (Q)≤nλ 2.
The local martingaleLn·Mis therefore anL^2 (Q)-martingale. For eachn
we define a sequence of stopping times (Tn,i)i≥ 0. We start withTn, 0 =0and
put (eventually the value is +∞)
Tn,i=inf

{

t|t≥Tn,i− 1 and|(Ln·M)t−(Ln·M)Tn,i− 1 |≥ 1

}

.

We then may estimate

∥

∥(Ln·M)Tn,i−(Ln·M)Tn,i− 1

∥

∥

L^2 (Q)≤1+

∥

∥∆(Ln·M)Tn,i

∥

∥

L^2 (Q)

≤1+

3 λ

n^2

≤1+α≤2 for alln≥N.

Letknbe the integer part ofnα 4. We claim that fori=1,...,knand all
n≥N,wehaveQ[Tn,i<∞]> 6 α. An inequality of this type is suggested by
the fact that the variablesfn,i=(Ln·M)Tn,i−(Ln·M)Tn,i− 1 are bounded
by 2 inL^2 (Q) but their sum has to be large, so we need many of them. To
prove that for eachi≤knwe haveQ[Tn,i<∞]> 6 α, it is of course sufficient
to prove that

Q[Tn,kn<∞]=Q[|(Ln·M)Tn,kn−(Ln·M)Tn,kn− 1 |≥1]> 6 α.

PutB={Tn,kn <∞}and estimate, forn≥N,theL^2 (Q)-norm of

(Ln·M)∗ (^1) Bc:
‖(Ln·M)∗ (^1) Bc‖L (^2) (Q)

≤

∥

∥

∥

∥

∥

∑kn

i=1

(Ln (^1) ]]Tn,i− 1 ,Tn,i]]·M)∗ (^1) Bc

∥

∥

∥

∥

∥

L^2 (Q)

≤

∑kn

i=1

∥

∥(Ln (^1) ]]Tn,i− 1 ,Tn,i]]·M)∗ (^1) Bc

∥

∥

L^2 (Q)

≤

∑kn

i=1

∥

∥(Ln (^1) ]]Tn,i− 1 ,Tn,i]]·M)∗

∥

∥

L^2 (Q)

≤ 2

∑kn

i=1

∥

∥(Ln (^1) ]]T
n,i− 1 ,Tn,i]]·M)∞

∥

∥

L^2 (Q) (by Doob’s inequality)

≤ 4 kn

≤nα.